SymbolicFunction

class SymbolicFunction(*args)

Symbolic function.

Parameters:
inputssequence of str, or str

List of input variables names of the function.

formulassequence of str, or str

List of analytical formulas between the inputs and the outputs. The function is defined by outputs = formulas(inputs).

Available functions:
  • sin

  • cos

  • tan

  • asin

  • acos

  • atan

  • sinh

  • cosh

  • tanh

  • asinh

  • acosh

  • atanh

  • log2

  • log10

  • log

  • ln

  • lngamma

  • gamma

  • exp

  • erf

  • erfc

  • sqrt

  • cbrt

  • besselJ0

  • besselJ1

  • besselY0

  • besselY1

  • sign

  • rint

  • abs

  • min

  • max

  • sum

  • avg

  • floor

  • ceil

  • trunc

  • round

Available operators:
  • <= (less or equal)

  • >= (greater or equal)

  • != (not equal)

  • == (equal)

  • > (greater than)

  • < (less than)

  • + (addition)

  • - (subtraction)

  • * (multiplication)

  • / (division)

  • ^ (raise x to the power of y)

Available constants:
  • e_ (Euler’s constant)

  • pi_ (Pi)

Notes

Up to version 1.10, OpenTURNS relied on muParser to parse analytical formulas. Since version 1.11, ExprTk is used by default, but both parsers can be used if their support have been compiled. This is controlled by the SymbolicParser-Backend ResourceMap entry.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x0', 'x1'], ['x0 + x1', 'x0 - x1'])
>>> print(f([1, 2]))
[3,-1]

ExprTk allows one to write multiple outputs; in this case, the constructor has a special syntax, it contains input variables names, but also output variables names, and formula is a string:

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x0', 'x1'], ['y0', 'y1'], 'y0 := x0 + x1; y1 := x0 - x1')
>>> print(f([1, 2]))
[3,-1]

The following example uses the min and sqrt functions:

>>> formula = 'min(-x1 - x2 - x3 + 3 * sqrt(3), -x3 + 3)'
>>> limitStateFunction = ot.SymbolicFunction(['x1', 'x2', 'x3'], [formula])
>>> print(limitStateFunction([1, 2, 3]))
[-0.803848]

The following example splits the formula into four parts to manage its length:

>>> formula = '15.59 * 1e4 - x1 *x2^3 / (2 * x3^3) *'
>>> formula += '((x4^2 - 4 * x5 * x6 * x7^2 + '
>>> formula += 'x4 * (x6 + 4 * x5 + 2 *x6 * x7)) / '
>>> formula += '(x4 * x5 * (x4 + x6 + 2 *x6 *x7)))'
>>> input_variables = ['x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7']
>>> limitStateFunction = ot.SymbolicFunction(input_variables, [formula])
>>> print(limitStateFunction([1, 2, 3, 4, 5, 6, 7]))
[155900]

ExprTk allows one to manage intermediate variables with the var keyword. This is convenient in the situation where several outputs require the same intermediate calculation or if the output is a complex function of the input. In the following example, we compute the alpha variable which is the slope of the river in the flooding example. This slope is then used in the computation of the height H.

>>> import openturns as ot
>>> inputs = ['Q', 'Ks', 'Zv', 'Zm', 'Hd', 'Zb', 'L', 'B']
>>> outputs = ['H', 'S']
>>> formula = 'var alpha := (Zm - Zv)/L;'
>>> formula += 'H := (Q / (Ks * B * sqrt(alpha)))^(3.0 / 5.0);'
>>> formula += 'var Zc := H + Zv;'
>>> formula += 'var Zd := Zb + Hd;'
>>> formula += 'S := Zc - Zd'
>>> myFunction = ot.SymbolicFunction(inputs, outputs, formula)
>>> X = [1013.0, 30.0, 50.0, 55.0, 8, 55.5, 5000.0, 300.0]
>>> print(myFunction(X))
[2.142,-11.358]

The following example illustrates a function for a system of two components.

>>> equations = ['var g1 := x1^2 -8 * x2 + 16']
>>> equations.append('var g2 := -16 * x1 + x2 + 32')
>>> equations.append('gsys := max(g1, g2)')
>>> formula = ';'.join(equations)
>>> limitStateFunction = ot.SymbolicFunction(['x1', 'x2'], ['gsys'], formula)
>>> print(limitStateFunction([1, 2]))
[18]

See the ExprTk documentation for details.

