ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Polynomial-chaos coefficients group by variable support

Example

Let $X_{1}, X_{2}$ be independent symmetric two-point inputs on ${- 1, 1}$ , and take the full degree-one tensor index set $Λ = {(0, 0), (1, 0), (0, 1), (1, 1)}$ . Its basis is $1, X_{1}, X_{2}, X_{1} X_{2} .$ For the exact finite expansion $f = 2 + 3 X_{1} + 4 X_{1} X_{2}$ , the coefficients are $c_{00} = 2$ , $c_{10} = 3$ , $c_{01} = 0$ , and $c_{11} = 4$ . The variance is $25$ , with $S_{1} = 9 / 25$ , $S_{2} = 0$ , $S_{T_{1}} = 1$ , and $S_{T_{2}} = 16 / 25$ .

Facts & Assumptions

Given: Independent symmetric two-point inputs, the full degree-one tensor index set, and the exact finite expansion in the Example.

[F1]

Finite product-law polynomial-chaos input data: finite independent real inputs use their product law and a finite admissible degree box; the degree condition alone asserts no polynomial completeness.

[F2]

Positive moment Gram matrices produce orthonormal polynomial families: for the symmetric two-point law with cutoff one, the orthonormal degree-one polynomials are $1$ and $x$ .

[F3]

Finite orthonormal polynomial tensor basis: a finite tensor basis consists of products of the univariate polynomials over a declared finite multi-index set.

[F4]

Product-coordinate averaging operators: $P_{J}$ averages over coordinates outside $J$ and lifts the result to the full product space.

[F5]

Independent product inputs have an orthogonal Hoeffding decomposition: coordinate averages define components that reconstruct $f$ , are orthogonal, and contribute their squared norms to the variance.

[F6]

First-order and total Sobol indices: first and total indices divide variance-component sums by positive total variance.

[F7]

Moments, variance, and covariance on a probability space: variance is the expected square of the output after subtracting its mean.

Verification

Given: The finite product law, index set and exact expansion of the Example.

1.1F1F2F3givenalgebra

Each coordinate has mean zero and second moment one, so its degree-one monomial Gram matrix is the identity. The construction [F2] therefore gives $1$ and $x_{j}$ , whose products in [F3] are the four displayed functions. On the four equiprobable outcomes $(1, 1), (1, - 1), (- 1, 1), (- 1, - 1)$ , their value vectors are $(1, 1, 1, 1)$ , $(1, 1, - 1, - 1)$ , $(1, - 1, 1, - 1)$ , and $(1, - 1, - 1, 1)$ . These vectors are pairwise orthogonal and each has squared length four, so under weights $1 / 4$ the functions are orthonormal. Four independent functions on a four-point support form a basis of its four-dimensional $L^{2}$ space.

2.1F7step 1.1givenalgebra

The functions in step 1.1 are linearly independent, so the exact expansion has unique coefficients: $c_{00} = 2$ , $c_{10} = 3$ , $c_{01} = 0$ , and $c_{11} = 4$ . The centered output is $3 X_{1} + 4 X_{1} X_{2}$ . Orthogonality gives $V = E [(3 X_{1} + 4 X_{1} X_{2})^{2}] = 9 + 16 = 25$ by [F7]. The exact displayed finite expansion needs no infinite Parseval assertion.

2.2F1F4F5givenstep 1.1algebra

Product averaging [F4] and $E X_{j} = 0$ give $P_{\emptyset} f = 2$ , $P_{{1}} f = 2 + 3 X_{1}$ , and $P_{{2}} f = 2$ . The component definition in [F5] yields $f_{{1}} = 3 X_{1}$ , $f_{{2}} = 0$ , and $f_{{1, 2}} = 4 X_{1} X_{2}$ . Their variance contributions are $9, 0, 16$ because $E X_{j}^{2} = 1$ , matching the coefficient sums grouped by support.

3.1F6step 1.1step 2.1step 2.2algebra∎

Dividing these components by $V = 25$ according to [F6] gives $S_{1} = 9 / 25$ , $S_{2} = 0$ , $S_{T_{1}} = (9 + 16) / 25 = 1$ , and $S_{T_{2}} = 16 / 25$ . The family is complete here by the finite four-point argument in step 1.1. Finite cutoff and positive moment Gram hypotheses alone do not imply completeness for a general input law.

Remarks

This example's full degree-one basis is complete because its input support has four points. In general the finite cutoff and positive moment Gram hypotheses of UQ1 do not imply completeness; an infinite coefficient formula needs a separate law-specific completeness hypothesis.

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Sources

Dongbin Xiu and George Em Karniadakis, The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations, §§4–5 (standard reference, not scraped)