ONBasis

An ONBasis (orthonormal basis) encapsulates information about the orthogonal polynomials associated with a given probability distribution. This includes norms, quadrature rules, and cached evaluations at quadrature points. The ONBasis serves as a fundamental building block for constructing the tensorized basis linked to the multi-indices in the stochastic discretization.

ExtendableASGFEM.ONBasis — Type

ONBasis(
    OBT::Type{<:OrthogonalPolynomialType},
    maxorder;
    ...
) -> ONBasis{Float64}
ONBasis(
    OBT::Type{<:OrthogonalPolynomialType},
    maxorder,
    maxquadorder;
    T
) -> ONBasis{Float64}

Constructs an ONBasis for the given orthogonal polynomial type and associated weight function, up to the specified maximum polynomial order.

Arguments

OBT: The type of orthogonal polynomial (subtype of OrthogonalPolynomialType).
maxorder: The highest polynomial degree to include in the basis.
maxquadorder: The quadrature order to use for integration (default: 2 * maxorder).
T: The floating-point type for computations (default: Float64).

Returns

An ONBasis object containing:

The norms of all basis functions.
A workspace for storing evaluations.
The quadrature rule (points and weights).
Precomputed values of all basis functions at the quadrature points.

ExtendableASGFEM.ONBasis — Type

struct ONBasis{T<:Real, OBT<:OrthogonalPolynomialType, npoly, nquad}

A structure that stores all relevant information for an orthonormal polynomial basis (ONBasis), including:

The norms of all basis functions.
The quadrature rule (points and weights) used for integration.
Precomputed values of all basis functions at the quadrature points.
A workspace for storing the result of the last evaluation.

This structure enables efficient evaluation, integration, and manipulation of orthogonal polynomial bases for stochastic Galerkin methods and related applications.

ExtendableASGFEM.OrthogonalPolynomialType — Method

OrthogonalPolynomialType(ONB::ONBasis{T, OBT}) -> Any

returns the OrthogonalPolynomialType

ExtendableASGFEM.distribution — Method

distribution(ONB::ONBasis{T, OBT}) -> Any

returns the distribution associated to the orthogonal polynomials

ExtendableASGFEM.evaluate — Method

evaluate(
    ONB::ONBasis{T, OBT, maxorder},
    x;
    normalize
) -> Vector

Evaluates all basis polynomials at point x

ExtendableASGFEM.integral — Method

integral(ONB::ONBasis, j; normalize) -> Any

Evaluates the integral of a single basis function.

ExtendableASGFEM.norm4poly — Method

norm4poly(ONB::ONBasis, p) -> Any

returns the norm of the p-th polynomial

ExtendableASGFEM.qp — Method

qp(
    ONB::ONBasis
) -> StaticArraysCore.SVector{nquad, T} where {T<:Real, nquad}

returns the quadrature points

ExtendableASGFEM.qw — Method

qw(
    ONB::ONBasis
) -> StaticArraysCore.SVector{nquad, T} where {T<:Real, nquad}

returns the quadrature weights

ExtendableASGFEM.scalar_product — Method

scalar_product(ONB::ONBasis, j, k; normalize) -> Any

Evaluates the scalar product of two basis functions.

ExtendableASGFEM.triple_product — Method

triple_product(ONB::ONBasis, j, k, l; normalize) -> Any

Evaluates the triple product of three basis functions.

ExtendableASGFEM.triple_product_y — Method

triple_product_y(ONB::ONBasis, j, k; normalize) -> Any

Evaluates the triple product of two basis functions and y.

ExtendableASGFEM.vals4poly — Method

vals4poly(ONB::ONBasis, p) -> Any

returns the values of the p-th polynomial at all quadrature points

ExtendableASGFEM.vals4qp — Method

vals4qp(ONB::ONBasis, k) -> Any

returns the values of all polynomials at the k-th quadrature point

ExtendableASGFEM.vals4xref — Method

vals4xref(
    ONB::ONBasis
) -> StaticArraysCore.SMatrix{nquad, npoly, T} where {T<:Real, npoly, nquad}

returns the values of all polynomials at the quadrature points