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PySMO Tutorial#

Author: Mayo Amusat
Maintainer: Mayo Amusat
Updated: 2023-06-01

Python-based Surrogate Modelling Objects (PySMO) provides tools for generating different types of reduced order models. PySMO currently provides tools for sampling and surrogate model generation.

Installation#

PySMO is installed by default as part of IDAES. For instructions on installing IDAES, see the online documentation.

One-Shot Sampling with PySMO#

The PySMO package offers five common sampling methods for one-shot design:

  • Latin Hypercube Sampling (LHS)

  • Full-Factorial Sampling

  • Halton Sampling

  • Hammersley Sampling

  • Centroidal voronoi tessellation (CVT) sampling

PySMO provides two modes for data sampling: creation and selection.

  • In creation mode, PySMO creates a specified number of sample points from the bounds provided by the user.

  • In selection mode, PySMO selects a specified number of data points from a user-supplied dataset or file.

Generating samples:#

For demonstration purposes, let us consider a problem for which we need twenty-five (25) samples of temperature and pressure from within the ranges T = 273K - 373K, P = 1 MPa - 50 MPa. Let us generate these samples in PySMO.

Step 1: Import PySMO’s sampling tool#

For this demonstration, we will attempt to generate the samples using the Hammersley sampling method.

from idaes.core.surrogate.pysmo.sampling import HammersleySampling

Step 2: Specify sampling information and initialize class#

All the sampling tools (except full-factorial sampling) require the same keyword arguments:

      -  data_input             : must be a list of lists containing the problem bounds (when creating points), 
                                  or an input dataset (when selecting points from a dataset)
      -  number_of_samples      : number of samples to be created or selected.
      -  sampling_type          : "creation" or "selection".

For full factorial sampling, the user needs to enter a list of points in each dimension in place of the number of samples. Full-factorial sampling requires other inputs - details may be found in the documentation.

For our example, we will create the bounds and then initialize the class with the number of samples.

bounds_info = [[273, 1], [373, 50]]
init_data = HammersleySampling(
    data_input=bounds_info, number_of_samples=25, sampling_type="creation"
)
Sampling type:  creation 

Step 3: Create the samples#

The samples are created by calling the sample_points method on the initialized class.

samples = init_data.sample_points()
print(samples)
[[273.        1.     ]
 [277.       25.5    ]
 [281.       13.25   ]
 [285.       37.75   ]
 [289.        7.125  ]
 [293.       31.625  ]
 [297.       19.375  ]
 [301.       43.875  ]
 [305.        4.0625 ]
 [309.       28.5625 ]
 [313.       16.3125 ]
 [317.       40.8125 ]
 [321.       10.1875 ]
 [325.       34.6875 ]
 [329.       22.4375 ]
 [333.       46.9375 ]
 [337.        2.53125]
 [341.       27.03125]
 [345.       14.78125]
 [349.       39.28125]
 [353.        8.65625]
 [357.       33.15625]
 [361.       20.90625]
 [365.       45.40625]
 [369.        5.59375]]

Simple as that, the samples have been created!

Now, let us visualize the samples in a 2-D plot.

Step 4: Visualize samples with matplotlib#

from matplotlib import pyplot as plt

plt.plot(samples[:, 0], samples[:, 1], "o")
plt.xlabel(r"Temperature", fontsize=11)
plt.xlabel(r"Pressure", fontsize=11)
plt.xlim(272, 374)
plt.ylim(0, 50)
plt.show()
../../../_images/pysmo_basics_doc_14_0.png

Generating surrogates with PySMO#

PySMO currently provides tools for generating three types of surrogates:

  • Polynomial surrogates

  • Radial basis function (RBF) surrogates, and

  • Kriging surrogates

Details about thee various methods may be found in the documentation.

Generating polynomial models#

The PolynomialRegression class trains polynomial models from data.

As an example, let us generate a surrogate for the Brainin function.

The true Brainin function is given by the expression:

\begin{gather} \hat{y}(x_{1},x_{2})=\left(x_{2}-\frac{5.1x_{1}^{2}}{4\pi^{2}}+\frac{5x_{1}}{\pi}-6\right)^{2}+10\left[\left(1-\frac{1}{8\pi}\right)\cos\left(x_{1}\right)+1\right]+5x_{1}\nonumber \ x_{1}\in\left[-5,10\right];x_{2}\in\left[0,15\right] \end{gather}

We have generated 30 points from the function and saved the information in a text file called “brainin_30.txt”. We will use this data to train a simple polynomial model. The data is in XY format, with the outputs \(y\) in the third column.

Step 1: Import and visualize the data#

import numpy as np

brainin_data = np.loadtxt("brainin_30.txt")
print(brainin_data, "\n\nDataset shape:", brainin_data.shape)
[[ 3.15107413e+00  4.17554078e+00  4.03815759e+00]
 [ 1.36776386e+00  1.26716420e+01  8.60127077e+01]
 [-4.92921716e+00  1.82353681e+00  2.41888089e+02]
 [ 5.06123627e+00  1.23877913e+01  1.37242426e+02]
 [-2.94940115e+00  8.62639994e+00  1.07587601e+01]
 [ 8.36982931e+00  3.13803183e+00  7.24964869e+00]
 [-2.22007671e+00  1.62565336e+00  7.71988248e+01]
 [-1.70453761e+00  1.46793568e+01  3.99820398e+01]
 [ 7.17524724e+00  2.57911519e+00  1.78453527e+01]
 [ 7.24337123e+00  4.11110621e+00  2.36937839e+01]
 [ 1.47556275e+00  1.41004473e+01  1.14292050e+02]
 [ 7.26474068e+00  5.04167925e+00  2.96703147e+01]
 [-2.36884319e+00  5.59248069e+00  2.71582680e+01]
 [-4.91467239e+00  3.78639530e+00  1.84993457e+02]
 [ 6.93493763e+00  2.28824569e-01  1.85302363e+01]
 [ 3.98265065e+00  9.05706809e+00  5.75698037e+01]
 [-3.42278472e+00  5.72915167e+00  5.30702398e+01]
 [-4.45285915e+00  1.33561735e+01  1.27882794e+01]
 [ 9.71381286e+00  8.99129832e-01  4.14567857e+00]
 [ 8.35818917e+00  8.65352249e+00  5.34055688e+01]
 [ 6.13719534e+00  9.45275905e+00  8.93001840e+01]
 [ 3.72763289e+00  3.06586980e-01  4.42072359e+00]
 [-1.84960133e+00  8.17027317e+00  8.83431235e+00]
 [ 6.53672757e+00  3.76042844e+00  2.62862798e+01]
 [-7.11564644e-01  1.27859263e+01  4.84986100e+01]
 [ 9.62509740e+00  1.32727994e+01  1.13457908e+02]
 [ 3.92617659e-01  8.98288419e+00  3.17439666e+01]
 [ 3.21934175e-01  5.10285323e+00  1.92673649e+01]
 [-2.32878516e+00  3.56541313e+00  5.02034858e+01]
 [-4.32706576e+00  7.58147144e+00  6.60457721e+01]] 

Dataset shape: (30, 3)

Let us visualize the data:

from mpl_toolkits.mplot3d import Axes3D
from matplotlib import pyplot as plt

fig1 = plt.figure(figsize=(6, 4), tight_layout=True)
ax = fig1.add_subplot(111, projection="3d")
ax.scatter3D(
    brainin_data[:, 0], brainin_data[:, 1], brainin_data[:, 2], cmap=brainin_data[:, 2]
)
ax.set_xlabel(r"$x_{1}$", fontsize=11)
ax.set_ylabel(r"$x_{2}$", fontsize=11)
ax.set_zlabel(r"$y$", fontsize=11)
C:\Users\dkgun\AppData\Local\Temp\ipykernel_31700\142152307.py:6: UserWarning: No data for colormapping provided via 'c'. Parameters 'cmap' will be ignored
  ax.scatter3D(
Text(0.5, 0, '$y$')
../../../_images/pysmo_basics_doc_20_2.png

Step 2: Import the polynomial model tool#

from idaes.core.surrogate.pysmo.polynomial_regression import PolynomialRegression

Step 3: Specify the regression settings and initialize the PolynomialRegression class#

The PolynomialRegression class takes a keyword arguments:

  -  original_data_input           : The dataset for regression training. training_data is expected to contain xy_data, 
                                     with the output values (y) in the last column.
  -  regression_data_input         : same as above
  -  maximum_polynomial_order      : maximum order of the polynomial to be generated  

It also takes a number of optional arguments:

  - multinomials                  : True/False option for specifying second-order bi-variate terms. default is False
  - training_split                : The training/cross-validation split of training data. Must be between 0 and 1. 
                                    Default is 0.75
  - fname                         : Filename for saving results (.pickle extension). 
  - overwrite                     : Option determining whether any existing file with the same name supplied in 'fname'  
                                    should be overwritten.

For this example, let us consider a 4th order polynomial with interaction terms. We will split the data 80/20 betweeen training and cross-validation.

poly_class = PolynomialRegression(
    original_data_input=brainin_data,
    regression_data_input=brainin_data,
    maximum_polynomial_order=4,
    multinomials=1,
    training_split=0.8,
    number_of_crossvalidations=10,
    overwrite=True,
)
===========================Polynomial Regression===============================================

Warning: solution.pickle already exists; previous file will be overwritten.

No iterations will be run.
Default parameter estimation method is used.
Parameter estimation method:  pyomo 

Step 4: Extract variable names#

Next, we extract Pyomo variable names from the dataset. This should be done always.

vars = poly_class.get_feature_vector()

We can view the variables using Pyomo’s pprint function:

vars.pprint()
IndexedParam : Size=2, Index={0, 1}, Domain=Any, Default=None, Mutable=True
    Key : Value
      0 :     0
      1 :     0

Step 5: Specify additional regression terms, if required.#

This is one of the unique features of PySMO - it allows the user to specify additional regression features if they want. The additional features must be specified in terms of the Pyomo variables created when calling the get_feature_vector()

For this example, let us create three additional features: \(x_{1}^{2}x_{2}^{2}\), \(exp(x_1)\) and \(exp(x_2)\). We do this by calling the set_additional_terms function:

from pyomo.environ import exp

poly_class.set_additional_terms(
    [vars[0] * vars[0] * vars[1] * vars[1], exp(vars[0]), exp(vars[1])]
)

That’s it - those features will now exist in the model.

Note that set_additional_terms an optional call - the regression process works just fine without it.

Step 6: Train the surrogate and view results#

Next, we train the polynomial surrogate by calling training:

poly_class.training()
No iterations will be run.
Best surrogate model is of order 3  with a cross-val S.S. Error  of 19.737617

------------------------------------------------------------
The final coefficients of the regression terms are: 

k               | 57.607003
(x_ 1 )^ 1      | -19.078561
(x_ 2 )^ 1      | -17.191563
(x_ 1 )^ 2      | 3.889924
(x_ 2 )^ 2      | 1.163328
(x_ 1 )^ 3      | -0.351928
(x_ 2 )^ 3      | 0.018398
x_ 1 .x_ 2      | 3.100135

The coefficients of the extra terms in additional_regression_features are:

Coeff. additional_regression_features[ 1 ]:  -0.017933
Coeff. additional_regression_features[ 2 ]:  0.004746
Coeff. additional_regression_features[ 3 ]:  -1.3e-05

Regression model performance on training data:
Order:  3  / MAE: 6.154513  / MSE: 65.995248  / R^2: 0.978531

Results saved in  solution.pickle
========================================================================================================================
Results of polynomial regression run:

Polynomial order                   : 3
Number of terms in polynomial model: 11

Polynomial Expression:
--------------------------

57.60700349249289 - 19.078560846595344*IndexedParam[0] - 17.191563035875806*IndexedParam[1] + 3.8899241750021663*IndexedParam[0]**2 + 1.1633280319958954*IndexedParam[1]**2 - 0.3519284134546862*IndexedParam[0]**3 + 0.018398180902055014*IndexedParam[1]**3 + 3.1001348356698335*(IndexedParam[1]*IndexedParam[0]) - 0.017933037213866673*(IndexedParam[0]*IndexedParam[0]*IndexedParam[1]*IndexedParam[1]) + 0.00474571104342994*exp(IndexedParam[0]) - 1.2975320261052192e-05*exp(IndexedParam[1])
--------------------------

Model training errors:
-----------------------
Mean Squared Error (MSE)         : 65.99524781717177
Root Mean Squared Error (RMSE)   : 8.123745922736122
Mean Absolute error (MSE)        : 6.154513199255071
Goodness of fit (R2)             : 0.97853053384543

========================================================================================================================

The polynomial model seems to fit well based on the \(R^2\). It should be noted that the metrics are only an indication of how well of how well the model fit the training data - the user needs to verify the model’s performance on a test data set if possible.

