tensap.approximation.bases package¶

Submodules¶

tensap.approximation.bases.full_tensor_product_functional_basis module¶

Module full_tensor_product_functional_basis.

class tensap.approximation.bases.full_tensor_product_functional_basis.FullTensorProductFunctionalBasis(bases)¶

Bases: tensap.approximation.bases.functional_basis.FunctionalBasis

Class FullTensorProductFunctionalBasis.

Attributes

baseslist or tensap.FunctionalBases: The bases associated with the object.

Methods

`adaptation_path`()	Return the adaptation path of the functional basis.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`(dims, *args)	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x)	Return the evaluation of the basis functions at the points x.
`expectation`()	Return the expectation of the basis functions.
`gram_matrix`([dims])	Return the gram matrix of each basis of self, or of a selection of them if dims is provided.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`keep_bases`(ind)	Keep only the bases of self of index ind.
`kron`()	Compute the Kronecker product of two bases.
`length`()	Return the number of bases in self.bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`(*args)	Return the mean of the basis functions.
`ndim`()	Return the dimension n for f defined in R^n.
`orthonormalize`()	Orthonormalize the basis.
`plot`([indices, n])	Plot the functions of the basis.
`projection`(fun, I)	Compute the projection of the function fun on the basis functions of self.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`remove_bases`(ind)	Remove bases of self of index ind.
`storage`()	Return the storage requirement of the FunctionalBasis.
`tensor_product_interpolation`(fun[, grid])	Return the interpolation of function fun on a product grid.
`transpose`(perm)	Return self with the basis permutation perm.

christoffel
keep_mapping
optimal_sampling_measure
remove_mapping

cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

conditional_expectation(dims, *args)¶

Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)). The expectation with respect to other variables (in the complementary set of dims) is taken with respect to the probability measure given by tensap.RandomVector XdimsC if provided, or with respect to the probability measure associated with the corresponding bases of the function.

Parameters

dimsnumpy.ndarray: The dimensions in which the expectation is computed.
XdimsC: tensap.RandomVector, optional: The random vector used for the computation of the conditional expectation. The default is None, indicating to use the
probability measure associated with the basis.

Returns

ftensap.FunctionalBasisArray: The conditional expectation of the function.

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

eval(x)¶

Return the evaluation of the basis functions at the points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluted.

Returns

numpy.ndarray: The evaluations of the basis functions at the points x.

gram_matrix(dims=None)¶

Return the gram matrix of each basis of self, or of a selection of them if dims is provided.

Parameters

dimslist or numpy.ndarray, optional: The dimensions of the bases for which the gram matrix is computed. The default is None, indicating all the bases.

Returns

list: The gram matrix of the selected bases.

interpolation_points(*args)¶

Return the interpolation points for the basis.

tensap.approximation.bases.functional_bases module¶

Module functional_bases.

class tensap.approximation.bases.functional_bases.FunctionalBases(bases=None)¶

Bases: object

Class FunctionalBases.

Attributes

baseslist or numpy.ndarray, or tensap.FunctionalBases, optional: List or numpy.ndarray containing objects of type FunctionalBasis. The default is None.
measuretensap.Measure: The measure associated with the functional bases. By default, a tensap.ProductMeasure constituted of the attribute measure of each basis of bases.

Methods

`adaptation_path`()	Compute the adaptationPath for each basis in self.
`cardinals`([ind])	Return the number of functions in each basis of self, or in basis ind if provided.
`derivative`(n)	Compute the n-derivative of the basis functions of self.
`domain`()	Return the domain of each basis of self.
`duplicate`(basis, dim)	Create a FunctionalBases with bases created with a duplication of basis d times.
`eval`(x[, dims, nargout])	Computes evaluations of the basis functions of self at points x in dimensions dims if provided, in all the dimensions if not.
`eval_derivative`(n, x[, dims, nargout])	Compute evaluations of the n-derivative of the basis functions of
`get_random_vector`()	Returns the random vector associated with self.
`gram_matrix`([dims])	Return the gram matrix of each basis of self, or of a selection of them if dims is provided.
`interpolation_points`([x])	Return the interpolation points for the bases.
`keep_bases`(ind)	Keep only the bases of self of index ind.
`kron`(g)	Return the bases obtained by the Kronecker product of two bases.
`length`()	Return the number of bases in self.
`magic_points`(x[, J])	Provide the magic points associated with the functional bases selected in a given set of points x.
`mean`([dims, measure])	Compute the mean of self in the dimensions in dims according to the RandomVector measure if provided, or to the standard RandomVector associated with each basis if not.
`ndim`()	Return the dimension of each basis of self.
`one`([dims])	Return the coefficients associated with the FunctionalBases so that it returns one.
`orthonormalize`()	Orthonormalize the basis functions of self.
`random`(args, *kwargs)	Compute random evaluations of the bases in self.
`random_dims`(dims[, n, measure, nargout])	Evaluate the bases in dimensions dims of the bases of self using n points drawn randomly according to measure if provided, or to self.measure.marginal(dims) otherwise.
`remove_bases`(ind)	Remove bases of self of index ind.
`storage`()	Return the storage requirement of the FunctionalBases.
`tensor_product_interpolation`(*args)	Interpolate a function on a product grid.
`transpose`(perm)	Return self with the basis permutation perm.

adaptation_path()¶

Compute the adaptationPath for each basis in self.