Methods

GetValidConstants()

Return the list of valid constants.

GetValidFunctions()

Return the list of valid functions.

GetValidOperators()

Return the list of valid operators.

GetValidParsers()

Return the list of built-in parsers.

draw(*args)

Draw the output of function as a Graph.

getCallsNumber()

Accessor to the number of direct calls to the function.

getClassName()

Accessor to the object's name.

getDescription()

Accessor to the description of the inputs and outputs.

getEvaluation()

Accessor to the evaluation function.

getEvaluationCallsNumber()

Accessor to the number of times the evaluation of the function has been called.

getFormulas()

Formulas accessor.

getGradient()

Accessor to the gradient function.

getGradientCallsNumber()

Accessor to the number of times the gradient of the function has been called.

getHessian()

Accessor to the hessian function.

getHessianCallsNumber()

Accessor to the number of times the hessian of the function has been called.

getId()

Accessor to the object's id.

getImplementation()

Accessor to the underlying implementation.

getInputDescription()

Accessor to the description of the input vector.

getInputDimension()

Accessor to the dimension of the input vector.

getMarginal(*args)

Accessor to marginal.

getName()

Accessor to the object's name.

getOutputDescription()

Accessor to the description of the output vector.

getOutputDimension()

Accessor to the number of the outputs.

getParameter()

Accessor to the parameter values.

getParameterDescription()

Accessor to the parameter description.

getParameterDimension()

Accessor to the dimension of the parameter.

gradient(inP)

Return the Jacobian transposed matrix of the function at a point.

hessian(inP)

Return the hessian of the function at a point.

isLinear()

Accessor to the linearity of the function.

isLinearlyDependent(index)

Accessor to the linearity of the function with regard to a specific variable.

parameterGradient(inP)

Accessor to the gradient against the parameter.

setDescription(description)

Accessor to the description of the inputs and outputs.

setEvaluation(evaluation)

Accessor to the evaluation function.

setGradient(gradient)

Accessor to the gradient function.

setHessian(hessian)

Accessor to the hessian function.

setInputDescription(inputDescription)

Accessor to the description of the input vector.

setName(name)

Accessor to the object's name.

setOutputDescription(inputDescription)

Accessor to the description of the output vector.

setParameter(parameter)

Accessor to the parameter values.

setParameterDescription(description)

Accessor to the parameter description.

setStopCallback(callBack[, state])

Set up a stop callback.

__init__(*args)
static GetValidConstants()

Return the list of valid constants.

Returns:
list_constantsDescription

List of the available constants.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidConstants()[0])
e_ -> Euler's constant (2.71828...)
static GetValidFunctions()

Return the list of valid functions.

Returns:
list_functionsDescription

List of the available functions.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidFunctions()[0])
sin(arg) -> sine function
static GetValidOperators()

Return the list of valid operators.

Returns:
list_operatorsDescription

List of the available operators.

Examples

>>> import openturns as ot
>>> print(ot.SymbolicFunction.GetValidOperators()[0])
= -> assignment, can only be applied to variable names (priority -1)
static GetValidParsers()

Return the list of built-in parsers.

Analytical formulas can be parsed by ‘MuParser’ or ‘ExprTk’ parsers, but this support may be disabled at build-time. This method returns the list of parsers available at run-time. Parser can be switched by changing ‘SymbolicParser-Backend’ ResourceMap entry.

Returns:
list_constantsDescription

List of the available parsers.

draw(*args)

Draw the output of function as a Graph.

Available usages:

draw(inputMarg, outputMarg, centralPoint, xiMin, xiMax, ptNb, scale)

draw(firstInputMarg, secondInputMarg, outputMarg, centralPoint, xiMin_xjMin, xiMax_xjMax, ptNbs, scale, isFilled)

draw(xiMin, xiMax, ptNb, scale)

draw(xiMin_xjMin, xiMax_xjMax, ptNbs, scale)

Parameters:
outputMarg, inputMargint, outputMarg, inputMarg \geq 0

outputMarg is the index of the marginal to draw as a function of the marginal with index inputMarg.

firstInputMarg, secondInputMargint, firstInputMarg, secondInputMarg \geq 0

In the 2D case, the marginal outputMarg is drawn as a function of the two marginals with indexes firstInputMarg and secondInputMarg.

centralPointsequence of float

Central point with dimension equal to the input dimension of the function.

xiMin, xiMaxfloat

Define the interval where the curve is plotted.

xiMin_xjMin, xiMax_xjMaxsequence of float of dimension 2.