We can view the parity and residual plots for the fit:

poly_class.parity_residual_plots()
../../../_images/pysmo_basics_doc_36_0.png

PySMO is also able to compute the confidence intervals on the regression coefficients obtained by calling confint_regression(). This is left as an exercise for the user.

Step 8 (Optional): Generate Pyomo expression#

If the user wishes, they can generate the Pyomo expression for the polynomial fit using PySMO’s generate_expression. To do this, the user must pass in a list of Pyomo variables corresponding to each variable in the input dataset.

As a demonstration, let us create the variables \(x_1\) and \(x_2\) and generate the pyomo expression based on them:

from pyomo.environ import Var, ConcreteModel

m = ConcreteModel()
m.x = Var([1, 2])
print(poly_class.generate_expression([m.x[1], m.x[2]]))
57.60700349249289 - 19.078560846595344*x[1] - 17.191563035875806*x[2] + 3.8899241750021663*x[1]**2 + 1.1633280319958954*x[2]**2 - 0.3519284134546862*x[1]**3 + 0.018398180902055014*x[2]**3 + 3.1001348356698335*(x[2]*x[1]) - 0.017933037213866673*(x[1]*x[1]*x[2]*x[2]) + 0.00474571104342994*exp(x[1]) - 1.2975320261052192e-05*exp(x[2])

Step 9 (Optional): Predict output at any unsampled point#

Based on the model we trained, we can predict the surrogate value at any previously unsampled point.

Let us evaluate the surrogate at three points:

  • \(x_{1}=5\), \(x_{2}=8\) (true function value: 57.9908)

  • \(x_{1}=-3\), \(x_{2}=10\) (true function value: 4.2461)

  • \(x_{1}=-2\), \(x_{2}=3\). (true function value: 50.8899)

We will pass the points in as an array.

unsampled_points = np.array([[5, 8], [-3, 10], [-2, 3]])
ys = poly_class.predict_output(unsampled_points)
print(ys)
[[57.7897913 ]
 [12.74008178]
 [54.28524655]]

The model performs fairly well in predicting the value at two of our sampled points but is off on the value at [-3, 2]. For better model performance, additional training data is needed in this region. We will leave this to the user to try.

Further information about using PySMO’s polynomial regression tool can be found in the documentation.

Generating RBF models#

The RadialBasisFunction class trains RBF models from data. For details about RBF models, the user should consult the documentation.

As an example, we will again consider the Brainin function. The same dataset loaded previously will be used.

Step 1: Import the data and the RBF tool#

import numpy as np

brainin_data = np.loadtxt("brainin_30.txt")
from idaes.core.surrogate.pysmo.radial_basis_function import RadialBasisFunctions

Step 2: Specify the RBF settings and initialize the RadialBasisFunctions class#

The RadialBasisFunctions class takes a number of keyword arguments:

  -  XY_data                       : The dataset forRBF training. training_data is expected to contain xy_data, 
                                     with the output values (y) in the last column.

It also takes a number of optional arguments:

  -  regularization                : Boolean variable determining whether regularization is done. Default is True.
  -  basis_function                : Basis function transformation to be applied to the training data. PySMO offers 
                                     six basis function types including the Gaussian and Spline transformations. User 
                                     should consult documentation for full list of options.   
  -  fname                         : Filename for saving (.pickle extension)
  - overwrite                      : Option determining whether any existing file with the same name supplied in 'fname'  
                                     should be overwritten.

For this demonstration, we will train an RBF model with a Gaussian basis function:

rbf_class = RadialBasisFunctions(
    XY_data=brainin_data, basis_function="gaussian", overwrite=True
)
Warning: solution.pickle already exists; previous file will be overwritten.

Default parameter estimation method is used.

Parameter estimation method:  algebraic
Basis function:  gaussian
Regularization done:  True

Step 3: Extract variable names#

Next, we extract Pyomo variable names from the dataset.

vars = rbf_class.get_feature_vector()

Step 4: Train the RBF surrogate#

Next, we train the RBF surrogate by calling training:

rbf_class.training()
===========================================================================================================
0.001    |     1e-05    |     1.330188999805357    |     7.936658808585865e+18    |     1762.2922695772181    |     2999999.7996930624    |     6.661337702980191e-10
0.001    |     2e-05    |     1.3231321486358596    |     7.936658808585865e+18    |     1762.2922695772181    |     1500000.3998815753    |     3.3306699617909336e-10
0.001    |     5e-05    |     1.3204882439270513    |     7.936658808585865e+18    |     1762.2922695772181    |     600000.7598810853    |     1.3322693168251415e-10
0.001    |     7.5e-05    |     1.3203184940690464    |     7.936658808585865e+18    |     1762.2922695772181    |     400000.83992750663    |     8.88180284713839e-11
0.001    |     0.0001    |     1.3203209127587061    |     7.936658808585865e+18    |     1762.2922695772181    |     300000.87994025816    |     6.661357686349637e-11
0.001    |     0.0002    |     1.320453829126744    |     7.936658808585865e+18    |     1762.2922695772181    |     150000.93996799184    |     3.330689945357609e-11
0.001    |     0.0005    |     1.3206180157500875    |     7.936658808585865e+18    |     1762.2922695772181    |     60000.975986938625    |     1.3322893008136078e-11
0.001    |     0.00075    |     1.3206637488446602    |     7.936658808585865e+18    |     1762.2922695772181    |     40000.98399132156    |     8.882002686965498e-12
0.001    |     0.001    |     1.3206877851838068    |     7.936658808585865e+18    |     1762.2922695772181    |     30000.987993464787    |     6.661557526369496e-12
0.001    |     0.002    |     1.3207246211571246    |     7.936658808585865e+18    |     1762.2922695772181    |     15000.993996724195    |     3.3308897854853903e-12
0.001    |     0.005    |     1.3207441010685603    |     7.936658808585865e+18    |     1762.2922695772181    |     6000.997598686373    |     1.3324891409563772e-12
0.001    |     0.0075    |     1.3207457685491724    |     7.936658808585865e+18    |     1762.2922695772181    |     4000.9983991229724    |     8.884001088389431e-13
0.001    |     0.01    |     1.3207447379585568    |     7.936658808585865e+18    |     1762.2922695772181    |     3000.998799341891    |     6.663555927803635e-13
0.001    |     0.02    |     1.320733880721723    |     7.936658808585865e+18    |     1762.2922695772181    |     1500.9993996709732    |     3.332888186926504e-13
0.001    |     0.05    |     1.320692242454931    |     7.936658808585865e+18    |     1762.2922695772181    |     600.999759868342    |     1.334487542400047e-13
0.001    |     0.075    |     1.3206569542287017    |     7.936658808585865e+18    |     1762.2922695772181    |     400.9998399122189    |     8.903985102830944e-14
0.001    |     0.1    |     1.320622227525564    |     7.936658808585865e+18    |     1762.2922695772181    |     300.99987993416397    |     6.68353994224633e-14
0.001    |     0.2    |     1.3204909253945512    |     7.936658808585865e+18    |     1762.2922695772181    |     150.9999399670793    |     3.352872201369357e-14
0.001    |     0.5    |     1.3201734897566477    |     7.936658808585865e+18    |     1762.2922695772181    |     60.99997598683129    |     1.354471556843235e-14
0.001    |     0.75    |     1.3199938819612356    |     7.936658808585865e+18    |     1762.2922695772181    |     40.999983991220795    |     9.10382524726323e-15
0.001    |     1    |     1.3198876280547236    |     7.936658808585865e+18    |     1762.2922695772181    |     30.999987993415566    |     6.883380086678673e-15
0.002    |     1e-05    |     1.362387822278568    |     5.782309804024167e+18    |     1283.9306959886815    |     2999996.19772378    |     6.661329705001728e-10
0.002    |     2e-05    |     1.343994630873922    |     5.782309804024167e+18    |     1283.9306959886815    |     1499998.5987316754    |     3.3306659624347544e-10
0.002    |     5e-05    |     1.3272246485698753    |     5.782309804024167e+18    |     1283.9306959886815    |     600000.0394937922    |     1.3322677172440228e-10
0.002    |     7.5e-05    |     1.3235440099208564    |     5.782309804024167e+18    |     1283.9306959886815    |     400000.3596555643    |     8.881792182959021e-11
0.002    |     0.0001    |     1.3220037168855518    |     5.782309804024167e+18    |     1283.9306959886815    |     300000.51974200836    |     6.66134968834183e-11
0.002    |     0.0002    |     1.3204880868279008    |     5.782309804024167e+18    |     1283.9306959886815    |     150000.75986912134    |     3.330685946359354e-11
0.002    |     0.0005    |     1.3203538501482557    |     5.782309804024167e+18    |     1283.9306959886815    |     60000.903947554114    |     1.3322877012179405e-11
0.002    |     0.00075    |     1.3204389361297426    |     5.782309804024167e+18    |     1283.9306959886815    |     40000.935965030716    |     8.881992022986721e-12
0.002    |     0.001    |     1.320500039765387    |     5.782309804024167e+18    |     1283.9306959886815    |     30000.95197376373    |     6.661549528389206e-12
0.002    |     0.002    |     1.320615846455827    |     5.782309804024167e+18    |     1283.9306959886815    |     15000.975986864492    |     3.330885786493208e-12
0.002    |     0.005    |     1.3206969267656394    |     5.782309804024167e+18    |     1283.9306959886815    |     6000.990394744541    |     1.3324875413599593e-12
0.002    |     0.0075    |     1.3207137929011277    |     5.782309804024167e+18    |     1283.9306959886815    |     4000.9935964959222    |     8.883990424415172e-13
0.002    |     0.01    |     1.3207205715037762    |     5.782309804024167e+18    |     1283.9306959886815    |     3000.9951973718266    |     6.663547929823436e-13
0.002    |     0.02    |     1.3207216922930982    |     5.782309804024167e+18    |     1283.9306959886815    |     1500.997598685877    |     3.3328841879362624e-13
0.002    |     0.05    |     1.320687393868504    |     5.782309804024167e+18    |     1283.9306959886815    |     600.9990394743187    |     1.334485942803984e-13
0.002    |     0.075    |     1.320653752364362    |     5.782309804024167e+18    |     1283.9306959886815    |     400.9993596495438    |     8.90397443885735e-14
0.002    |     0.1    |     1.3206198511578064    |     5.782309804024167e+18    |     1283.9306959886815    |     300.9995197371585    |     6.683531944266152e-14
0.002    |     0.2    |     1.320489789943987    |     5.782309804024167e+18    |     1283.9306959886815    |     150.9997598685783    |     3.352868202379307e-14
0.002    |     0.5    |     1.3201731005146329    |     5.782309804024167e+18    |     1283.9306959886815    |     60.99990394743119    |     1.3544699572472217e-14
0.002    |     0.75    |     1.3199936589276795    |     5.782309804024167e+18    |     1283.9306959886815    |     40.99993596495411    |     9.103814583289818e-15
0.002    |     1    |     1.319887488406783    |     5.782309804024167e+18    |     1283.9306959886815    |     30.99995197371558    |     6.88337208869862e-15
0.005    |     1e-05    |     1.398281393989088    |     4.7315568285138035e+19    |     10506.166666676816    |     2999970.9840130205    |     6.661273719317286e-10
0.005    |     2e-05    |     1.388722431615681    |     4.7315568285138035e+19    |     10506.166666676816    |     1499985.9920165162    |     3.330637969903885e-10
0.005    |     5e-05    |     1.3683630926262451    |     4.7315568285138035e+19    |     10506.166666676816    |     599994.9967708938    |     1.3322565201498855e-10
0.005    |     7.5e-05    |     1.3574022549722211    |     4.7315568285138035e+19    |     10506.166666676816    |     399996.9978457239    |     8.881717535785236e-11
0.005    |     0.0001    |     1.349656526521845    |     4.7315568285138035e+19    |     10506.166666676816    |     299997.99838577345    |     6.661293702986925e-11
0.005    |     0.0002    |     1.3338899892280682    |     4.7315568285138035e+19    |     10506.166666676816    |     149999.49919096037    |     3.330657953680935e-11
0.005    |     0.0005    |     1.3231314606154727    |     4.7315568285138035e+19    |     10506.166666676816    |     60000.399676413464    |     1.3322765041493204e-11
0.005    |     0.00075    |     1.3213658709059117    |     4.7315568285138035e+19    |     10506.166666676816    |     40000.599784124155    |     8.88191737583014e-12
0.005    |     0.001    |     1.3207434325086935    |     4.7315568285138035e+19    |     10506.166666676816    |     30000.699838091627    |     6.661493543023506e-12
0.005    |     0.002    |     1.3203109441737189    |     4.7315568285138035e+19    |     10506.166666676816    |     15000.84991903379    |     3.3308577938115457e-12
0.005    |     0.005    |     1.3204473592564139    |     4.7315568285138035e+19    |     10506.166666676816    |     6000.939967612918    |     1.3324763442874405e-12
0.005    |     0.0075    |     1.3205266147948784    |     4.7315568285138035e+19    |     10506.166666676816    |     4000.95997840818    |     8.883915777265062e-13
0.005    |     0.01    |     1.3205722511348426    |     4.7315568285138035e+19    |     10506.166666676816    |     3000.9699838059914    |     6.66349194446079e-13
0.005    |     0.02    |     1.3206416674457822    |     4.7315568285138035e+19    |     10506.166666676816    |     1500.9849919029054    |     3.3328561952548196e-13
0.005    |     0.05    |     1.3206543062552778    |     4.7315568285138035e+19    |     10506.166666676816    |     600.9939967611574    |     1.3344747457314674e-13
0.005    |     0.075    |     1.3206317178276619    |     4.7315568285138035e+19    |     10506.166666676816    |     400.9959978407689    |     8.903899791707224e-14
0.005    |     0.1    |     1.3206034290426678    |     4.7315568285138035e+19    |     10506.166666676816    |     300.9969983805781    |     6.683475958903575e-14
0.005    |     0.2    |     1.3204818942961338    |     4.7315568285138035e+19    |     10506.166666676816    |     150.99849919028685    |     3.3528402096979903e-14
0.005    |     0.5    |     1.3201703838941503    |     4.7315568285138035e+19    |     10506.166666676816    |     60.99939967611459    |     1.3544587601746947e-14
0.005    |     0.75    |     1.3199921011576614    |     4.7315568285138035e+19    |     10506.166666676816    |     40.999599784076366    |     9.103739936139636e-15
0.005    |     1    |     1.3198865127506896    |     4.7315568285138035e+19    |     10506.166666676816    |     30.999699838057264    |     6.883316103335982e-15
0.0075    |     1e-05    |     1.404300593878447    |     1.7372731761309637e+18    |     385.75213604085417    |     2999933.4645628342    |     6.661190409402349e-10
0.0075    |     2e-05    |     1.3994032633183124    |     1.7372731761309637e+18    |     385.75213604085417    |     1499967.232282145    |     3.3305963149258154e-10
0.0075    |     5e-05    |     1.386805647947958    |     1.7372731761309637e+18    |     385.75213604085417    |     599987.4928880802    |     1.332239858182938e-10
0.0075    |     7.5e-05    |     1.3783815287748937    |     1.7372731761309637e+18    |     385.75213604085417    |     399991.9952527292    |     8.881606455906726e-11
0.0075    |     0.0001    |     1.3714017808048378    |     1.7372731761309637e+18    |     385.75213604085417    |     299994.24643613514    |     6.661210392969411e-11
0.0075    |     0.0002    |     1.352783451657262    |     1.7372731761309637e+18    |     385.75213604085417    |     149997.6232191162    |     3.330616298738236e-11
0.0075    |     0.0005    |     1.3320471009302473    |     1.7372731761309637e+18    |     385.75213604085417    |     59999.64928707162    |     1.3322598421588254e-11
0.0075    |     0.00075    |     1.3264972569267337    |     1.7372731761309637e+18    |     385.75213604085417    |     40000.09952468137    |     8.881806295919807e-12
0.0075    |     0.001    |     1.3239147582316242    |     1.7372731761309637e+18    |     385.75213604085417    |     30000.32464348631    |     6.6614102330855985e-12
0.0075    |     0.002    |     1.3209497744222265    |     1.7372731761309637e+18    |     385.75213604085417    |     15000.66232173369    |     3.33081613884316e-12
0.0075    |     0.005    |     1.32030667526022    |     1.7372731761309637e+18    |     385.75213604085417    |     6000.864928690638    |     1.332459682299589e-12
0.0075    |     0.0075    |     1.3203572891448485    |     1.7372731761309637e+18    |     385.75213604085417    |     4000.9099524600265    |     8.883804697346124e-13
0.0075    |     0.01    |     1.3204149453606089    |     1.7372731761309637e+18    |     385.75213604085417    |     3000.932464345096    |     6.663408634522074e-13
0.0075    |     0.02    |     1.3205391985944162    |     1.7372731761309637e+18    |     385.75213604085417    |     1500.9662321724334    |     3.332814540285408e-13
0.0075    |     0.05    |     1.3206077953595612    |     1.7372731761309637e+18    |     385.75213604085417    |     600.9864928689655    |     1.3344580837436958e-13
0.0075    |     0.075    |     1.320600144100872    |     1.7372731761309637e+18    |     385.75213604085417    |     400.9909952459681    |     8.90378871178861e-14
0.0075    |     0.1    |     1.320579675219363    |     1.7372731761309637e+18    |     385.75213604085417    |     300.99324643447636    |     6.683392648964589e-14
0.0075    |     0.2    |     1.3204703143593972    |     1.7372731761309637e+18    |     385.75213604085417    |     150.99662321723713    |     3.352798554728523e-14
0.0075    |     0.5    |     1.320166367422988    |     1.7372731761309637e+18    |     385.75213604085417    |     60.998649286894484    |     1.3544420981869028e-14
0.0075    |     0.75    |     1.3199897942613739    |     1.7372731761309637e+18    |     385.75213604085417    |     40.99909952459634    |     9.103628856221033e-15
0.0075    |     1    |     1.3198850669656137    |     1.7372731761309637e+18    |     385.75213604085417    |     30.9993246434473    |     6.8832327933970424e-15
0.01    |     1e-05    |     1.4061992061470945    |     9.15881260810045e+18    |     2033.66492714806    |     2999880.9389639026    |     6.661073779143717e-10
0.01    |     2e-05    |     1.4035355143448425    |     9.15881260810045e+18    |     2033.66492714806    |     1499940.969350007    |     3.330537999501908e-10
0.01    |     5e-05    |     1.395627182515116    |     9.15881260810045e+18    |     2033.66492714806    |     599976.9877232183    |     1.3322165320311236e-10
0.01    |     7.5e-05    |     1.389760075244399    |     9.15881260810045e+18    |     2033.66492714806    |     399984.9918138873    |     8.881450948325649e-11
0.01    |     0.0001    |     1.3845164268287486    |     9.15881260810045e+18    |     2033.66492714806    |     299988.99385887664    |     6.66109376232519e-11
0.01    |     0.0002    |     1.368340200656151    |     9.15881260810045e+18    |     2033.66492714806    |     149994.9969292092    |     3.3305579833877544e-11
0.01    |     0.0005    |     1.3439837129673218    |     9.15881260810045e+18    |     2033.66492714806    |     59998.598771298515    |     1.3322365160228448e-11