See also tensap.FunctionalBasis.adaptation_path.

Returns

list: List of adaptation paths for each basis.

cardinals(ind=None)¶

Return the number of functions in each basis of self, or in basis ind if provided.

Parameters

indind, optional: The index of the selected basis. The default is None.

Returns

numpy.ndarray: The number of functions in each basis of self, or in basis ind if provided.

derivative(n)¶

Compute the n-derivative of the basis functions of self.

Parameters

nlist or numpy.ndarray: The order of derivation in each dimension.

Returns

outFunctionalBases: A Functionalbases with the n-derivative of the basis functions of self.

domain()¶

Return the domain of each basis of self.

Returns

list: The domain of each basis of self.

static duplicate(basis, dim)¶

Create a FunctionalBases with bases created with a duplication of basis d times.

Parameters

basistensap.FunctionalBasis: The basis to be duplicated.
dimint: The number of times basis is duplicated.

Returns

tensap.FunctionalBases: The obtained FunctionalBases.

eval(x, dims=None, nargout=1)¶

Computes evaluations of the basis functions of self at points x in dimensions dims if provided, in all the dimensions if not.

Parameters

xnumpy.ndarray: The input points.
dimslist or numpy.ndarray, optional: The dimensions of the bases to be evaluated. The default is None, indicating all the dimensions.
nargoutint, optional: Indicates the number of expected outputs. The default is 1, indicating to return only the evaluations of the basis functions.

Returns

outlist: Evaluations of the basis functions of self.
xnumpy.ndarray: The input points, grouped by basis.

eval_derivative(n, x, dims=None, nargout=1)¶

Compute evaluations of the n-derivative of the basis functions of self at points x in each dimension in dims if provided, in all the dimensions otherwise.

Parameters

nlist or numpy.ndarray: The order of derivation in each dimension (in dims if provided).
xnumpy.ndarray: The input points.
dimslist or numpy.ndarray, optional: The dimensions of the bases for which the n-derivative is to be computed. The default is None, indicating all the dimensions.
nargoutint, optional: Indicates the number of expected outputs. The default is 1, indicating to return only the evaluations of the n-derivative of the basis functions.

Returns

outlist: Evaluations of the n-derivative of the basis functions of self.
xnumpy.ndarray: The input points, grouped by basis.

get_random_vector()¶

Returns the random vector associated with self.

Returns

measuretensap.RandomVector: The random vector associated with self.

gram_matrix(dims=None)¶

Return the gram matrix of each basis of self, or of a selection of them if dims is provided.

Parameters

dimslist or numpy.ndarray, optional: The dimensions of the bases for which the gram matrix is computed. The default is None, indicating all the bases.

Returns

list: The gram matrix of the selected bases.

interpolation_points(x=None)¶

Return the interpolation points for the bases.

Parameters

xtensap.FullTensorGrid or list or numpy.ndarray: The set of points in which the interpolation points are selected.

Returns

pointslist: The interpolation points.

keep_bases(ind)¶

Keep only the bases of self of index ind.

Parameters

indint or list or numpy.ndarray: The indices of the bases to keep.

Returns

outtensap.FunctionalBases: The FunctionalBases with kept bases.

kron(g)¶

Return the bases obtained by the Kronecker product of two bases.

Parameters

gFunctionalBases: The second bases of the product.

Returns

outFunctionalBases: The bases obtained by the Kronecker product of two bases.

length()¶

Return the number of bases in self.

Returns

int: The number of bases in self.

magic_points(x, J=None)¶

Provide the magic points associated with the functional bases selected in a given set of points x.

Parameters

xtensap.FullTensorGrid or list or numpy.ndarray: The set of points in which the magic points are selected.
Jnumpy.ndarray, optional: The default is None. If not none, selected the magic indices with tensap.magic_indices(F[:, J], self.cardinal(), ‘left’)[0]

Returns

pointslist: The magic points associated with each basis of self.
indlist: The locations of the magic points in x for each basis of self.

mean(dims=None, measure=None)¶

Compute the mean of self in the dimensions in dims according to the RandomVector measure if provided, or to the standard RandomVector associated with each basis if not.

