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Tensor Gaussian rules recover polynomial-chaos coefficients exactly

Statement

Suppose each marginal law $ρ_{j}$ is the positive measure $μ_{j}$ in Gaussian data from Coordinatewise product-moment functional for tensor polynomials, with $m_{j} \geq p_{j} + 1$ , where $p$ is the admissible cutoff vector of Finite product-law polynomial-chaos input data. Let $Q_{\otimes}$ be the tensor rule of Tensor-product Gaussian quadrature, exact through degree $2 m_{j} - 1$ in coordinate $j$ . For every $u \in P_{Λ}$ and $α \in Λ$ , the quadrature coefficient $Q_{\otimes} (u ψ_{α})$ equals the exact coefficient $⟨ u, ψ_{α} ⟩$ . Hence ${\hat{P}}_{Λ} u = P_{Λ} u$ and the coefficient quadrature term in Projection and quadrature errors split in L2 is zero.

Facts & Assumptions

Given: A product law with the stated marginal Gaussian moment data, $m_{j} \geq p_{j} + 1$ , and $u \in P_{Λ}$ .

[F1]

Finite product-law polynomial-chaos input data bounds every basis coordinate degree by $p_{j}$ .

[F2]

Finite orthonormal polynomial tensor basis defines each $ψ_{β}$ as a product of one-dimensional polynomials of degree $β_{j}$ in coordinate $j$ .

[F3]

Independent random elements have product joint law identifies the joint law with the product of its marginal laws.

[F4]

Coordinatewise product-moment functional for tensor polynomials defines the algebraic product-moment functional; Tensor product moments equal integration against the finite product measure proves that it equals integration under the finite product of the marginal measures for these coordinatewise tensor polynomials.

[F5]

Tensor-product Gaussian quadrature preserves coordinatewise exactness states exactness of the tensor Gaussian functional for coordinatewise degrees at most $2 m_{j} - 1$ .

[F6]

Independent polynomial-chaos tensor products are orthonormal gives $\int ψ_{β} ψ_{α} d ρ = δ_{α β}$ .

[F7]

Quadrature estimates of polynomial-chaos coefficients defines ${\hat{c}}_{α} = Q_{\otimes} (u ψ_{α})$ .

Proof

1.1F1F2givenalgebra

Expand $u = \sum_{β \in Λ} c_{β} ψ_{β}$ . By [F1] and [F2], each $ψ_{β}$ has coordinate degree $β_{j} \leq p_{j}$ , so each product $ψ_{β} ψ_{α}$ has degree at most $2 p_{j} \leq 2 m_{j} - 2 \leq 2 m_{j} - 1$ in coordinate $j$ .

2.1F3F4F5F6step 1.1algebra

By [F3] and [F4], the product-law integral of this finite polynomial equals its coordinatewise product-moment functional. Its degrees satisfy the limits checked in step 1.1, so [F5] gives $Q_{\otimes} (ψ_{β} ψ_{α}) = \int ψ_{β} ψ_{α} d ρ = δ_{α β}$ by [F6]. Finite linearity then gives $Q_{\otimes} (u ψ_{α}) = c_{α} = ⟨ u, ψ_{α} ⟩$ .

3.1F7step 2.1algebra∎

The equality in step 2.1 holds for every $α \in Λ$ , so all coefficient differences vanish and the finite sums defining ${\hat{P}}_{Λ} u$ and $P_{Λ} u$ coincide. The quadrature contribution in the error split is therefore zero.

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Sources

Dongbin Xiu, Fast Numerical Methods for Stochastic Computations, §4 (standard reference, not scraped)
Bruno Sudret, Polynomial Chaos Expansions and Stochastic Finite Element Methods, §3.4.2 (standard reference, not scraped)