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TensorizedBasis

Each multi-index $\mu = [\mu_1, \mu_2, \ldots, \mu_M]$ defines a tensorized basis function for the parameter space of the form $H_\mu = \prod_{k=1}^M H_k$, where each $H_k$ is an orthogonal polynomial.

The TensorizedBasis object collects all information required to evaluate these basis functions, including the set of multi-indices and the triple products of the form $(y_m H_\mu, H_\lambda)$ for each $m$ and $\mu, \lambda$ in the set of multi-indices, stored as a sparse matrix. While analytic formulas exist to compute these triple products using recurrence coefficients, storing them can significantly speed up evaluations.

ExtendableASGFEM.TensorizedBasis — Type
TensorizedBasis(
    OBT::Type{<:OrthogonalPolynomialType},
    M,
    order,
    maxorder;
    ...
) -> TensorizedBasis{Float64, ONBasis{T, OBT, npoly, nquad}} where {T<:Real, OBT<:OrthogonalPolynomialType, npoly, nquad}
TensorizedBasis(
    OBT::Type{<:OrthogonalPolynomialType},
    M,
    order,
    maxorder,
    maxquadorder;
    T,
    multi_indices
) -> TensorizedBasis{Float64, ONBasis{T, OBT, npoly, nquad}} where {T<:Real, OBT<:OrthogonalPolynomialType, npoly, nquad}

constructor for a tensorized basis for the given OrthogonalPolynomialType. If no multi-indices are provided it automatically generates all multi-indices up to support length M and polynomial order maxorder.

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ExtendableASGFEM.TensorizedBasis — Type
struct TensorizedBasis{T<:Real, ONBType<:ONBasis, MIType}

A structure representing a tensorized orthogonal polynomial basis for stochastic Galerkin methods, associated with a given set of multi-indices.

Fields

  • ONB::ONBType: The underlying univariate orthogonal basis (e.g., Hermite, Legendre).
  • vals::Vector{Vector{T}}: Cached evaluations of the univariate basis functions at the most recent sample(s).
  • nmodes::Int: Number of multi-indices (i.e., the number of stochastic modes).
  • G::ExtendableSparseMatrix{T, Int64}: Coupling matrix encoding triple products or recurrence relations between basis functions.
  • multi_indices::MIType: Collection of multi-indices (typically an array of integer arrays), each representing a multi-dimensional polynomial degree.

Description

The TensorizedBasis type encapsulates all information required to evaluate and manipulate a tensor-product polynomial basis in multiple stochastic dimensions. It supports efficient evaluation, storage of basis values, and access to coupling coefficients for use in stochastic Galerkin finite element methods (SGFEM).

Example

# Construct a tensorized Hermite basis with 3 variables and total degree 2
TB = TensorizedBasis(HermitePolynomials, 3, 2, 2)

# Evaluate all basis functions at a sample point
set_sample!(TB, [0.1, -0.2, 0.3])

# Get the value of the 5th basis function at the current sample
val = evaluate(TB, 5)
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Base.show — Method
show(io::IO, TB::TensorizedBasis)

shows information on the tensorized basis.

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ExtendableASGFEM.evaluate — Method
evaluate(TB::TensorizedBasis{T}, j) -> Any

evaluates the basis function for the j-th multi-index at the sample that was set with set_sample.

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ExtendableASGFEM.sample_distribution — Method
sample_distribution(
    TB::TensorizedBasis,
    nsamples;
    M,
    Mweights
) -> Tuple{Matrix{Float64}, Any}

Generate samples and weights for the distribution of the ON basis that the tensorized basis is based upon (that can be used for a Monte-Carlo estimator).

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ExtendableASGFEM.set_sample! — Method
set_sample!(
    TB::TensorizedBasis,
    x;
    normalize
) -> Array{Vector{T}, 1} where T<:Real

evaluates all basis functions at sample vector x; call this before using evaluate!

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