Parameters

dimslist or numpy.ndarray, optional: The dimensions of the bases for which the mean is to be computed. The default is None, indicating all the bases.
measuretensap.RandomVector or tensap.RandomVariable, optional: The probability measure according to which the mean is computed. The default is None, indicating to use the self.measure.

Returns

outlist: The mean of each basis functions.

ndim()¶

Return the dimension of each basis of self.

Returns

list: The dimension of each basis of self.

one(dims=None)¶

Return the coefficients associated with the FunctionalBases so that it returns one.

Parameters

dimslist or numpy.ndarray, optional: The dimensions of the bases that need to return one. The default is None, indicating all the bases.

Returns

list: The list of coefficients so that the selected bases return one.

orthonormalize()¶

Orthonormalize the basis functions of self.

Returns

tensap.FunctionalBases: The FunctionalBases with orthonormalized basis functions.

random(*args, **kwargs)¶

Compute random evaluations of the bases in self.

Parameters

*argsmisc: Additional parameters for the random generation. See random_dims.

Returns

list or numpy.ndarray: Random evaluations of the basis functions of self.
numpy.ndarray: The input points, grouped by basis.

random_dims(dims, n=1, measure=None, nargout=1)¶

Evaluate the bases in dimensions dims of the bases of self using n points drawn randomly according to measure if provided, or to self.measure.marginal(dims) otherwise.

Parameters

dimslist or numpy.ndarray: The dimensions of the bases to be evaluated.
nint, optional: The number of random evaluations. The default is 1.
measuretensap.ProbabilityMeasure, optional: The probability measure used for the generation of the input points. The default is None, indicating to use self.measure.marginal(dims).

Returns

bases_evallist or numpy.ndarray: Random evaluations of the basis functions of self.
xnumpy.ndarray: The input points, grouped by basis.

remove_bases(ind)¶

Remove bases of self of index ind.

Parameters

indint or list or numpy.ndarray: The indices of the bases to remove.

Returns

outtensap.FunctionalBases: The FunctionalBases with removed bases.

storage()¶

Return the storage requirement of the FunctionalBases.

Returns

int: The storage requirement of the FunctionalBases.

tensor_product_interpolation(*args)¶

Interpolate a function on a product grid.

See also tensap.FullTensorProductFunctionalBasis.tensorProductInterpolation.

Parameters

*argstuple: Parameters of the method tensorProductInterpolation of tensap.FullTensorProductFunctionalBasis.

Returns

tensap.FunctionalTensor: The interpolation of the function on a product grid.

transpose(perm)¶

Return self with the basis permutation perm.

Parameters

permlist or numpy.ndarray: The permutation of the bases.

Returns

outtensap.FunctionalBases: The FunctionalBases with permuted bases.

tensap.approximation.bases.functional_basis module¶

Module functional_basis.

class tensap.approximation.bases.functional_basis.FunctionalBasis¶

Bases: object

Class FunctionalBasis.

Attributes

measuretensap.Measure: The measure associated with the FunctionalBasis.
is_orthonormalbool: Indicates if the basis is orthonormal with respect to the associated measure.

Methods

`adaptation_path`()	Return the adaptation path of the functional basis.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`()	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x)	Return the evaluation of the basis functions at the points x.
`expectation`()	Return the expectation of the basis functions.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`kron`()	Compute the Kronecker product of two bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`()	Return the mean of the basis functions.
`ndim`()	Return the dimension n for f defined in R^n.
`orthonormalize`()	Orthonormalize the basis.
`plot`([indices, n])	Plot the functions of the basis.
`projection`(fun, G)	Compute the projection of the function fun onto the functional basis using the integration rule G.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`storage`()	Return the storage requirement of the FunctionalBasis.

christoffel
optimal_sampling_measure

adaptation_path()¶

Return the adaptation path of the functional basis.

Returns

numpy.ndarray: Boolean array, where n is the dimension of the functional basis, and m is the number of elements in the adaptation path, column P[:,i] corresponds to a sparsity pattern.

abstract cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

christoffel(x)¶

static conditional_expectation()¶

Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)). The expectation with respect to other variables (in the complementary set of dims) is taken with respect to the probability measure given by tensap.RandomVector XdimsC if provided, or with respect to the probability measure associated with the corresponding bases of the function.

Parameters

dimsnumpy.ndarray: The dimensions in which the expectation is computed.
XdimsC: tensap.RandomVector, optional: The random vector used for the computation of the conditional expectation. The default is None, indicating to use the
probability measure associated with the basis.

Returns

ftensap.FunctionalBasisArray: The conditional expectation of the function.