In the 2D case, define the intervals where the curves are plotted.

ptNbint ptNb > 0

The number of points to draw the curves.

ptNbslist of int of dimension 2 ptNbs_k > 0, k=1,2

The number of points to draw the contour in the 2D case.

scalebool

scale indicates whether the logarithmic scale is used either for one or both axes:

  • ot.GraphImplementation.NONE or 0: no log scale is used,

  • ot.GraphImplementation.LOGX or 1: log scale is used only for horizontal data,

  • ot.GraphImplementation.LOGY or 2: log scale is used only for vertical data,

  • ot.GraphImplementation.LOGXY or 3: log scale is used for both data.

isFilledbool

isFilled indicates whether the contour graph is filled or not

Notes

We note f: \Rset^n \rightarrow \Rset^p where \vect{x} = (x_1, \dots, x_n) and f(\vect{x}) = (f_1(\vect{x}), \dots, f_p(\vect{x})), with n\geq 1 and p\geq 1.

  • In the first usage:

Draws graph of the given 1D outputMarg marginal f_k: \Rset^n \rightarrow \Rset as a function of the given 1D inputMarg marginal with respect to the variation of x_i in the interval [x_i^{min}, x_i^{max}], when all the other components of \vect{x} are fixed to the corresponding components of the centralPoint \vect{c}. Then OpenTURNS draws the graph:

y = f_k^{(i)}(s)

for any s \in [x_i^{min}, x_i^{max}] where f_k^{(i)}(s) is defined by the equation:

f_k^{(i)}(s) = f_k(c_1, \dots, c_{i-1}, s,  c_{i+1} \dots, c_n).

  • In the second usage:

Draws the iso-curves of the given outputMarg marginal f_k as a function of the given 2D firstInputMarg and secondInputMarg marginals with respect to the variation of (x_i, x_j) in the interval [x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}], when all the other components of \vect{x} are fixed to the corresponding components of the centralPoint \vect{c}. Then OpenTURNS draws the graph:

y = f_k^{(i,j)}(s, t)

for any (s, t) \in [x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}] where f_k^{(i,j)} is defined by the equation:

f_k^{(i,j)}(s,t) = f_k(c_1, \dots, c_{i-1}, s, c_{i+1}, \dots, c_{j-1}, t,  c_{j+1} \dots, c_n).

  • In the third usage:

The same as the first usage but only for function f: \Rset \rightarrow \Rset.

  • In the fourth usage:

The same as the second usage but only for function f: \Rset^2 \rightarrow \Rset.

Examples

>>> import openturns as ot
>>> from openturns.viewer import View
>>> f = ot.SymbolicFunction('x', 'sin(2*pi_*x)*exp(-x^2/2)')
>>> graph = f.draw(-1.2, 1.2, 100)
>>> View(graph).show()
getCallsNumber()

Accessor to the number of direct calls to the function.

Returns:
calls_numberint

Integer that counts the number of times the function has been called directly through the () operator.

getClassName()

Accessor to the object’s name.

Returns:
class_namestr

The object class name (object.__class__.__name__).

getDescription()

Accessor to the description of the inputs and outputs.

Returns:
descriptionDescription

Description of the inputs and the outputs.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getDescription())
[x1,x2,y0]
getEvaluation()

Accessor to the evaluation function.

Returns:
functionEvaluationImplementation

The evaluation function.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getEvaluation())
[x1,x2]->[2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6]
getEvaluationCallsNumber()

Accessor to the number of times the evaluation of the function has been called.

Returns:
evaluation_calls_numberint

Integer that counts the number of times the evaluation of the function has been called since its creation. This may include indirect calls via finite-difference gradient or Hessian.

getFormulas()

Formulas accessor.

Returns:
list_functionsDescription

List of the formulas.

getGradient()

Accessor to the gradient function.

Returns:
gradientGradientImplementation

The gradient function.

getGradientCallsNumber()

Accessor to the number of times the gradient of the function has been called.

Returns:
gradient_calls_numberint

Integer that counts the number of times the gradient of the Function has been called since its creation. Note that if the gradient is implemented by a finite difference method, the gradient calls number is equal to 0 and the different calls are counted in the evaluation calls number.

getHessian()

Accessor to the hessian function.