0.01    |     0.00075    |     1.335093711991605    |     9.15881260810045e+18    |     2033.66492714806    |     39999.399180849905    |     8.881650788350438e-12
0.01    |     0.001    |     1.3301839804871138    |     9.15881260810045e+18    |     2033.66492714806    |     29999.799385624003    |     6.661293602411079e-12
0.01    |     0.002    |     1.3231293127499864    |     9.15881260810045e+18    |     2033.66492714806    |     15000.399692799341    |     3.3307578235051907e-12
0.01    |     0.005    |     1.3204830758443515    |     9.15881260810045e+18    |     2033.66492714806    |     6000.7598771195135    |     1.3324363561649818e-12
0.01    |     0.0075    |     1.3203100371670242    |     9.15881260810045e+18    |     2033.66492714806    |     4000.839918079025    |     8.883649189781518e-13
0.01    |     0.01    |     1.320308943205694    |     9.15881260810045e+18    |     2033.66492714806    |     3000.879938558849    |     6.663292003847519e-13
0.01    |     0.02    |     1.3204273423452193    |     9.15881260810045e+18    |     2033.66492714806    |     1500.9399692794002    |     3.3327562249483305e-13
0.01    |     0.05    |     1.3205480387004698    |     9.15881260810045e+18    |     2033.66492714806    |     600.9759877117588    |     1.3344347576088794e-13
0.01    |     0.075    |     1.3205583477737104    |     9.15881260810045e+18    |     2033.66492714806    |     400.9839918078323    |     8.903633204223211e-14
0.01    |     0.1    |     1.320547778680883    |     9.15881260810045e+18    |     2033.66492714806    |     300.98799385587427    |     6.683276018290535e-14
0.01    |     0.2    |     1.3204544411843475    |     9.15881260810045e+18    |     2033.66492714806    |     150.99399692793642    |     3.3527402393915034e-14
0.01    |     0.5    |     1.3201607967208802    |     9.15881260810045e+18    |     2033.66492714806    |     60.99759877117422    |     1.3544187720520954e-14
0.01    |     0.75    |     1.3199865871021215    |     9.15881260810045e+18    |     2033.66492714806    |     40.99839918078277    |     9.103473348655637e-15
0.01    |     1    |     1.3198830550759415    |     9.15881260810045e+18    |     2033.66492714806    |     30.998799385587116    |     6.8831161627229944e-15
0.02    |     1e-05    |     1.3991967821771687    |     4.596172752109427e+18    |     1020.5553629093316    |     2999520.8077985244    |     6.660274127320341e-10
0.02    |     2e-05    |     1.4033527820009084    |     4.596172752109427e+18    |     1020.5553629093316    |     1499760.9038967583    |     3.3301381738776355e-10
0.02    |     5e-05    |     1.4040547659095226    |     4.596172752109427e+18    |     1020.5553629093316    |     599904.9615311394    |     1.3320566017574796e-10
0.02    |     7.5e-05    |     1.4028711082098146    |     4.596172752109427e+18    |     1020.5553629093316    |     399936.9743506041    |     8.880384746459227e-11
0.02    |     0.0001    |     1.4014091897441256    |     4.596172752109427e+18    |     1020.5553629093316    |     299952.98076160415    |     6.660294110929591e-11
0.02    |     0.0002    |     1.3952513762994323    |     4.596172752109427e+18    |     1020.5553629093316    |     149976.99038130237    |     3.3301581577061505e-11
0.02    |     0.0005    |     1.3796701173934471    |     4.596172752109427e+18    |     1020.5553629093316    |     59991.39615212075    |     1.3320765857498696e-11
0.02    |     0.00075    |     1.3699232085655064    |     4.596172752109427e+18    |     1020.5553629093316    |     39994.5974347125    |     8.880584586526408e-12
0.02    |     0.001    |     1.362292110296485    |     4.596172752109427e+18    |     1020.5553629093316    |     29996.198076027966    |     6.660493951044614e-12
0.02    |     0.002    |     1.3439496124690997    |     4.596172752109427e+18    |     1020.5553629093316    |     14998.599038011473    |     3.330357997824212e-12
0.02    |     0.005    |     1.3272089427378684    |     4.596172752109427e+18    |     1020.5553629093316    |     6000.039615202919    |     1.332276425892269e-12
0.02    |     0.0075    |     1.323532461862641    |     4.596172752109427e+18    |     1020.5553629093316    |     4000.3597434684293    |     8.88258298796447e-13
0.02    |     0.01    |     1.3219923714906017    |     4.596172752109427e+18    |     1020.5553629093316    |     3000.5198076010715    |     6.662492352485109e-13
0.02    |     0.02    |     1.3204676186157385    |     4.596172752109427e+18    |     1020.5553629093316    |     1500.7599038004946    |     3.332356399267088e-13
0.02    |     0.05    |     1.3202928659769584    |     4.596172752109427e+18    |     1020.5553629093316    |     600.9039615201905    |     1.334274827336369e-13
0.02    |     0.075    |     1.3203431410941724    |     4.596172752109427e+18    |     1020.5553629093316    |     400.9359743467912    |     8.902567002406575e-14
0.02    |     0.1    |     1.320369937260628    |     4.596172752109427e+18    |     1020.5553629093316    |     300.95198076008995    |     6.68247636692798e-14
0.02    |     0.2    |     1.320356005679441    |     4.596172752109427e+18    |     1020.5553629093316    |     150.97599038004486    |     3.352340413710239e-14
0.02    |     0.5    |     1.3201242272878075    |     4.596172752109427e+18    |     1020.5553629093316    |     60.990396152017844    |     1.3542588417795952e-14
0.02    |     0.75    |     1.3199652973344642    |     4.596172752109427e+18    |     1020.5553629093316    |     40.99359743467856    |     9.102407146838978e-15
0.02    |     1    |     1.3198696409051938    |     4.596172752109427e+18    |     1020.5553629093316    |     30.99519807600895    |     6.882316511360498e-15
0.05    |     1e-05    |     1.1802626763387565    |     2.292482447793158e+19    |     5090.333594178005    |     2997002.1306760353    |     6.654681540654373e-10
0.05    |     2e-05    |     1.2638668839634342    |     2.292482447793158e+19    |     5090.333594178005    |     1498501.5652584448    |     3.327341880373524e-10
0.05    |     5e-05    |     1.3399914336674565    |     2.292482447793158e+19    |     5090.333594178005    |     599401.2260877036    |     1.3309380843822352e-10
0.05    |     7.5e-05    |     1.3607917455523473    |     2.292482447793158e+19    |     5090.333594178005    |     399601.150725315    |     8.872927964039045e-11
0.05    |     0.0001    |     1.3717399149142016    |     2.292482447793158e+19    |     5090.333594178005    |     299701.11304101965    |     6.654701524078536e-11
0.05    |     0.0002    |     1.388357196418246    |     2.292482447793158e+19    |     5090.333594178005    |     149851.0565198782    |     3.3273618642554895e-11
0.05    |     0.0005    |     1.3960809195486679    |     2.292482447793158e+19    |     5090.333594178005    |     59941.02260797523    |     1.330958068379023e-11
0.05    |     0.00075    |     1.395825918790288    |     2.292482447793158e+19    |     5090.333594178005    |     39961.015071850336    |     8.87312780403223e-12
0.05    |     0.001    |     1.3944270862078088    |     2.292482447793158e+19    |     5090.333594178005    |     29971.011303897183    |     6.6549013641774975e-12
0.05    |     0.002    |     1.3868038656182429    |     2.292482447793158e+19    |     5090.333594178005    |     14986.005651941321    |     3.327561704389597e-12
0.05    |     0.005    |     1.3676102367336447    |     2.292482447793158e+19    |     5090.333594178005    |     5995.002260777155    |     1.331157908518933e-12
0.05    |     0.0075    |     1.3569104532323966    |     2.292482447793158e+19    |     5090.333594178005    |     3997.0015071830744    |     8.875126205472204e-13
0.05    |     0.01    |     1.3492964425395066    |     2.292482447793158e+19    |     5090.333594178005    |     2998.001130387283    |     6.656899765616415e-13
0.05    |     0.02    |     1.3337278030826976    |     2.292482447793158e+19    |     5090.333594178005    |     1499.500565193526    |     3.3295601058325763e-13
0.05    |     0.05    |     1.3230623528240189    |     2.292482447793158e+19    |     5090.333594178005    |     600.4002260774155    |     1.333156309962592e-13
0.05    |     0.075    |     1.3212917568278908    |     2.292482447793158e+19    |     5090.333594178005    |     400.6001507182743    |     8.895110219914721e-14
0.05    |     0.1    |     1.3206495137172813    |     2.292482447793158e+19    |     5090.333594178005    |     300.70011303870496    |     6.67688378005915e-14
0.05    |     0.2    |     1.3201079927565518    |     2.292482447793158e+19    |     5090.333594178005    |     150.85005651935137    |     3.349544120275802e-14
0.05    |     0.5    |     1.3199430633401974    |     2.292482447793158e+19    |     5090.333594178005    |     60.940022607740424    |     1.3531403244058199e-14
0.05    |     0.75    |     1.3198490982380757    |     2.292482447793158e+19    |     5090.333594178005    |     40.96001507182699    |     9.094950364347151e-15
0.05    |     1    |     1.3197937467706273    |     2.292482447793158e+19    |     5090.333594178005    |     30.970011303870223    |     6.876723924491617e-15
0.075    |     1e-05    |     1.0289125856997683    |     2.5655434447895905e+18    |     569.6650806163085    |     2993261.340844146    |     6.646375318651079e-10
0.075    |     2e-05    |     1.06833574230173    |     2.5655434447895905e+18    |     569.6650806163085    |     1496631.170356869    |     3.323188769403782e-10
0.075    |     5e-05    |     1.1788194485853987    |     2.5655434447895905e+18    |     569.6650806163085    |     598653.0681396328    |     1.3292768400222262e-10
0.075    |     7.5e-05    |     1.229861581346619    |     2.5655434447895905e+18    |     569.6650806163085    |     399102.3787592882    |     8.861853001624636e-11
0.075    |     0.0001    |     1.2625315302908773    |     2.5655434447895905e+18    |     569.6650806163085    |     299327.0340661838    |     6.646395302260717e-11
0.075    |     0.0002    |     1.3241269630070216    |     2.5655434447895905e+18    |     569.6650806163085    |     149664.01703276392    |     3.323208753353322e-11
0.075    |     0.0005    |     1.3699287056814906    |     2.5655434447895905e+18    |     569.6650806163085    |     59866.20681304865    |     1.3292968240163605e-11
0.075    |     0.00075    |     1.3804567145238036    |     2.5655434447895905e+18    |     569.6650806163085    |     39911.137875334636    |     8.862052841637133e-12
0.075    |     0.001    |     1.3852806670848645    |     2.5655434447895905e+18    |     569.6650806163085    |     29933.60340651576    |     6.646595142382363e-12
0.075    |     0.002    |     1.389766784950012    |     2.5655434447895905e+18    |     569.6650806163085    |     14967.301703250861    |     3.3234085934920857e-12
0.075    |     0.005    |     1.3829509021399737    |     2.5655434447895905e+18    |     569.6650806163085    |     5987.520681298484    |     1.3294966641593762e-12
0.075    |     0.0075    |     1.3758254841268682    |     2.5655434447895905e+18    |     569.6650806163085    |     3992.013787532067    |     8.864051243078357e-13
0.075    |     0.01    |     1.3694975238618983    |     2.5655434447895905e+18    |     569.6650806163085    |     2994.2603406490575    |     6.648593543821096e-13
0.075    |     0.02    |     1.3518655278680678    |     2.5655434447895905e+18    |     569.6650806163085    |     1497.630170324489    |     3.325406994935085e-13
0.075    |     0.05    |     1.3317271553627905    |     2.5655434447895905e+18    |     569.6650806163085    |     599.6520681297745    |     1.3314950656035374e-13
0.075    |     0.075    |     1.3262949589526654    |     2.5655434447895905e+18    |     569.6650806163085    |     400.1013787531858    |     8.884035257521146e-14
0.075    |     0.1    |     1.3237555826770928    |     2.5655434447895905e+18    |     569.6650806163085    |     300.3260340648898    |     6.668577558263996e-14
0.075    |     0.2    |     1.3207755491672837    |     2.5655434447895905e+18    |     569.6650806163085    |     150.6630170324437    |     3.345391009378222e-14
0.075    |     0.5    |     1.3198839335916897    |     2.5655434447895905e+18    |     569.6650806163085    |     60.86520681297757    |     1.3514790800467928e-14
0.075    |     0.75    |     1.3197723726186081    |     2.5655434447895905e+18    |     569.6650806163085    |     40.91013787531835    |     9.083875401953622e-15
0.075    |     1    |     1.3197346497546036    |     2.5655434447895905e+18    |     569.6650806163085    |     30.932603406488766    |     6.868417702696476e-15
0.1    |     1e-05    |     1.0798981068160698    |     1.4534741232350866e+18    |     322.73608746249107    |     2988038.7413779353    |     6.634778818299514e-10
0.1    |     2e-05    |     1.034411147287902    |     1.4534741232350866e+18    |     322.73608746249107    |     1494019.8706289497    |     3.3173905192395153e-10
0.1    |     5e-05    |     1.0485665416883234    |     1.4534741232350866e+18    |     322.73608746249107    |     597608.5482298866    |     1.3269575399152668e-10
0.1    |     7.5e-05    |     1.0859812540943814    |     1.4534741232350866e+18    |     322.73608746249107    |     398406.0321496727    |     8.846391000842339e-11
0.1    |     0.0001    |     1.1199271289309896    |     1.4534741232350866e+18    |     322.73608746249107    |     298804.7741134506    |     6.634798801773436e-11
0.1    |     0.0002    |     1.2092033599799092    |     1.4534741232350866e+18    |     322.73608746249107    |     149402.88705538897    |     3.317410503087292e-11
0.1    |     0.0005    |     1.305803183492356    |     1.4534741232350866e+18    |     322.73608746249107    |     59761.75482211882    |     1.3269775239103958e-11
0.1    |     0.00075    |     1.3347573633949794    |     1.4534741232350866e+18    |     322.73608746249107    |     39841.5032147429    |     8.846590840936952e-12
0.1    |     0.001    |     1.3503821090957047    |     1.4534741232350866e+18    |     322.73608746249107    |     29881.37741105147    |     6.634998641852679e-12
0.1    |     0.002    |     1.374194887133274    |     1.4534741232350866e+18    |     322.73608746249107    |     14941.188705527122    |     3.3176103432291097e-12
0.1    |     0.005    |     1.3835828525725824    |     1.4534741232350866e+18    |     322.73608746249107    |     5977.075482208381    |     1.327177364054051e-12
0.1    |     0.0075    |     1.381710463377333    |     1.4534741232350866e+18    |     322.73608746249107    |     3985.050321472298    |     8.848589242376854e-13
0.1    |     0.01    |     1.378484030173927    |     1.4534741232350866e+18    |     322.73608746249107    |     2989.0377411040554    |     6.63699704329458e-13
0.1    |     0.02    |     1.365360961026239    |     1.4534741232350866e+18    |     322.73608746249107    |     1495.018870551993    |     3.319608744671838e-13
0.1    |     0.05    |     1.3428726584962276    |     1.4534741232350866e+18    |     322.73608746249107    |     598.607548220793    |     1.329175765498276e-13