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

abstract eval(x)¶

Return the evaluation of the basis functions at the points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluted.

Returns

numpy.ndarray: The evaluations of the basis functions at the points x.

expectation()¶

Return the expectation of the basis functions. Equivalent to self.mean().

Returns

numpy.ndarray: The expectation of the basis functions.

interpolate(y, x=None)¶

Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.

Parameters

yfunction or list or numpy.ndarray: The function to interpolate, or values of it.
xlist or numpy.ndarray, optional: The interpolation points. The default is None, indicating to deduce them from the basis.

Returns

ftensap.FunctionalBasisArray: The computed interpolation.

interpolation_points(*args)¶

Return the interpolation points for the basis.

tensap.approximation.bases.functional_basis_array module¶

Module functional_basis_array.

class tensap.approximation.bases.functional_basis_array.FunctionalBasisArray(data=None, basis=None, shape=None)¶

Bases: tensap.functions.function.Function

Class FunctionalBasisArray.

Attributes

datanumpy.ndarray, optional: The coefficents of the function on the basis. The default is None.
basistensap.FunctionalBasis, optional: The basis. The default is None.
shapelist or numpy.ndarray, optional: Array such that the function is with values in R^(shape[0] x shape[1] x …). The default is 1.

Methods

`__call__`(x[, return_f])	Call self as a function.
`conditional_expectation`(dims, *args)	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`derivative`(n)	Compute the n-derivative of the function.
`dot`(g[, dim])	Compute the dot product between the arrays self.data and g.data treated as collections of vectors.
`dot_product_expectation`(g[, dims, measure])	Compute the expectation of self(X)g(X), where X is the probability measure associated with the underlying basis, or measure if provided.
`eval`(x, *args)	Evaluate the function at the points x.
`eval_derivative`(n, x)	Compute the n-derivative of the function at points x in R^d, with n a multi-index of size d.
`eval_on_tensor_grid`(x)	Evaluate the Function on a grid x.
`eval_with_bases_evals`(H)	Compute the evaluations of the function using the evaluations of the basis H.
`expectation`([measure])	Compute the expectation of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.
`fplot`([support, n_points])	Plot the function on a support using a given number of points.
`get_coefficients`()	Return the coefficients of the object.
`get_random_vector`()	Return the random vector associated with the basis functions of the object.
`is_random`()	Determine if the object is random.
`matdiv`(v)	Compute the matrix multiplication of self.data with the inverse of v.
`matmul`(v)	Compute the matrix multiplication of self.data with v.
`mean`([measure])	Compute the expectation of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.
`norm`([p])	Compute the p-norm of the array self.data.
`norm_expectation`([measure])	Compute the L^2 norm of self(measure).
`partial_evaluation`(not_alpha, x_not_alpha)	Return the partial evaluation of a function f(x) = f(x_alpha,x_not_alpha), a function f_alpha(.) = f(., x_not_alpha) for fixed values x_not_alpha of the variables with indices not_alpha.
`projection`(basis[, indices])	Projection of the object on a functional basis using multi-indices indices if provided, or the multi-indices associated with the functional basis if not.
`random`([n, measure])	Compute evaluations of the function at an array of points of size n, drawn randomly according to the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.
`sparse_storage`()	The storage complexity of the object, taking into account the sparsity.
`std`(*args)	Compute the standard deviation of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.
`storage`()	The storage complexity of the object.
`store_eval`(x)	Evaluate the function, reuising previous evaluations if possible, and storing the new evaluations in self.
`sub_functional_basis`()	Converts the FunctionalBasisArray into a tensap.SubFunctionalbasis.
`surf`([n])	Surface plot of the bivariate function.
`test_error`(g[, n, measure])	Compute the test error associated with the function, using a function g or some of its evaluations as a reference.
`variance`([measure])	Compute the variance of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.
`variance_conditional_expectation`(alpha)	Compute the variance of the conditional expectation of the function in dimensions in alpha.

conditional_expectation(dims, *args)¶

Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)). The expectation with respect to other variables (in the complementary set of dims) is taken with respect to the probability measure given by tensap.RandomVector XdimsC if provided, or with respect to the probability measure associated with the corresponding bases of the function.

Parameters

dimsnumpy.ndarray: The dimensions in which the expectation is computed.
*argstuple: Additional parameters. See also the method conditional_expectation of the underlying basis.

Returns

ftensap.FunctionalBasisArray: The conditional expectation of the function.

derivative(n)¶

Compute the n-derivative of the function.

Parameters

nint: The derivation order.

Returns

dftensap.FunctionalBasisArray(): The n-derivative of the function.

dot(g, dim=None)¶

Compute the dot product between the arrays self.data and g.data treated as collections of vectors. The function calculates the dot product of corresponding vectors along the first array dimension whose size does not equal 1.