Returns:
hessianHessianImplementation

The hessian function.

getHessianCallsNumber()

Accessor to the number of times the hessian of the function has been called.

Returns:
hessian_calls_numberint

Integer that counts the number of times the hessian of the Function has been called since its creation. Note that if the hessian is implemented by a finite difference method, the hessian calls number is equal to 0 and the different calls are counted in the evaluation calls number.

getId()

Accessor to the object’s id.

Returns:
idint

Internal unique identifier.

getImplementation()

Accessor to the underlying implementation.

Returns:
implImplementation

A copy of the underlying implementation object.

getInputDescription()

Accessor to the description of the input vector.

Returns:
descriptionDescription

Description of the input vector.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getInputDescription())
[x1,x2]
getInputDimension()

Accessor to the dimension of the input vector.

Returns:
inputDimint

Dimension of the input vector d.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getInputDimension())
2
getMarginal(*args)

Accessor to marginal.

Parameters:
indicesint or list of ints

Set of indices for which the marginal is extracted.

Returns:
marginalFunction

Function corresponding to either f_i or (f_i)_{i \in indices}, with f:\Rset^n \rightarrow \Rset^p and f=(f_0 , \dots, f_{p-1}).

getName()

Accessor to the object’s name.

Returns:
namestr

The name of the object.

getOutputDescription()

Accessor to the description of the output vector.

Returns:
descriptionDescription

Description of the output vector.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getOutputDescription())
[y0]
getOutputDimension()

Accessor to the number of the outputs.

Returns:
number_outputsint

Dimension of the output vector d'.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getOutputDimension())
1
getParameter()

Accessor to the parameter values.

Returns:
parameterPoint

The parameter values.

getParameterDescription()

Accessor to the parameter description.

Returns:
parameterDescription

The parameter description.

getParameterDimension()

Accessor to the dimension of the parameter.

Returns:
parameterDimensionint

Dimension of the parameter.

gradient(inP)

Return the Jacobian transposed matrix of the function at a point.

Parameters:
pointsequence of float

Point where the Jacobian transposed matrix is calculated.

Returns:
gradientMatrix

The Jacobian transposed matrix of the function at point.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2'])
>>> print(f.gradient([3.14, 4]))
[[ 13.5345   1       ]
 [  4.00001  1       ]]
hessian(inP)

Return the hessian of the function at a point.

Parameters:
pointsequence of float

Point where the hessian of the function is calculated.

Returns:
hessianSymmetricTensor

Hessian of the function at point.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2'])
>>> print(f.hessian([3.14, 4]))
sheet #0
[[ 20          -0.00637061 ]
 [ -0.00637061  0          ]]
sheet #1
[[  0           0          ]
 [  0           0          ]]
isLinear()

Accessor to the linearity of the function.

Returns:
linearbool

True if the function is linear, False otherwise.

isLinearlyDependent(index)

Accessor to the linearity of the function with regard to a specific variable.

Parameters:
indexint

The index of the variable with regard to which linearity is evaluated.

Returns:
linearbool

True if the function is linearly dependent on the specified variable, False otherwise.

parameterGradient(inP)

Accessor to the gradient against the parameter.

Returns:
gradientMatrix

The gradient.

setDescription(description)

Accessor to the description of the inputs and outputs.

Parameters:
descriptionsequence of str

Description of the inputs and the outputs.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> print(f.getDescription())
[x1,x2,y0]
>>> f.setDescription(['a','b','y'])
>>> print(f.getDescription())
[a,b,y]
setEvaluation(evaluation)

Accessor to the evaluation function.

Parameters:
functionEvaluationImplementation

The evaluation function.

setGradient(gradient)

Accessor to the gradient function.

Parameters:
gradient_functionGradientImplementation

The gradient function.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                          ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> f.setGradient(ot.CenteredFiniteDifferenceGradient(
...  ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceGradient-DefaultEpsilon'),
...  f.getEvaluation()))
setHessian(hessian)

Accessor to the hessian function.

Parameters:
hessian_functionHessianImplementation

The hessian function.

Examples

>>> import openturns as ot
>>> f = ot.SymbolicFunction(['x1', 'x2'],
...                         ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6'])
>>> f.setHessian(ot.CenteredFiniteDifferenceHessian(
...  ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceHessian-DefaultEpsilon'),
...  f.getEvaluation()))
setInputDescription(inputDescription)

Accessor to the description of the input vector.