0.1    |     0.075    |     1.3344025162997288    |     1.4534741232350866e+18    |     322.73608746249107    |     399.4050321471928    |     8.868573256819285e-14
0.1    |     0.1    |     1.3296947739898706    |     1.4534741232350866e+18    |     322.73608746249107    |     299.80377411039274    |     6.656981057737548e-14
0.1    |     0.2    |     1.3228751118106652    |     1.4534741232350866e+18    |     322.73608746249107    |     150.40188705519589    |     3.339592759115015e-14
0.1    |     0.5    |     1.320130553011318    |     1.4534741232350866e+18    |     322.73608746249107    |     60.76075482207832    |     1.3491597799415071e-14
0.1    |     0.75    |     1.319826836642748    |     1.4534741232350866e+18    |     322.73608746249107    |     40.84050321471888    |     9.068413401251725e-15
0.1    |     1    |     1.3197458467857655    |     1.4534741232350866e+18    |     322.73608746249107    |     30.880377411039163    |     6.856821202170052e-15
0.2    |     1e-05    |     1.034878071426437    |     8.562996015832429e+17    |     190.13670673101288    |     2952676.851384146    |     6.556259649368781e-10
0.2    |     2e-05    |     1.099054270537908    |     8.562996015832429e+17    |     190.13670673101288    |     1476338.9256587334    |     3.278130934833386e-10
0.2    |     5e-05    |     1.118668157567818    |     8.562996015832429e+17    |     190.13670673101288    |     590536.170254422    |     1.3112537061808416e-10
0.2    |     7.5e-05    |     1.106463291346616    |     8.562996015832429e+17    |     190.13670673101288    |     393691.1135038531    |     8.741698776045872e-11
0.2    |     0.0001    |     1.091806769704812    |     8.562996015832429e+17    |     190.13670673101288    |     295268.58512797963    |     6.556279633151521e-11
0.2    |     0.0002    |     1.0475901909841485    |     8.562996015832429e+17    |     190.13670673101288    |     147634.79256333216    |     3.278150918791404e-11
0.2    |     0.0005    |     1.0237060377513256    |     8.562996015832429e+17    |     190.13670673101288    |     59054.517025191155    |     1.3112736901897105e-11
0.2    |     0.00075    |     1.04286126411308    |     8.562996015832429e+17    |     190.13670673101288    |     39370.011350120825    |     8.741898616131577e-12
0.2    |     0.001    |     1.0675482361224131    |     8.562996015832429e+17    |     190.13670673101288    |     29527.758512602293    |     6.556479473252506e-12
0.2    |     0.002    |     1.150251960798731    |     8.562996015832429e+17    |     190.13670673101288    |     14764.379256294229    |     3.2783507589271796e-12
0.2    |     0.005    |     1.2613461172689404    |     8.562996015832429e+17    |     190.13670673101288    |     5906.351702517787    |     1.311473530333848e-12
0.2    |     0.0075    |     1.2986164432111689    |     8.562996015832429e+17    |     190.13670673101288    |     3937.9011350116157    |     8.743897017574866e-13
0.2    |     0.01    |     1.3193653278045645    |     8.562996015832429e+17    |     190.13670673101288    |     2953.6758512586707    |     6.558477874693371e-13
0.2    |     0.02    |     1.3511101649720016    |     8.562996015832429e+17    |     190.13670673101288    |     1477.3379256293263    |     3.2803491603712903e-13
0.2    |     0.05    |     1.361365740618009    |     8.562996015832429e+17    |     190.13670673101288    |     591.5351702517382    |     1.3134719317780835e-13
0.2    |     0.075    |     1.3577244256580792    |     8.562996015832429e+17    |     190.13670673101288    |     394.69011350115363    |     8.763881032017942e-14
0.2    |     0.1    |     1.35321206907301    |     8.562996015832429e+17    |     190.13670673101288    |     296.26758512586605    |     6.578461889136601e-14
0.2    |     0.2    |     1.3396807959929746    |     8.562996015832429e+17    |     190.13670673101288    |     148.6337925629331    |     3.300333174814554e-14
0.2    |     0.5    |     1.3258785979549303    |     8.562996015832429e+17    |     190.13670673101288    |     60.05351702517299    |     1.3334559462213177e-14
0.2    |     0.75    |     1.3227927985529944    |     8.562996015832429e+17    |     190.13670673101288    |     40.369011350115315    |     8.96372117645046e-15
0.2    |     1    |     1.321535717952325    |     8.562996015832429e+17    |     190.13670673101288    |     30.5267585125865    |     6.778302033569106e-15
0.5    |     1e-05    |     0.24096794492402857    |     980258980477233.9    |     0.21766121804428137    |     2725466.7390964394    |     6.051751853189823e-10
0.5    |     2e-05    |     0.24490838803503873    |     980258980477233.9    |     0.21766121804428137    |     1362733.8714523853    |     3.025877041046033e-10
0.5    |     5e-05    |     0.26603719418936833    |     980258980477233.9    |     0.21766121804428137    |     545094.1490353851    |     1.2103521496950822e-10
0.5    |     7.5e-05    |     0.3041435170925932    |     980258980477233.9    |     0.21766121804428137    |     363396.432758199    |     8.069021734296e-11
0.5    |     0.0001    |     0.3466455959864901    |     980258980477233.9    |     0.21766121804428137    |     272547.57459426456    |     6.051771852405897e-11
0.5    |     0.0002    |     0.48965105730966685    |     980258980477233.9    |     0.21766121804428137    |     136274.2873159813    |     3.025897028851727e-11
0.5    |     0.0005    |     0.7000258094502446    |     980258980477233.9    |     0.21766121804428137    |     54510.31493093633    |     1.2103721343178792e-11
0.5    |     0.00075    |     0.7803152404537564    |     980258980477233.9    |     0.21766121804428137    |     36340.54328796332    |     8.069221577136814e-12
0.5    |     0.001    |     0.8291864833507758    |     980258980477233.9    |     0.21766121804428137    |     27255.657466223165    |     6.0519716940595025e-12
0.5    |     0.002    |     0.9148557625310408    |     980258980477233.9    |     0.21766121804428137    |     13628.32873330063    |     3.0260968693741908e-12
0.5    |     0.005    |     0.9605054768523522    |     980258980477233.9    |     0.21766121804428137    |     5451.931493366453    |     1.21057197452289e-12
0.5    |     0.0075    |     0.9664986040888508    |     980258980477233.9    |     0.21766121804428137    |     3634.9543289176663    |     8.071219978850555e-13
0.5    |     0.01    |     0.9715770597539694    |     980258980477233.9    |     0.21766121804428137    |     2726.465746690786    |     6.053970095655861e-13
0.5    |     0.02    |     1.0046221692212978    |     980258980477233.9    |     0.21766121804428137    |     1363.7328733473048    |     3.0280952708568005e-13
0.5    |     0.05    |     1.1047795554227993    |     980258980477233.9    |     0.21766121804428137    |     546.0931493393778    |     1.2125703759732826e-13
0.5    |     0.075    |     1.158142451657815    |     980258980477233.9    |     0.21766121804428137    |     364.395432892986    |     8.091203993320883e-14
0.5    |     0.1    |     1.1943040342674953    |     980258980477233.9    |     0.21766121804428137    |     273.5465746697651    |     6.073954110114357e-14
0.5    |     0.2    |     1.2650403625958067    |     980258980477233.9    |     0.21766121804428137    |     137.27328733490157    |     3.048079285303852e-14
0.5    |     0.5    |     1.3129959855689677    |     980258980477233.9    |     0.21766121804428137    |     55.50931493396519    |     1.2325543904171442e-14
0.5    |     0.75    |     1.320563153538016    |     980258980477233.9    |     0.21766121804428137    |     37.3395432893108    |     8.29104413775612e-15
0.5    |     1    |     1.322880946564286    |     980258980477233.9    |     0.21766121804428137    |     28.254657466983357    |     6.273794254548405e-15
0.75    |     1e-05    |     0.2224499701913163    |     3532627464900.7993    |     0.0007844008697912129    |     2442727.9074404994    |     5.423945531469741e-10
0.75    |     2e-05    |     0.20530606055267492    |     3532627464900.7993    |     0.0007844008697912129    |     1221364.876011224    |     2.711974813632221e-10
0.75    |     5e-05    |     0.2085533760821295    |     3532627464900.7993    |     0.0007844008697912129    |     488546.6517436233    |     1.084791482738597e-10
0.75    |     7.5e-05    |     0.21981047089552586    |     3532627464900.7993    |     0.0007844008697912129    |     325698.1161762267    |     7.23195095311772e-11
0.75    |     0.0001    |     0.22929183435173858    |     3532627464900.7993    |     0.0007844008697912129    |     244273.84276196134    |     5.4239688909598923e-11
0.75    |     0.0002    |     0.25206556710849465    |     3532627464900.7993    |     0.0007844008697912129    |     122137.4256038074    |     2.7119956414757816e-11
0.75    |     0.0005    |     0.29114658168992563    |     3532627464900.7993    |     0.0007844008697912129    |     48855.571254952665    |     1.084811601769268e-11
0.75    |     0.00075    |     0.32523987895143874    |     3532627464900.7993    |     0.0007844008697912129    |     32570.714320096464    |     7.232151393331879e-12
0.75    |     0.001    |     0.35990828966432387    |     3532627464900.7993    |     0.0007844008697912129    |     24428.285796379587    |     5.424169068652859e-12
0.75    |     0.002    |     0.4758051392903338    |     3532627464900.7993    |     0.0007844008697912129    |     12214.642940416235    |     2.7121955660050457e-12
0.75    |     0.005    |     0.6515260879776209    |     3532627464900.7993    |     0.0007844008697912129    |     4886.457186300838    |     1.0850114554152496e-12
0.75    |     0.0075    |     0.7197356993725796    |     3532627464900.7993    |     0.0007844008697912129    |     3257.9714590352405    |     7.234149854785078e-13
0.75    |     0.01    |     0.7617582568840171    |     3532627464900.7993    |     0.0007844008697912129    |     2443.7285948394756    |     5.426167503851333e-13
0.75    |     0.02    |     0.8408992984214783    |     3532627464900.7993    |     0.0007844008697912129    |     1222.3642978419875    |     2.714193975887874e-13
0.75    |     0.05    |     0.9210906761308861    |     3532627464900.7993    |     0.0007844008697912129    |     489.5457192381444    |     1.0870098582097407e-13
0.75    |     0.075    |     0.9624492708631208    |     3532627464900.7993    |     0.0007844008697912129    |     326.69714617377673    |     7.254133875229146e-14
0.75    |     0.1    |     0.9968207350967749    |     3532627464900.7993    |     0.0007844008697912129    |     245.2728596359627    |     5.4461515216699997e-14
0.75    |     0.2    |     1.0921914353907192    |     3532627464900.7993    |     0.0007844008697912129    |     123.13642982220418    |     2.734177991175017e-14
0.75    |     0.5    |     1.2113887072395224    |     3532627464900.7993    |     0.0007844008697912129    |     49.85457192989512    |     1.1069938727880118e-14
0.75    |     0.75    |     1.2502765485666594    |     3532627464900.7993    |     0.0007844008697912129    |     33.56971462008021    |     7.453974020261758e-15
0.75    |     1    |     1.2715113409155712    |     3532627464900.7993    |     0.0007844008697912129    |     25.427285965116454    |     5.645991666440076e-15
1.0    |     1e-05    |     0.3201790446663786    |     63914578843.30032    |     1.4191887408210385e-05    |     2130369.1131565426    |     4.730369680753338e-10
1.0    |     2e-05    |     0.30074023142892836    |     63914578843.30032    |     1.4191887408210385e-05    |     1065202.8089900897    |     2.3652253688723805e-10
1.0    |     5e-05    |     0.26987296047072223    |     63914578843.30032    |     1.4191887408210385e-05    |     426085.98427822563    |     9.461009404315171e-11
1.0    |     7.5e-05    |     0.255971387187336    |     63914578843.30032    |     1.4191887408210385e-05    |     284058.28740671236    |     6.307361020290444e-11
1.0    |     0.0001    |     0.24652041427480517    |     63914578843.30032    |     1.4191887408210385e-05    |     213044.20226336323    |     4.730531572313695e-11
1.0    |     0.0002    |     0.2298542106226902    |     63914578843.30032    |     1.4191887408210385e-05    |     106522.77866349413    |     2.365280830385211e-11
1.0    |     0.0005    |     0.23919407237621151    |     63914578843.30032    |     1.4191887408210385e-05    |     42609.754073134274    |     9.461266009121844e-12
1.0    |     0.00075    |     0.25539979901780313    |     63914578843.30032    |     1.4191887408210385e-05    |     28406.842361022733    |     6.307586089220936e-12
1.0    |     0.001    |     0.2695794334088568    |     63914578843.30032    |     1.4191887408210385e-05    |     21305.384137872177    |     4.730745603669856e-12
1.0    |     0.002    |     0.31063876807450386    |     63914578843.30032    |     1.4191887408210385e-05    |     10653.193844259556    |     2.3654842183383887e-12
1.0    |     0.005    |     0.3936609595496504    |     63914578843.30032    |     1.4191887408210385e-05    |     4261.877963782238    |     9.46327008706724e-13
1.0    |     0.0075    |     0.4469915900961469    |     63914578843.30032    |     1.4191887408210385e-05    |     2841.5853723107916    |     6.309587013554977e-13
1.0    |     0.01    |     0.49016736300399305    |     63914578843.30032    |     1.4191887408210385e-05    |     2131.4390529041125    |     4.732745424238766e-13
1.0    |     0.02    |     0.6017532446914196    |     63914578843.30032    |     1.4191887408210385e-05    |     1066.21954420533    |     2.3674829745641947e-13
1.0    |     0.05    |     0.7405769938099301    |     63914578843.30032    |     1.4191887408210385e-05    |     427.0878219429178    |     9.48325466916073e-14
1.0    |     0.075    |     0.7971634633262175    |     63914578843.30032    |     1.4191887408210385e-05    |     285.05854859317236    |     6.32957128028738e-14
1.0    |     0.1    |     0.8372760302121777    |     63914578843.30032    |     1.4191887408210385e-05    |     214.04391168158926    |     4.752729580594678e-14
1.0    |     0.2    |     0.9403435142235629    |     63914578843.30032    |     1.4191887408210385e-05    |     107.52195601832736    |     2.387467024485609e-14
1.0    |     0.5    |     1.0888716539372494    |     63914578843.30032    |     1.4191887408210385e-05    |     43.60878244993884    |     9.68309487035831e-15
1.0    |     0.75    |     1.1499813978771314    |     63914578843.30032    |     1.4191887408210385e-05    |     29.40585497293815    |     6.529411449948819e-15
1.0    |     1    |     1.1884279444711843    |     63914578843.30032    |     1.4191887408210385e-05    |     22.304391232070714    |     4.952569739218474e-15
2.0    |     1e-05    |     0.3771752708028818    |     4995157.201079173    |     1.1091477072520501e-09    |     931785.0004902228    |     2.0689783230892163e-10
2.0    |     2e-05    |     0.3655473904437711    |     4995157.201079173    |     1.1091477072520501e-09    |     513815.8738667831    |     1.1409004271695957e-10
2.0    |     5e-05    |     0.3556659370171325    |     4995157.201079173    |     1.1091477072520501e-09    |     219045.86744179655    |     4.8637953096574494e-11