Parameters

gtensap.FunctionalBasisArray: The second object of the dot product.
dimint or list or numpy.ndarray, optional: The dimension along which the dot product is computed. The default is None, indicating all the dimensions.

Returns

float or numpy.ndarray: The result of the dot product.

dot_product_expectation(g, dims=None, measure=None)¶

Compute the expectation of self(X)g(X), where X is the probability measure associated with the underlying basis, or measure if provided.

For vector-valued functions of X, dims specifies the dimensions of self and g corresponding to the RandomVector measure.

Parameters

gtensap.FunctionalBasisArray: The second function of the product.
dimslist or numpy.ndarray, optional: The dimensions of self and g corresponding to the RandomVector measure. The default is None.
measuretensap.ProbabilityMeasure, optional: The probability measure used for the computation of the expectation. The default is None, indicating to use the probability measure associated with the underlying basis.

Returns

float or numpy.ndarray: The result of the dot product.

Raises

NotImplementedError: If the method is not implemented.

eval(x, *args)¶

Evaluate the function at the points x.

Parameters

xlist or numpy.ndarray: The points at which the function is to be evaluated.

Returns

numpy.ndarray: The evaluations of the function at the points x.

eval_derivative(n, x)¶

Compute the n-derivative of the function at points x in R^d, with n a multi-index of size d.

Parameters

nint or list or numpy.ndarray: The derivation order in all the dimensions, or the derivation orders for each dimension.
xnumpy.ndarray: The points used for the evaluation of the derivative.

Returns

numpy.ndarray: The evaluation of the n-derivative of the function at the points x.

Raises

NotImplementedError: If the method is not implemented for the basis.

eval_with_bases_evals(H)¶

Compute the evaluations of the function using the evaluations of the basis H.

Parameters

Hnumpy.ndarray: The evaluations of the basis.

Returns

numpy.ndarray: The evaluations of the function.

expectation(measure=None)¶

Compute the expectation of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.

Parameters

measuretensap.ProbabilityMeasure, optional: The probability measure used for the computation of the expectation. The default is None, indicating to use the standard tensap.ProbabilityMeasure associated with each polynomial.

Returns

numpy.ndarray: The expectation of the function.

get_coefficients()¶

Return the coefficients of the object.

Returns

numpy.ndarray: The coefficients of the object.

get_random_vector()¶

Return the random vector associated with the basis functions of the object.

Returns

tensap.RandomVector: The random vector associated with the basis functions of the object.

static is_random()¶

Determine if the object is random.

Returns

bool: Boolean equal to True if the object is random.

matdiv(v)¶

Compute the matrix multiplication of self.data with the inverse of v.

Parameters

vnumpy.ndarray: The array used in the matrix multiplication.

Returns

tensap.FunctionalBasisArray: The result of the matrix multiplication.

matmul(v)¶

Compute the matrix multiplication of self.data with v.

Parameters

vnumpy.ndarray: The array used in the matrix multiplication.

Returns

tensap.FunctionalBasisArray: The result of the matrix multiplication.

mean(measure=None)¶

Compute the expectation of the function, according to the measure associated with the tensap.ProbabilityMeasure measure if provided, or to the standard tensap.ProbabilityMeasure associated with each polynomial if not.

Parameters

measuretensap.ProbabilityMeasure, optional: The probability measure used for the computation of the expectation. The default is None, indicating to use the standard tensap.ProbabilityMeasure associated with each polynomial.

Returns

numpy.ndarray: The expectation of the function.

norm(p='fro')¶

Compute the p-norm of the array self.data.

tensap.approximation.bases.polynomial_functional_basis module¶

Module polynomial_functional_basis.

class tensap.approximation.bases.polynomial_functional_basis.PolynomialFunctionalBasis(basis, indices)¶

Bases: tensap.approximation.bases.functional_basis.FunctionalBasis

Class PolynomialFunctionalBasis.

Attributes

basistensap.UnivariatePolynomials: The polynomials associated with the basis.
indiceslist or numpy.ndarray: The indices of the selected polynomials.
measuretensap.Measure: The measure associated with basis.
is_orthonormalbool: Boolean equal to true, indicating that the basis is orthonormal with respect to measure.