Parameters:
descriptionDescription

Description of the input vector.

setName(name)

Accessor to the object’s name.

Parameters:
namestr

The name of the object.

setOutputDescription(inputDescription)

Accessor to the description of the output vector.

Parameters:
descriptionDescription

Description of the output vector.

setParameter(parameter)

Accessor to the parameter values.

Parameters:
parametersequence of float

The parameter values.

setParameterDescription(description)

Accessor to the parameter description.

Parameters:
parameterDescription

The parameter description.

setStopCallback(callBack, state=None)

Set up a stop callback.

Can be used to programmatically stop an evaluation.

Parameters:
callbackcallable

Returns a bool deciding whether to stop or continue.

Examples using the class

Estimate moments from sample

Estimate moments from sample

Sample manipulation

Sample manipulation

Estimate Wilks and empirical quantile

Estimate Wilks and empirical quantile

Compare unconditional and conditional histograms

Compare unconditional and conditional histograms

Model a singular multivariate distribution

Model a singular multivariate distribution

Estimate a GEV on the Port Pirie sea-levels data

Estimate a GEV on the Port Pirie sea-levels data

Estimate a GPD on the daily rainfall data

Estimate a GPD on the daily rainfall data

Estimate a GEV on race times data

Estimate a GEV on race times data

Estimate a GEV on the Fremantle sea-levels data

Estimate a GEV on the Fremantle sea-levels data

Create a conditional distribution

Create a conditional distribution

Create your own distribution given its quantile function

Create your own distribution given its quantile function

Create a Bayes distribution

Create a Bayes distribution

Transform a distribution

Transform a distribution

Overview of univariate distribution management

Overview of univariate distribution management

Composite random vector

Composite random vector

Create a functional basis process

Create a functional basis process

Add a trend to a process

Add a trend to a process

Use the Box-Cox transformation

Use the Box-Cox transformation

Create a process from random vectors and processes

Create a process from random vectors and processes

Trend computation

Trend computation

Compare covariance models

Compare covariance models

Create a mesh

Create a mesh

Export a metamodel

Export a metamodel

Create a linear least squares model

Create a linear least squares model

Create a general linear model metamodel

Create a general linear model metamodel

Taylor approximations

Taylor approximations

Create a linear model

Create a linear model

Perform stepwise regression

Perform stepwise regression

Over-fitting and model selection

Over-fitting and model selection

Fit a distribution from an input sample

Fit a distribution from an input sample

Polynomial chaos exploitation

Polynomial chaos exploitation

Create a full or sparse polynomial chaos expansion

Create a full or sparse polynomial chaos expansion

Create a polynomial chaos metamodel by integration on the cantilever beam

Create a polynomial chaos metamodel by integration on the cantilever beam

Advanced polynomial chaos construction

Advanced polynomial chaos construction

Create a polynomial chaos metamodel from a data set

Create a polynomial chaos metamodel from a data set

Compute Sobol’ indices confidence intervals

Compute Sobol' indices confidence intervals

Kriging : multiple input dimensions

Kriging : multiple input dimensions

Kriging : draw the likelihood

Kriging : draw the likelihood

Kriging : cantilever beam model

Kriging : cantilever beam model

Kriging: choose an arbitrary trend

Kriging: choose an arbitrary trend

Kriging the cantilever beam model using HMAT

Kriging the cantilever beam model using HMAT

Kriging : generate trajectories from a metamodel

Kriging : generate trajectories from a metamodel

Kriging: choose a polynomial trend on the beam model

Kriging: choose a polynomial trend on the beam model

Kriging : quick-start

Kriging : quick-start

Sequentially adding new points to a kriging

Sequentially adding new points to a kriging

Kriging: configure the optimization solver

Kriging: configure the optimization solver

Kriging: choose a polynomial trend

Kriging: choose a polynomial trend

Advanced Kriging

Advanced Kriging

Metamodel of a field function

Metamodel of a field function

Analyse the central tendency of a cantilever beam

Analyse the central tendency of a cantilever beam

Estimate moments from Taylor expansions

Estimate moments from Taylor expansions

Simulate an Event

Simulate an Event

Estimate a probability with Monte Carlo

Estimate a probability with Monte Carlo

Use a randomized QMC algorithm

Use a randomized QMC algorithm

Use the Adaptive Directional Stratification Algorithm

Use the Adaptive Directional Stratification Algorithm

Use the post-analytical importance sampling algorithm

Use the post-analytical importance sampling algorithm

Use the Directional Sampling Algorithm

Use the Directional Sampling Algorithm

Create a threshold event

Create a threshold event

Specify a simulation algorithm

Specify a simulation algorithm

Use the Importance Sampling algorithm

Use the Importance Sampling algorithm

Estimate a probability with Monte-Carlo on axial stressed beam: a quick start guide to reliability