2.0    |     7.5e-05    |     0.3535036870211835    |     4995157.201079173    |     1.1091477072520501e-09    |     148197.11674353815    |     3.290637023834767e-11
2.0    |     0.0001    |     0.3528173534980046    |     4995157.201079173    |     1.1091477072520501e-09    |     111978.62460515015    |     2.4864249460498955e-11
2.0    |     0.0002    |     0.35343475255308104    |     4995157.201079173    |     1.1091477072520501e-09    |     56624.48346399592    |     1.2573161059846943e-11
2.0    |     0.0005    |     0.3550242415453222    |     4995157.201079173    |     1.1091477072520501e-09    |     22805.49604335586    |     5.063837359066316e-12
2.0    |     0.00075    |     0.35371635738709434    |     4995157.201079173    |     1.1091477072520501e-09    |     15227.168148581388    |     3.3811105356787747e-12
2.0    |     0.001    |     0.35159094176904937    |     4995157.201079173    |     1.1091477072520501e-09    |     11429.335035838723    |     2.537822182588628e-12
2.0    |     0.002    |     0.3437425980245058    |     4995157.201079173    |     1.1091477072520501e-09    |     5721.711678609065    |     1.2704752091716875e-12
2.0    |     0.005    |     0.3384378232843943    |     4995157.201079173    |     1.1091477072520501e-09    |     2290.858149957779    |     5.086726928466632e-13
2.0    |     0.0075    |     0.34349118443995413    |     4995157.201079173    |     1.1091477072520501e-09    |     1527.8054038403636    |     3.3924094729806145e-13
2.0    |     0.01    |     0.35145093865438215    |     4995157.201079173    |     1.1091477072520501e-09    |     1146.1915618887438    |     2.545056525279907e-13
2.0    |     0.02    |     0.3879545312712362    |     4995157.201079173    |     1.1091477072520501e-09    |     573.6614252411312    |     1.2737842452839735e-13
2.0    |     0.05    |     0.47107485080566824    |     4995157.201079173    |     1.1091477072520501e-09    |     230.08032761773362    |     5.108809544690143e-14
2.0    |     0.075    |     0.5172647100633155    |     4995157.201079173    |     1.1091477072520501e-09    |     153.72255305569738    |     3.413326356131949e-14
2.0    |     0.1    |     0.5526131904880445    |     4995157.201079173    |     1.1091477072520501e-09    |     115.54279030330187    |     2.5655653224832402e-14
2.0    |     0.2    |     0.6463104722195465    |     4995157.201079173    |     1.1091477072520501e-09    |     58.27205179784529    |     1.2938994719623518e-14
2.0    |     0.5    |     0.7972439807925051    |     4995157.201079173    |     1.1091477072520501e-09    |     23.908978317115814    |     5.308859644585121e-15
2.0    |     0.75    |     0.8774340306364916    |     4995157.201079173    |     1.1091477072520501e-09    |     16.272675559455422    |     3.613259815672492e-15
2.0    |     1    |     0.9386486865700188    |     4995157.201079173    |     1.1091477072520501e-09    |     12.454515425128454    |     2.765457957105356e-15
5.0    |     1e-05    |     0.7290963744458004    |     555.636804006467    |     1.23376154627423e-13    |     554.9108780960391    |     1.2321496669543722e-13
5.0    |     2e-05    |     0.729116226423653    |     555.636804006467    |     1.23376154627423e-13    |     554.1868499306469    |     1.230542001474981e-13
5.0    |     5e-05    |     0.7291757523892822    |     555.636804006467    |     1.23376154627423e-13    |     552.0260777777667    |     1.225744123484788e-13
5.0    |     7.5e-05    |     0.729225323038207    |     555.636804006467    |     1.23376154627423e-13    |     550.2382870339197    |     1.2217744305907268e-13
5.0    |     0.0001    |     0.7292748625029514    |     555.636804006467    |     1.23376154627423e-13    |     548.462059660397    |     1.217830413536618e-13
5.0    |     0.0002    |     0.7294727088701933    |     555.636804006467    |     1.23376154627423e-13    |     541.470568779268    |     1.2023061852312455e-13
5.0    |     0.0005    |     0.7300632679255015    |     555.636804006467    |     1.23376154627423e-13    |     521.5279990898952    |     1.1580247851525786e-13
5.0    |     0.00075    |     0.7305520056062911    |     555.636804006467    |     1.23376154627423e-13    |     505.99987151524925    |     1.1235454156272012e-13
5.0    |     0.001    |     0.731037681432382    |     555.636804006467    |     1.23376154627423e-13    |     491.37136147682907    |     1.0910635983059726e-13
5.0    |     0.002    |     0.7329502858896307    |     555.636804006467    |     1.23376154627423e-13    |     440.4522822110097    |     9.780005299187204e-14
5.0    |     0.005    |     0.7384152522426066    |     555.636804006467    |     1.23376154627423e-13    |     336.0726653659447    |     7.462312220728345e-14
5.0    |     0.0075    |     0.742685910400422    |     555.636804006467    |     1.23376154627423e-13    |     280.7085388287958    |     6.232981660332277e-14
5.0    |     0.01    |     0.7467295294070299    |     555.636804006467    |     1.23376154627423e-13    |     241.04573230445237    |     5.3522904398406983e-14
5.0    |     0.02    |     0.7610911745523715    |     555.636804006467    |     1.23376154627423e-13    |     154.16834802636305    |     3.4232249929458513e-14
5.0    |     0.05    |     0.7938460924485602    |     555.636804006467    |     1.23376154627423e-13    |     74.4352551993876    |     1.6527946833241902e-14
5.0    |     0.075    |     0.815067856911206    |     555.636804006467    |     1.23376154627423e-13    |     52.21726682813808    |     1.1594562383118863e-14
5.0    |     0.1    |     0.8333792535955515    |     555.636804006467    |     1.23376154627423e-13    |     40.32070591377204    |     8.952995214921886e-15
5.0    |     0.2    |     0.8919888448352009    |     555.636804006467    |     1.23376154627423e-13    |     21.382869766048398    |     4.747950869365613e-15
5.0    |     0.5    |     1.0120030229593937    |     555.636804006467    |     1.23376154627423e-13    |     9.336977975122132    |     2.0732255856797125e-15
5.0    |     0.75    |     1.0827038377848375    |     555.636804006467    |     1.23376154627423e-13    |     6.585973686347976    |     1.4623799252317884e-15
5.0    |     1    |     1.1385693673981578    |     555.636804006467    |     1.23376154627423e-13    |     5.2000553828745195    |     1.1546442430786551e-15
7.5    |     1e-05    |     1.1579727162941067    |     65.42062274366073    |     1.4526296331065664e-14    |     65.40750197167736    |     1.4523382934434306e-14
7.5    |     2e-05    |     1.157977684282372    |     65.42062274366073    |     1.4526296331065664e-14    |     65.39438654331288    |     1.452047072432469e-14
7.5    |     5e-05    |     1.1579925874856083    |     65.42062274366073    |     1.4526296331065664e-14    |     65.35507228730188    |     1.4511741205880808e-14
7.5    |     7.5e-05    |     1.1580050059490283    |     65.42062274366073    |     1.4526296331065664e-14    |     65.32234705876328    |     1.4504474745438874e-14
7.5    |     0.0001    |     1.1580174236191763    |     65.42062274366073    |     1.4526296331065664e-14    |     65.28965509555218    |     1.449721567138344e-14
7.5    |     0.0002    |     1.1580670863672333    |     65.42062274366073    |     1.4526296331065664e-14    |     65.1592188836095    |     1.446825301423471e-14
7.5    |     0.0005    |     1.158215998465057    |     65.42062274366073    |     1.4526296331065664e-14    |     64.77106671380919    |     1.4382065919040608e-14
7.5    |     0.00075    |     1.1583400046414278    |     65.42062274366073    |     1.4526296331065664e-14    |     64.45117618545733    |     1.431103595305346e-14
7.5    |     0.001    |     1.1584639315271195    |     65.42062274366073    |     1.4526296331065664e-14    |     64.13447892974611    |     1.424071503602822e-14
7.5    |     0.002    |     1.1589588465897578    |     65.42062274366073    |     1.4526296331065664e-14    |     62.89868505684161    |     1.3966313673750365e-14
7.5    |     0.005    |     1.1604360020625992    |     65.42062274366073    |     1.4526296331065664e-14    |     59.46547650580346    |     1.320398823740986e-14
7.5    |     0.0075    |     1.161658314493383    |     65.42062274366073    |     1.4526296331065664e-14    |     56.88254043457721    |     1.2630461217927815e-14
7.5    |     0.01    |     1.162872831495904    |     65.42062274366073    |     1.4526296331065664e-14    |     54.518170529658384    |     1.2105465636493481e-14
7.5    |     0.02    |     1.1676546346371224    |     65.42062274366073    |     1.4526296331065664e-14    |     46.77181528449087    |     1.0385429246471316e-14
7.5    |     0.05    |     1.1813220595206992    |     65.42062274366073    |     1.4526296331065664e-14    |     32.913913723848474    |     7.3083569693485e-15
7.5    |     0.075    |     1.192025682675384    |     65.42062274366073    |     1.4526296331065664e-14    |     26.48423006114886    |     5.8806804006714365e-15
7.5    |     0.1    |     1.2021902002798601    |     65.42062274366073    |     1.4526296331065664e-14    |     22.210881996573836    |     4.931806517965728e-15
7.5    |     0.2    |     1.2385142386896435    |     65.42062274366073    |     1.4526296331065664e-14    |     13.695471638717006    |     3.0410055892808887e-15
7.5    |     0.5    |     1.3202321940460664    |     65.42062274366073    |     1.4526296331065664e-14    |     6.759169742060505    |     1.5008371749970505e-15
7.5    |     0.75    |     1.3692070811077144    |     65.42062274366073    |     1.4526296331065664e-14    |     4.957375592564133    |     1.1007585049158958e-15
7.5    |     1    |     1.4074503211973    |     65.42062274366073    |     1.4526296331065664e-14    |     4.014324376831694    |     8.91359070294516e-16
10.0    |     1e-05    |     1.4570968886841438    |     19.17586490054359    |     4.257897345936976e-15    |     19.174489304841284    |     4.257591902332721e-15
10.0    |     2e-05    |     1.4570993187331154    |     19.17586490054359    |     4.257897345936976e-15    |     19.173113917340388    |     4.257286504958466e-15
10.0    |     5e-05    |     1.4571066086154152    |     19.17586490054359    |     4.257897345936976e-15    |     19.168989003573486    |     4.256370590110744e-15
10.0    |     7.5e-05    |     1.4571126832141397    |     19.17586490054359    |     4.257897345936976e-15    |     19.16555300557406    |     4.255607645292439e-15
10.0    |     0.0001    |     1.457118757537252    |     19.17586490054359    |     4.257897345936976e-15    |     19.162118306915016    |     4.2548449889856545e-15
10.0    |     0.0002    |     1.4571430520739157    |     19.17586490054359    |     4.257897345936976e-15    |     19.148392490952492    |     4.251797245602982e-15
10.0    |     0.0005    |     1.4572159092372132    |     19.17586490054359    |     4.257897345936976e-15    |     19.10733923905199    |     4.242681592503847e-15
10.0    |     0.00075    |     1.4572765932520457    |     19.17586490054359    |     4.257897345936976e-15    |     19.0732698075378    |     4.235116659043259e-15
10.0    |     0.001    |     1.4573372497498753    |     19.17586490054359    |     4.257897345936976e-15    |     19.039328340305325    |     4.227580139361048e-15
10.0    |     0.002    |     1.4575796008997444    |     19.17586490054359    |     4.257897345936976e-15    |     18.904827802739522    |     4.197715020635045e-15
10.0    |     0.005    |     1.4583040248763595    |     19.17586490054359    |     4.257897345936976e-15    |     18.513096874683345    |     4.110733281477895e-15
10.0    |     0.0075    |     1.4589047145245593    |     19.17586490054359    |     4.257897345936976e-15    |     18.199513783630795    |     4.041103847913962e-15
10.0    |     0.01    |     1.459502697662358    |     19.17586490054359    |     4.257897345936976e-15    |     17.89696296052529    |     3.973924069927757e-15
10.0    |     0.02    |     1.461867908452495    |     19.17586490054359    |     4.257897345936976e-15    |     16.786203891155246    |     3.72728601120259e-15
10.0    |     0.05    |     1.468716436218725    |     19.17586490054359    |     4.257897345936976e-15    |     14.185809264834264    |     3.149882413751973e-15
10.0    |     0.075    |     1.4741565641494008    |     19.17586490054359    |     4.257897345936976e-15    |     12.594249511471265    |     2.7964851571019056e-15
10.0    |     0.1    |     1.479371019681003    |     19.17586490054359    |     4.257897345936976e-15    |     11.345519722685252    |     2.5192114444927974e-15
10.0    |     0.2    |     1.4982457886579081    |     19.17586490054359    |     4.257897345936976e-15    |     8.230533331790685    |     1.8275455219797643e-15
10.0    |     0.5    |     1.541009865953393    |     19.17586490054359    |     4.257897345936976e-15    |     4.798974397071008    |     1.0655863740429724e-15
10.0    |     0.75    |     1.5663574352182952    |     19.17586490054359    |     4.257897345936976e-15    |     3.7223151344244276    |     8.265199934297368e-16
10.0    |     1    |     1.5858716852811883    |     19.17586490054359    |     4.257897345936976e-15    |     3.1211613132222835    |     6.930370307017338e-16
20.0    |     1e-05    |     1.728648675651949    |     2.1429162348750768    |     4.75822988760272e-16    |     2.142898417351135    |     4.758190324752077e-16
20.0    |     2e-05    |     1.728648764407431    |     2.1429162348750768    |     4.75822988760272e-16    |     2.142880600382717    |     4.758150763134943e-16
20.0    |     5e-05    |     1.7286490306649482    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1428271528103546    |     4.758032085684049e-16
20.0    |     7.5e-05    |     1.7286492525359833    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1427826169852673    |     4.757933196287184e-16
20.0    |     0.0001    |     1.7286494743977217    |     2.1429162348750768    |     4.75822988760272e-16    |     2.142738084631156    |     4.757834314597433e-16
20.0    |     0.0002    |     1.728650361751707    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1425599899163488    |     4.757438864891547e-16
20.0    |     0.0005    |     1.728653022921501    |     2.1429162348750768    |     4.75822988760272e-16    |     2.142026038687107    |     4.756253254994085e-16
20.0    |     0.00075    |     1.7286552395412744    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1415814604086187    |     4.755266092912033e-16
20.0    |     0.001    |     1.728657455232824    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1411372281343835    |     4.754279699113758e-16
20.0    |     0.002    |     1.7286663087282654    |     2.1429162348750768    |     4.75822988760272e-16    |     2.13936375101392    |     4.750341788848189e-16
20.0    |     0.005    |     1.7286927805265953    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1340762407576848    |     4.738601157589361e-16
20.0    |     0.0075    |     1.7287147392794118    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1297073264769604    |     4.728900219135213e-16