Methods

`adaptation_path`()	Return the adaptation path of the functional basis.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`()	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`derivative`(k[, measure])	Compute the k-th order derivative of the functions of the basis projected on itself.
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x)	Evaluate the polynomials of self.basis of degrees in self.indices at points x.
`eval_derivative`(k, x)	Evaluate the k-th order derivative of the functions of the basis at the points x.
`expectation`()	Return the expectation of the basis functions.
`gradient`([measure])	Compute the first order derivative of the functions of the basis projected on itself.
`gram_matrix`([measure])	Compute the Gram matrix of the basis.
`hessian`(measure)	Compute the second order derivative of the functions of the basis projected on itself.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`kron`(q)	Compute the Kronecker product of two bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`([measure])	Return the expectation of the basis functions, accorging to measure if provided, and to self.measure ortherwise.
`ndim`()	Return the dimension n for f defined in R^n.
`one`()	Return the coefficients associated with the basis so that it returns one.
`orthonormalize`()	Orthonormalize the basis.
`plot`([indices, n])	Plot the functions of the basis.
`projection`(fun, G)	Compute the projection of the function fun onto the functional basis using the integration rule G.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`random_variable`()	Return the tensap.RandomVariable associated with self if it exists.
`storage`()	Return the storage requirement of the FunctionalBasis.

christoffel
optimal_sampling_measure

cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

christoffel()¶

derivative(k, measure=None)¶

Compute the k-th order derivative of the functions of the basis projected on itself.

Parameters

kint: The order of the derivative.
measuretensap.Measure, optional: The measure used fot the projection. The default is None, indicating to use self.measure if it is a tensap.RandomVariable.

Returns

tensap.SubFunctionalBasis: The k-th order derivative of the functions of the basis projected on itself.

Raises

ValueError: If no Measure is provided or can be extracted from self.

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

eval(x)¶

Evaluate the polynomials of self.basis of degrees in self.indices at points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluated.

Returns

numpy.ndarray: The evaluations of the polynomials of self.basis of degrees in self.indices at points x.

eval_derivative(k, x)¶

Evaluate the k-th order derivative of the functions of the basis at the points x.

Parameters

kint: The order of the derivative.
xlist or numpy.ndarray: The points at which the k-th derivative of the basis functions are to be evaluated.

Returns

numpy.ndarray: The evaluations of the k-th derivative of the polynomials of self.basis of degrees in self.indices at points x.

gradient(measure=None)¶

Compute the first order derivative of the functions of the basis projected on itself.

Parameters

measuretensap.Measure, optional: The measure used fot the projection. The default is None, indicating to use self.measure if it is a tensap.RandomVariable.

Returns

tensap.SubFunctionalBasis

The first order derivative of the functions of the basis projected: on itself.

gram_matrix(measure=None)¶

Compute the Gram matrix of the basis. The Gram matrix is the matrix of the dot products between each possible couple of basis functions. The dot product in the dimension i is computed according to measure if provided, or according to self.measure otherwise.

Parameters

measuretensap.Measure, optional: The measure according to which the dot product is computed. The default is None, indicating to use self.measure.

Returns

numpy.ndarray: The Gram matrix of the basis.

Raises

NotImplementedError: If the attribute basis of self does not have a method moment.

hessian(measure)¶

Compute the second order derivative of the functions of the basis projected on itself.

Parameters

measuretensap.Measure, optional: The measure used fot the projection. The default is None, indicating to use self.measure if it is a tensap.RandomVariable.

Returns

tensap.SubFunctionalBasis

The second order derivative of the functions of the basis projected: on itself.

kron(q)¶

Compute the Kronecker product of two bases. For two functional bases f_i, i = 1, …, n and g_j, j = 1, …, m, return a functional basis h_k, k = 1, …, nm.

Returns

tensap.FunctionalBasis: The obtained basis.

mean(measure=None)¶

Return the expectation of the basis functions, accorging to measure if provided, and to self.measure ortherwise.

Parameters

measureNone or tensap.RandomVariable, optional: The measure according to which the mean is computed. The default is None, indicating to use self.measure.

Returns

numpy.ndarray: The mean of the basis functions.

ndim()¶

Return the dimension n for f defined in R^n.

Returns

int: The dimension n for f defined in R^n.

one()¶

Return the coefficients associated with the basis so that it returns one.

Returns

list: The list of coefficients so that the basis returns one.

optimal_sampling_measure()¶

random_variable()¶

Return the tensap.RandomVariable associated with self if it exists.

Returns

outtensap.RandomVariable or []: The tensap.RandomVariable associated with self if it exists, and [] otherwise.

tensap.approximation.bases.sparse_tensor_product_functional_basis module¶

Module sparse_tensor_product_functional_basis.

class tensap.approximation.bases.sparse_tensor_product_functional_basis.SparseTensorProductFunctionalBasis(bases, indices)¶

Bases: tensap.approximation.bases.functional_basis.FunctionalBasis

Class SparseTensorProductFunctionalBasis.

Attributes

baseslist or tensap.FunctionalBases: The bases associated with the object.
indicestensap.MultiIndices: The indices of the basis functions (the indices start at 0).