Estimate a probability with Monte-Carlo on axial stressed beam: a quick start guide to reliability

Estimate a buckling probability

Estimate a buckling probability

Exploitation of simulation algorithm results

Exploitation of simulation algorithm results

Use the FORM algorithm in case of several design points

Use the FORM algorithm in case of several design points

Subset Sampling

Subset Sampling

Use the FORM - SORM algorithms

Use the FORM - SORM algorithms

Create a domain event

Create a domain event

Test the design point with the Strong Maximum Test

Test the design point with the Strong Maximum Test

Time variant system reliability problem

Time variant system reliability problem

Create unions or intersections of events

Create unions or intersections of events

Axial stressed beam : comparing different methods to estimate a probability

Axial stressed beam : comparing different methods to estimate a probability

An illustrated example of a FORM probability estimate

An illustrated example of a FORM probability estimate

Cross Entropy Importance Sampling

Cross Entropy Importance Sampling

Using the FORM - SORM algorithms on a nonlinear function

Using the FORM - SORM algorithms on a nonlinear function

Estimate Sobol indices on a field to point function

Estimate Sobol indices on a field to point function

Parallel coordinates graph as sensitivity tool

Parallel coordinates graph as sensitivity tool

Estimate Sobol’ indices for a function with multivariate output

Estimate Sobol' indices for a function with multivariate output

Sobol’ sensitivity indices from chaos

Sobol' sensitivity indices from chaos

Use the ANCOVA indices

Use the ANCOVA indices

The HSIC sensitivity indices: the Ishigami model

The HSIC sensitivity indices: the Ishigami model

Compute the L2 error between two functions

Compute the L2 error between two functions

Use the Smolyak quadrature

Use the Smolyak quadrature

Create a composed function

Create a composed function

Create an aggregated function

Create an aggregated function

Create a linear combination of functions

Create a linear combination of functions

Create a symbolic function

Create a symbolic function

Increase the output dimension of a function

Increase the output dimension of a function

Increase the input dimension of a function

Increase the input dimension of a function

Defining Python and symbolic functions: a quick start introduction to functions

Defining Python and symbolic functions: a quick start introduction to functions

Create multivariate functions

Create multivariate functions

Create a multivariate basis of functions from scalar multivariable functions

Create a multivariate basis of functions from scalar multivariable functions

Value function

Value function

Vertex value function

Vertex value function

Define a connection function with a field output

Define a connection function with a field output

Logistic growth model

Logistic growth model

Function manipulation

Function manipulation

Calibration without observed inputs

Calibration without observed inputs

Calibration of the deflection of a tube

Calibration of the deflection of a tube

Sampling from an unnormalized probability density

Sampling from an unnormalized probability density

Posterior sampling using a PythonDistribution

Posterior sampling using a PythonDistribution

Bayesian calibration of a computer code

Bayesian calibration of a computer code

Customize your Metropolis-Hastings algorithm

Customize your Metropolis-Hastings algorithm

Estimate an integral

Estimate an integral

Save/load a study

Save/load a study

Iterated Functions System

Iterated Functions System

Compute confidence intervals of a regression model from data

Compute confidence intervals of a regression model from data

Compute confidence intervals of a univariate noisy function

Compute confidence intervals of a univariate noisy function

Mix/max search and sensitivity from design

Mix/max search and sensitivity from design

Optimization with constraints

Optimization with constraints

Optimization using NLopt

Optimization using NLopt

Mix/max search using optimization

Mix/max search using optimization

Control algorithm termination

Control algorithm termination

Optimization using bonmin

Optimization using bonmin

Multi-objective optimization using Pagmo

Multi-objective optimization using Pagmo

Quick start guide to optimization

Quick start guide to optimization

Optimization using dlib

Optimization using dlib

EfficientGlobalOptimization examples

EfficientGlobalOptimization examples

A quick start guide to contours

A quick start guide to contours

A quick start guide to graphs

A quick start guide to graphs