20.0    |     0.01    |     1.7287366067447667    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1253719446187813    |     4.719273737616228e-16
20.0    |     0.02    |     1.7288231748695113    |     2.1429162348750768    |     4.75822988760272e-16    |     2.1083581370698803    |     4.681495495861566e-16
20.0    |     0.05    |     1.7290745149199334    |     2.1429162348750768    |     4.75822988760272e-16    |     2.060269448440304    |     4.574717157180394e-16
20.0    |     0.075    |     1.7292748667352158    |     2.1429162348750768    |     4.75822988760272e-16    |     2.023271926743742    |     4.492566156297211e-16
20.0    |     0.1    |     1.7294674543189974    |     2.1429162348750768    |     4.75822988760272e-16    |     1.9887693614232311    |     4.4159550714422816e-16
20.0    |     0.2    |     1.7301684663530894    |     2.1429162348750768    |     4.75822988760272e-16    |     1.871261042321496    |     4.1550341885387885e-16
20.0    |     0.5    |     1.7317725169771812    |     2.1429162348750768    |     4.75822988760272e-16    |     1.64227232787048    |     3.6465771022131223e-16
20.0    |     0.75    |     1.7327288355012886    |     2.1429162348750768    |     4.75822988760272e-16    |     1.5268757896947271    |     3.390345314923609e-16
20.0    |     1    |     1.7334658393718347    |     2.1429162348750768    |     4.75822988760272e-16    |     1.446630124465455    |     3.212164144595808e-16
50.0    |     1e-05    |     1.7391411403978456    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013610614050856    |     2.2234682126700225e-16
50.0    |     2e-05    |     1.739141140461826    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013610477854815    |     2.2234681824284264e-16
50.0    |     5e-05    |     1.7391411406537574    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013610069283048    |     2.2234680917072697e-16
50.0    |     7.5e-05    |     1.739141140813692    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013609728825315    |     2.223468016110467e-16
50.0    |     0.0001    |     1.7391411409736177    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013609388384612    |     2.2234679405174456e-16
50.0    |     0.0002    |     1.7391411416132447    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013608026792107    |     2.2234676381831757e-16
50.0    |     0.0005    |     1.7391411435313564    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.00136039436488    |     2.223466731543233e-16
50.0    |     0.00075    |     1.7391411451289047    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013600542900682    |     2.223465976425461e-16
50.0    |     0.001    |     1.7391411467256548    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013597143852402    |     2.2234652216851285e-16
50.0    |     0.002    |     1.739141153104692    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013583564632105    |     2.2234622064925224e-16
50.0    |     0.005    |     1.739141172165665    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013542989220954    |     2.2234531969413837e-16
50.0    |     0.0075    |     1.7391411879631304    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013509361089248    |     2.2234457299961647e-16
50.0    |     0.01    |     1.7391412036824188    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013475899545612    |     2.223438300040928e-16
50.0    |     0.02    |     1.7391412657893084    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0013343694737178    |     2.22340894467647e-16
50.0    |     0.05    |     1.7391414450146112    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.001296219922522    |     2.2233242356562365e-16
50.0    |     0.075    |     1.7391415867303393    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.001266056190751    |     2.223257258717195e-16
50.0    |     0.1    |     1.7391417220066685    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0012372643858902    |     2.223193328067841e-16
50.0    |     0.2    |     1.739142206764682    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0011341005544732    |     2.2229642583459456e-16
50.0    |     0.5    |     1.7391432733304768    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0009071775605214    |     2.2224603880805413e-16
50.0    |     0.75    |     1.7391438828572479    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0007775303834385    |     2.222172513518391e-16
50.0    |     1    |     1.739144340031251    |     1.0013610750249617    |     2.2234682429122227e-16    |     1.0006803060255958    |     2.2219566320771284e-16
75.0    |     1e-05    |     1.7391475402804037    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001490198573    |     2.2204463801408666e-16
75.0    |     2e-05    |     1.7391475402804104    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001490183668    |     2.220446380137557e-16
75.0    |     5e-05    |     1.7391475402804306    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001490138963    |     2.2204463801276305e-16
75.0    |     7.5e-05    |     1.7391475402804475    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001490101718    |     2.2204463801193603e-16
75.0    |     0.0001    |     1.739147540280464    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001490064463    |     2.220446380111088e-16
75.0    |     0.0002    |     1.73914754028053    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001489915489    |     2.220446380078009e-16
75.0    |     0.0005    |     1.7391475402807293    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001489468737    |     2.2204463799788104e-16
75.0    |     0.00075    |     1.7391475402808945    |     1.000000149021347    |     2.2204463801441744e-16    |     1.000000148909665    |     2.2204463798961905e-16
75.0    |     0.001    |     1.7391475402810592    |     1.000000149021347    |     2.2204463801441744e-16    |     1.000000148872475    |     2.220446379813612e-16
75.0    |     0.002    |     1.7391475402817205    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001487238994    |     2.220446379483708e-16
75.0    |     0.005    |     1.739147540283696    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001482799472    |     2.220446378497936e-16
75.0    |     0.0075    |     1.7391475402853336    |     1.000000149021347    |     2.2204463801441744e-16    |     1.000000147912007    |     2.220446377680945e-16
75.0    |     0.01    |     1.7391475402869625    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001475458882    |     2.2204463768679976e-16
75.0    |     0.02    |     1.7391475402933994    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001460993597    |     2.220446373656059e-16
75.0    |     0.05    |     1.7391475403119734    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001419250921    |     2.220446364387323e-16
75.0    |     0.075    |     1.7391475403266607    |     1.000000149021347    |     2.2204463801441744e-16    |     1.000000138624508    |     2.2204463570585544e-16
75.0    |     0.1    |     1.7391475403406793    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001354739512    |     2.2204463500629127e-16
75.0    |     0.2    |     1.7391475403909142    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000001241844543    |     2.220446324995194e-16
75.0    |     0.5    |     1.7391475405014305    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000000993475624    |     2.2204462698462155e-16
75.0    |     0.75    |     1.7391475405645829    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000000851550528    |     2.2204462383325136e-16
75.0    |     1    |     1.7391475406119472    |     1.000000149021347    |     2.2204463801441744e-16    |     1.0000000745106707    |     2.2204462146972376e-16
100.0    |     1e-05    |     1.7391475409434956    |     1.000000000000426    |     2.220446049251259e-16    |     1.000000000000426    |     2.220446049251259e-16
100.0    |     2e-05    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.000000000000426    |     2.220446049251259e-16
100.0    |     5e-05    |     1.7391475409434949    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004259    |     2.2204460492512587e-16
100.0    |     7.5e-05    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004259    |     2.2204460492512587e-16
100.0    |     0.0001    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004259    |     2.2204460492512587e-16
100.0    |     0.0002    |     1.739147540943495    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004259    |     2.2204460492512587e-16
100.0    |     0.0005    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004257    |     2.220446049251258e-16
100.0    |     0.00075    |     1.739147540943495    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004259    |     2.2204460492512587e-16
100.0    |     0.001    |     1.7391475409434949    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004254    |     2.2204460492512577e-16
100.0    |     0.002    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004252    |     2.2204460492512572e-16
100.0    |     0.005    |     1.7391475409434949    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004237    |     2.220446049251254e-16
100.0    |     0.0075    |     1.739147540943495    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004228    |     2.220446049251252e-16
100.0    |     0.01    |     1.7391475409434949    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004219    |     2.22044604925125e-16
100.0    |     0.02    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000004174    |     2.22044604925124e-16
100.0    |     0.05    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.000000000000406    |     2.2204460492512144e-16
100.0    |     0.075    |     1.7391475409434953    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000003963    |     2.220446049251193e-16
100.0    |     0.1    |     1.7391475409434949    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000003872    |     2.220446049251173e-16
100.0    |     0.2    |     1.739147540943496    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000003548    |     2.220446049251101e-16
100.0    |     0.5    |     1.739147540943496    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000002842    |     2.220446049250944e-16
100.0    |     0.75    |     1.7391475409434956    |     1.000000000000426    |     2.220446049251259e-16    |     1.0000000000002436    |     2.220446049250854e-16
100.0    |     1    |     1.7391475409434956    |     1.000000000000426    |     2.220446049251259e-16    |     1.000000000000213    |     2.220446049250786e-16
200.0    |     1e-05    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000002    |     2.2204460492503136e-16
200.0    |     2e-05    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000002    |     2.2204460492503136e-16
200.0    |     5e-05    |     1.7391475409434967    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     7.5e-05    |     1.7391475409434973    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.0001    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.0002    |     1.7391475409434969    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.0005    |     1.7391475409434973    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.00075    |     1.7391475409434967    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.001    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.002    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.005    |     1.7391475409434969    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.0075    |     1.7391475409434973    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.01    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.02    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.05    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.075    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0    |     2.220446049250313e-16
200.0    |     0.1    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000007    |     2.2204460492503146e-16
200.0    |     0.2    |     1.7391475409434973    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
200.0    |     0.5    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0    |     2.220446049250313e-16
200.0    |     0.75    |     1.739147540943497    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000002    |     2.2204460492503136e-16
200.0    |     1    |     1.7391475409434967    |     1.0000000000000004    |     2.220446049250314e-16    |     1.0000000000000004    |     2.220446049250314e-16
500.0    |     1e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     2e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     5e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     7.5e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.0001    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.0002    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.0005    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.00075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.001    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.002    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.005    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.0075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.01    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.02    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.1    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.2    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.5    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     0.75    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
500.0    |     1    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     1e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     2e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     5e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     7.5e-05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.0001    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.0002    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.0005    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.00075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.001    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.002    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.005    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.0075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.01    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.02    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.05    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.075    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.1    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.2    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.5    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     0.75    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16
1000.0    |     1    |     1.739147540943497    |     1.0    |     2.220446049250313e-16    |     1.0    |     2.220446049250313e-16