Methods

`adaptation_path`([p])	Create an adaptation path associated with increasing p-norm of multi-indices.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`(dims, *args)	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x)	Return the evaluation of the basis functions at the points x.
`expectation`()	Return the expectation of the basis functions.
`get_random_vector`()	Return the random vector associated with the basis functions of self.
`gram_matrix`()	Return the gram matrix of the basis.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`keep_bases`(ind)	Keep only the bases of self of index ind.
`kron`()	Compute the Kronecker product of two bases.
`length`()	Return the number of bases in self.bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`(*args)	Return the mean of the basis functions.
`ndim`()	Return the dimension n for f defined in R^n.
`orthonormalize`()	Orthonormalize the basis.
`plot`([indices, n])	Plot the functions of the basis.
`plot_multi_indices`(*args)	PLot the multi-index set of the object.
`projection`(fun, G)	Compute the projection of the function fun onto the functional basis using the integration rule G.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`remove_bases`(ind)	Remove bases of self of index ind.
`storage`()	Return the storage requirement of the FunctionalBasis.
`tensor_product_interpolation`(fun, *args)	Return the interpolation of function fun on a sparse grid.
`transpose`(perm)	Return self with the basis permutation perm.

christoffel
derivative
eval_derivative
eval_with_functional_bases_evals
keep_mapping
optimal_sampling_measure
remove_mapping

adaptation_path(p=1)¶

Create an adaptation path associated with increasing p-norm of multi-indices.

Parameters

pfloat, optional: The positive real scalar p of the p-norm. The default is 1.

Returns

Pnumpy.ndarray: The adaptation path.

cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

conditional_expectation(dims, *args)¶

Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)). The expectation with respect to other variables (in the complementary set of dims) is taken with respect to the probability measure given by tensap.RandomVector XdimsC if provided, or with respect to the probability measure associated with the corresponding bases of the function.

Parameters

dimsnumpy.ndarray: The dimensions in which the expectation is computed.
XdimsC: tensap.RandomVector, optional: The random vector used for the computation of the conditional expectation. The default is None, indicating to use the
probability measure associated with the basis.

Returns

ftensap.FunctionalBasisArray: The conditional expectation of the function.

derivative(n)¶

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

eval(x)¶

Return the evaluation of the basis functions at the points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluted.

Returns

numpy.ndarray: The evaluations of the basis functions at the points x.

eval_derivative(n, x)¶

eval_with_functional_bases_evals(Hx)¶

get_random_vector()¶

Return the random vector associated with the basis functions of self.

Returns

tensap.RandomVector: The random vector associated with the basis functions of self.

gram_matrix()¶

Return the gram matrix of the basis.

Returns

numpy.ndarray: The gram matrix of the basis.

keep_bases(ind)¶

Keep only the bases of self of index ind.

Parameters

indint or list or numpy.ndarray: The indices of the bases to keep.

Returns

tensap.SparseTensorProductFunctionalBasis: The SparseTensorProductFunctionalBasis with kept bases.

keep_mapping(ind)¶

length()¶

Return the number of bases in self.bases.

Returns

int: The number of bases in self.bases.

mean(*args)¶

Return the mean of the basis functions.

Returns

numpy.ndarray: The mean of the basis functions.

ndim()¶

Return the dimension n for f defined in R^n.

Returns

int: The dimension n for f defined in R^n.

plot_multi_indices(*args)¶

PLot the multi-index set of the object.

tensap.approximation.bases.sub_functional_basis module¶

Module sub_functional_basis.

class tensap.approximation.bases.sub_functional_basis.SubFunctionalBasis(underlying_basis=None, basis=None)¶

Bases: tensap.approximation.bases.functional_basis.FunctionalBasis

Class SubFunctionalBasis.

Attributes

underlying_basistensap.FunctionalBasis: The underlying basis.
basisnumpy.ndarray: Array of shape (n, m), where n is the number of elements in underlying_basis, which defines a set of m basis functions in the space generated by underlying_basis.

Methods

`adaptation_path`()	Return the adaptation path of the functional basis.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`()	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`derivative`(n)	Compute the k-th order derivative of the functions of the basis projected on itself.
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x[, indices])	Return the evaluation of the basis functions at the points x.
`eval_derivative`(n, x)	Evaluate the k-th order derivative of the functions of the basis at the points x.
`expectation`()	Return the expectation of the basis functions.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`kron`()	Compute the Kronecker product of two bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`()	Return the mean of the basis functions.
`ndim`()	Return the dimension n for f defined in R^n.
`orthonormalize`()	Orthonormalize the SubFunctionalBasis.
`plot`([indices, n])	Plot the functions of the basis.
`projection`(fun, G)	Compute the projection of the function fun onto the functional basis using the integration rule G.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`storage`()	Return the storage requirement of the FunctionalBasis.

christoffel
optimal_sampling_measure

cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

derivative(n)¶

Compute the k-th order derivative of the functions of the basis projected on itself.