Results saved in  solution.pickle
========================================================================================================================
Results of radial basis function run:

Basis function type               : gaussian
Shape parameter                    : 0.75
Regularization parameter           : 2e-05
Number of terms in RBF model       : 31

RBF Expression:
--------------------------

4.038157591240296 + 237.8499310618983*(-1551.5244569618876*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.5518182560031254)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.2731190901874679)**2)**0.5)**2) - 334.71891273048857*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.4300326512391472)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8610629174234938)**2)**0.5)**2) + 235.58220304204985*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.11035664436222248)**2)**0.5)**2) + 27.886153309561678*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.6822668136056202)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8414199967160326)**2)**0.5)**2) - 1415.9829578091171*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.13520535030959882)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5811256814195772)**2)**0.5)**2) - 990.5466782519585*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.908216841775921)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.20132180755786522)**2)**0.5)**2) - 343.10909375147367*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.18501228557936236)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.09666279198240106)**2)**0.5)**2) - 723.9922784849874*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.22021941796179972)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 1.0)**2)**0.5)**2) - 8.525226879211669*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8266365897830592)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.16264387978612238)**2)**0.5)**2) + 255.07386589590683*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.831288905218487)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.26866011456766936)**2)**0.5)**2) + 528.0399010886766*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.43739443999592303)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9599385359524552)**2)**0.5)**2) + 432.74002291691215*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8327482651149929)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3330572615505772)**2)**0.5)**2) - 761.4482839044426*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.17485274298531528)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3711736045917253)**2)**0.5)**2) + 854.4206928222548*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.0009932894634358945)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.2461895993357224)**2)**0.5)**2) + 158.4825871693355*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8102253955808789)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546)**2)**0.5)**2) - 2542.5018400558793*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.6086081776493459)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6109286075828757)**2)**0.5)**2) - 1366.554383948263*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.1028770983224648)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.38063145554753214)**2)**0.5)**2) + 342.1406582093882*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.0325313822086792)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9084335925839453)**2)**0.5)**2) + 107.6091733488148*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 1.0)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.046386199083146756)**2)**0.5)**2) + 49.67120676760413*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.9074219145138307)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5830026053427076)**2)**0.5)**2) + 897.3018668528205*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.7557460774371298)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6383110566137417)**2)**0.5)**2) - 8.627269390497531*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.5911925356865033)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.005381283583344741)**2)**0.5)**2) - 136.91234827602625*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.21031274399063923)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5495609776179707)**2)**0.5)**2) + 735.7407100245284*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.7830308832306688)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.24439265066740792)**2)**0.5)**2) + 541.730933783887*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.28803140536417743)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8689715765112118)**2)**0.5)**2) - 45.921399758259014*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.993941455353761)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9026639746834018)**2)**0.5)**2) + 1712.6788445573948*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.36343808704287006)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6057949629205177)**2)**0.5)**2) + 3266.9168170263265*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.3586109792561233)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3372905989731323)**2)**0.5)**2) - 827.2715689260567*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.17758838147122427)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.23089727866378493)**2)**0.5)**2) + 919.0325530603195*exp(- (0.75*(((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.04112204875223326)**2 + ((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5088149538788017)**2)**0.5)**2))
--------------------------

Model training errors:
-----------------------
Mean Squared Error (MSE)         : 0.0004443809827437673
Root Mean Squared Error (RMSE)   : 0.021080345887669095
Goodness of fit (R2)             : 0.9918215668806757

========================================================================================================================

Step 5: View model metrics#

We can view the Root Mean Square Error (RMSE) and \(R^2\) values of the RBF fit based on the training data:

print("R2: ", rbf_class.R2, "\nRMSE: ", rbf_class.rmse)
R2:  0.9918215668806757 
RMSE:  0.021080345887669095

Step 6 (Optional): Generate Pyomo expression output at any unsampled point#

Based on the model we trained, we can predict the surrogate value at any previously unsampled point. We do this by calling the function predict_output.

Let us again evaluate the RBF surrogate at the same set of points we considered for the polynomial model:

  • \(x_{1}=5\), \(x_{2}=8\) (true function value: 57.9908)

  • \(x_{1}=-3\), \(x_{2}=10\) (true function value: 4.2461)

  • \(x_{1}=-2\), \(x_{2}=3\) (true function value: 50.8899)

unsampled_points = np.array([[5, 8], [-3, 10], [-2, 3]])
ys = rbf_class.predict_output(unsampled_points)
print(ys)
[[61.06401281]
 [12.7142377 ]
 [50.97501439]]

The results from the RBF surrogate are similar to those obtained from the polynomial.

For RBF models, the Pyomo expression is generated by calling generate_expression on the results object, while parity plots may be viewed with the parity_residual_plots method.

Further information about using PySMO’s RBF tool and features can be found in the documentation.

Training Kriging models#

The KrigingModel class trains Kriging from data. For details about Kriging models, users should consult the documentation.

As an example, we will again consider the Brainin function. The same dataset loaded previously will be used.

Step 1: Load the data and import the Kriging tool#

import numpy as np

brainin_data = np.loadtxt("brainin_30.txt")
from idaes.core.surrogate.pysmo.kriging import KrigingModel

Step 2: Specify the Kriging settings and initialize the KrigingModel class#

The KrigingModel class takes a number of keyword arguments:

  -  XY_data                       : The dataset for Kriging training. training_data is expected to contain xy_data, 
                                     with the output values (y) in the last column.

It also takes a number of optional arguments:

  -  regularization                : Boolean variable determining whether regularization is done. Default is True.
  -  numerical_gradients           : Boolean variable which determines whether numerical gradients are used when
                                     solving the max. likelihood optimization problem. Default is True.
  -  fname                         : Filename for saving (.pickle extension)
  - overwrite                      : Option determining whether any existing file with the same name supplied in 'fname'  
                                     should be overwritten.

For this demonstration, we will train a Kriging model with regularization:

krg_class = KrigingModel(XY_data=brainin_data, overwrite=True)
Warning: solution.pickle already exists; previous file will be overwritten.

Step 3: Extract variable names (optional)#

Next, we extract Pyomo variable names from the dataset.

vars = krg_class.get_feature_vector()

Step 4: Train the Kriging surrogate#

Next, we train the RBF surrogate by calling training:

krg_class.training()
Optimizing kriging parameters using L-BFGS-B algorithm...
Final results
================
Theta: [6.01576437 0.22794622] 
Mean: [[386.67466251]] 
Regularization parameter: 1.000000000001e-06

Results saved in  solution.pickle
========================================================================================================================
Results of Kriging run:

Kriging mean                     : [[386.67466251]]
Kriging variance                 : [[70460.78166121]]
Kriging weights                  : [6.01576437 0.22794622]
Regularization parameter         : 1.000000000001e-06
Number of terms in Kriging model : 31

Kriging Expression:
--------------------

386.6746625106164 + (3469.0208871846553*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.5518182560031254)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.2731190901874679)**2)) - 141877.18052533641*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.4300326512391472)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8610629174234938)**2)) + 89858.53563122451*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.11035664436222248)**2)) - 17331.16845172411*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.6822668136056202)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8414199967160326)**2)) + 70866.88925537094*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.13520535030959882)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5811256814195772)**2)) + 84227.85506583657*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.908216841775921)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.20132180755786522)**2)) + 12240.144907206297*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.18501228557936236)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.09666279198240106)**2)) + 10862.880267933011*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.22021941796179972)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 1.0)**2)) - 88626.71938276757*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8266365897830592)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.16264387978612238)**2)) - 59560.382920601405*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.831288905218487)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.26866011456766936)**2)) + 108091.18288033456*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.43739443999592303)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9599385359524552)**2)) - 49286.85214255564*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8327482651149929)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3330572615505772)**2)) + 142911.70424503088*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.17485274298531528)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3711736045917253)**2)) - 121075.87439893931*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.0009932894634358945)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.2461895993357224)**2)) + 18631.616961394437*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.8102253955808789)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546)**2)) - 20558.20656931633*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.6086081776493459)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6109286075828757)**2)) - 147105.12800574303*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.1028770983224648)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.38063145554753214)**2)) - 20770.266188384965*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.0325313822086792)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9084335925839453)**2)) - 12552.899758738116*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 1.0)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.046386199083146756)**2)) - 15680.859052546322*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.9074219145138307)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5830026053427076)**2)) + 51439.576363677625*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.7557460774371298)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6383110566137417)**2)) - 7759.502903997432*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.5911925356865033)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.005381283583344741)**2)) - 105321.72080910672*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.21031274399063923)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5495609776179707)**2)) + 103559.18517672084*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.7830308832306688)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.24439265066740792)**2)) - 32907.053552615456*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.28803140536417743)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.8689715765112118)**2)) + 2458.404474788578*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.993941455353761)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.9026639746834018)**2)) + 99575.24403728923*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.36343808704287006)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.6057949629205177)**2)) - 18709.6442205254*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.3586109792561233)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.3372905989731323)**2)) - 47558.269057542086*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.17758838147122427)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.23089727866378493)**2)) + 108489.48778639734*exp(- (6.01576436749808*((IndexedParam[0] + 4.929217157135412)/14.643030012320084 - 0.04112204875223326)**2 + 0.22794621767031822*((IndexedParam[1] - 0.22882456869508516)/14.45053220191546 - 0.5088149538788017)**2)))
--------------------------

Model training errors:
-----------------------
Mean Squared Error (MSE)         : 0.005821749035738542
Root Mean Squared Error (RMSE)   : 0.07630038686493366
Goodness of fit (R2)             : 0.999998106078119

========================================================================================================================

The returned information correspond to the Kriging model parameters.

As we can see, the optimization problem was solved using SciPy’s L-BFGS algorithm which makes use of gradients. A different algorithm (Basinhopping) is used when no numerical gradients are computed (when numerical_gradients is set to False). The user should try this.

Step 5: View model metrics#

We can view the RMSE and \(R^2\) values of the Kriging fit based on the training data:

print("R2: ", krg_class.training_R2, "\nRMSE: ", krg_class.training_rmse)
R2:  0.999998106078119 
RMSE:  0.07630038686493366

Step 6 (Optional): Generate Pyomo expression output at any unsampled point#

Again, based on the model we trained, we evaluate the surrogate at a set of off-design points:

  • \(x_{1}=5\), \(x_{2}=8\) (true function value: 57.9908)

  • \(x_{1}=-3\), \(x_{2}=10\) (true function value: 4.2461)

  • \(x_{1}=-2\), \(x_{2}=3\) (true function value: 50.8899)

We do this by calling the function predict_output:

unsampled_points = np.array([[5, 8], [-3, 10], [-2, 3]])
ys = krg_class.predict_output(unsampled_points)
print(ys)
[[57.63225481]
 [ 4.44461901]
 [50.8024123 ]]

The Kriging model performs very well, predicting all three points fairly accurately.

For Kriging models, the Pyomo expression is generated by calling generate_expression on the results object:

print(krg_class.generate_expression([m.x[1], m.x[2]]))
3469.0208871846553*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.5518182560031254)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.2731190901874679)**2)) - 141877.18052533641*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.4300326512391472)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.8610629174234938)**2)) + 89858.53563122451*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.11035664436222248)**2)) - 17331.16845172411*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.6822668136056202)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.8414199967160326)**2)) + 70866.88925537094*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.13520535030959882)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.5811256814195772)**2)) + 84227.85506583657*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.908216841775921)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.20132180755786522)**2)) + 12240.144907206297*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.18501228557936236)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.09666279198240106)**2)) + 10862.880267933011*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.22021941796179972)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 1.0)**2)) - 88626.71938276757*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.8266365897830592)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.16264387978612238)**2)) - 59560.382920601405*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.831288905218487)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.26866011456766936)**2)) + 108091.18288033456*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.43739443999592303)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.9599385359524552)**2)) - 49286.85214255564*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.8327482651149929)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.3330572615505772)**2)) + 142911.70424503088*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.17485274298531528)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.3711736045917253)**2)) - 121075.87439893931*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.0009932894634358945)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.2461895993357224)**2)) + 18631.616961394437*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.8102253955808789)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546)**2)) - 20558.20656931633*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.6086081776493459)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.6109286075828757)**2)) - 147105.12800574303*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.1028770983224648)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.38063145554753214)**2)) - 20770.266188384965*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.0325313822086792)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.9084335925839453)**2)) - 12552.899758738116*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 1.0)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.046386199083146756)**2)) - 15680.859052546322*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.9074219145138307)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.5830026053427076)**2)) + 51439.576363677625*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.7557460774371298)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.6383110566137417)**2)) - 7759.502903997432*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.5911925356865033)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.005381283583344741)**2)) - 105321.72080910672*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.21031274399063923)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.5495609776179707)**2)) + 103559.18517672084*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.7830308832306688)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.24439265066740792)**2)) - 32907.053552615456*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.28803140536417743)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.8689715765112118)**2)) + 2458.404474788578*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.993941455353761)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.9026639746834018)**2)) + 99575.24403728923*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.36343808704287006)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.6057949629205177)**2)) - 18709.6442205254*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.3586109792561233)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.3372905989731323)**2)) - 47558.269057542086*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.17758838147122427)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.23089727866378493)**2)) + 108489.48778639734*exp(- (6.01576436749808*((x[1] + 4.929217157135412)/14.643030012320084 - 0.04112204875223326)**2 + 0.22794621767031822*((x[2] - 0.22882456869508516)/14.45053220191546 - 0.5088149538788017)**2)) + 386.6746625106164

As we can see, expressing a Kriging model algebraically is pretty complicated!

Parity plots for the Kriging model may be viewed with the parity_residual_plots method.

Further information about using PySMO’s Kriging tool and features can be found in the documentation.

Summary#

PySMO allows IDAES users to sample design spaces and generate different types of surrogate models. Further information about PySMO’s capabilities may be found in the documentation.