Parameters

kint: The order of the derivative.

Returns

tensap.SubFunctionalBasis: The k-th order derivative of the functions of the basis projected on itself.

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

eval(x, indices=None)¶

Return the evaluation of the basis functions at the points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluted.

Returns

numpy.ndarray: The evaluations of the basis functions at the points x.

eval_derivative(n, x)¶

Evaluate the k-th order derivative of the functions of the basis at the points x.

Parameters

kint: The order of the derivative.
xlist or numpy.ndarray: The points at which the k-th derivative of the basis functions are to be evaluated.

Returns

numpy.ndarray: The evaluations of the k-th derivative of the functions of self.basis of degrees in self.indices at points x.

mean()¶

Return the mean of the basis functions.

Returns

numpy.ndarray: The mean of the basis functions.

ndim()¶

Return the dimension n for f defined in R^n.

Returns

int: The dimension n for f defined in R^n.

orthonormalize()¶

Orthonormalize the SubFunctionalBasis.

Returns

tensap.SubFunctionalBasis: The orthonormalized SubFunctionalBasis.

storage()¶

Return the storage requirement of the FunctionalBasis.

Returns

int: The storage requirement of the FunctionalBasis.

tensap.approximation.bases.user_defined_functional_basis module¶

Module user_defined_functional_basis.

class tensap.approximation.bases.user_defined_functional_basis.UserDefinedFunctionalBasis(h_fun=None, measure=None, input_dim=None)¶

Bases: tensap.approximation.bases.functional_basis.FunctionalBasis

Class UserDefinedFunctionalBasis.

The basis is not L2-orthonormal a priori, hence the is_orthonormal attribute remains at its default value of False.

Attributes

handle_funnumpy.ndarray: The functions making the basis.
measuretensap.Measure: The measure associated with the basis. Can be a tensap.RandomVector or a tensap.RandomVariable to define a random generator and an expectation.
input_dimensionint: The dimension of the domain of the functions in handle_fun.

Methods

`adaptation_path`()	Return the adaptation path of the functional basis.
`cardinal`()	Return the number of basis functions.
`conditional_expectation`()	Compute the conditional expectation of self with respect to the random variables dims (a subset of range(d)).
`domain`()	Return the domain of the set of basis functions, which is the support of the associated measure.
`eval`(x)	Return the evaluation of the basis functions at the points x.
`expectation`()	Return the expectation of the basis functions.
`interpolate`(y[, x])	Provide an interpolation on a functional basis of a function (or values of the function) y associated with a set of n interpolation points x.
`interpolation_points`(*args)	Return the interpolation points for the basis.
`kron`()	Compute the Kronecker product of two bases.
`magic_points`([x, J])	Provide the magic points associated with a functional basis f selected in a given set of points x.
`mean`()	Return the mean of the basis functions.
`ndim`()	Return the dimension n for f defined in R^n.
`orthonormalize`()	Orthonormalize the basis.
`plot`([indices, n])	Plot the functions of the basis.
`projection`(fun, G)	Compute the projection of the function fun onto the functional basis using the integration rule G.
`random`([n, measure])	Evaluate the basis using n points drawn randomly according to measure if provided, or to self.measure otherwise.
`storage`()	Return the storage requirement of the FunctionalBasis.

christoffel
optimal_sampling_measure

cardinal()¶

Return the number of basis functions.

Returns

int: The number of basis functions.

domain()¶

Return the domain of the set of basis functions, which is the support of the associated measure.

Returns

numpy.ndarray: The domain of the set of basis functions

eval(x)¶

Return the evaluation of the basis functions at the points x.

Parameters

xlist or numpy.ndarray: The points at which the basis functions are to be evaluted.

Returns

numpy.ndarray: The evaluations of the basis functions at the points x.

ndim()¶

Return the dimension n for f defined in R^n.

Returns

int: The dimension n for f defined in R^n.

tensap.approximation.bases package¶

Submodules¶

tensap.approximation.bases.full_tensor_product_functional_basis module¶

tensap.approximation.bases.functional_bases module¶

tensap.approximation.bases.functional_basis module¶

tensap.approximation.bases.functional_basis_array module¶

tensap.approximation.bases.polynomial_functional_basis module¶

tensap.approximation.bases.sparse_tensor_product_functional_basis module¶

tensap.approximation.bases.sub_functional_basis module¶

tensap.approximation.bases.user_defined_functional_basis module¶

Module contents¶

tensap

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