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Polynomial chaos stochastic collocation and Galerkin UQ

1 · Prerequisites

2 · Summary

Finite product-law basis

The input is a finite family of independent real random variables with a specified product law and an explicit admissible moment cutoff. Gram–Schmidt gives the law-matched finite univariate polynomials; their tensor products are orthonormal by product integration. Total-degree and hyperbolic-cross truncations are intersected with the declared cutoff box. The construction proves finite orthogonality only. Polynomial completeness requires a separate law-specific density hypothesis, and moments alone do not supply it.

Projection and coefficient quadrature

For any finite chaos space, the projection leaves an exact orthogonal residual. Its squared norm, mean and variance retain that residual, so no completeness premise is hidden in Parseval-style identities. A cubature coefficient estimator, using a representative evaluable at its nodes, splits into projection error and coefficient quadrature error. Tensor Gaussian exactness applies when each marginal rule integrates the coordinate degrees of every product integrand; the coefficient result here assumes a polynomial input and the stated node count. The Smolyak bound applies only to the uniform cube when each coefficient integrand satisfies the supplied mixed fourth-derivative and endpoint hypotheses.

Christoffel-weighted least squares

The algorithm samples from the normalized Christoffel density, weights by the reciprocal leverage, and returns coefficients only when the empirical Gram spectrum is in $[1 / 2, 3 / 2]$ . The locally proved matrix concentration argument gives this event with probability at least $1 - δ$ when $M \geq 12 ∣ Λ ∣ \log (2 ∣ Λ ∣ / δ)$ . On the event, the fit error separates into the finite projection residual and the weighted sampled residual. This is a least-squares guarantee; no universal aliasing or collocation-equals-Galerkin statement is made.

Affine random diffusion and Galerkin approximation

For bounded product input, an affine coefficient uniformly bounded above and below for every supported parameter, and bounded affine forcing $F (y) = f_{0} + \sum_{j} y_{j} f_{j}$ with $f_{j} \in V^{*}$ , the pointwise weak solution is Lipschitz in the parameter and defines a Bochner $L^{2} (H_{0}^{1})$ solution. Countable Choice is used through the stated Sobolev, Bochner and Lax–Milgram interfaces. Integrating the form preserves its exact continuity and coercivity constants, so a finite tensor Galerkin space whose chaos indices lie in the declared admissible cutoff box satisfies Céa's estimate with factor $a^{*} / a_{*}$ . The finite-element rate is stated only on a shape-regular matching affine mesh and when the solution has an exact finite chaos expansion with spatially regular coefficients and a global conforming interpolant assembled from a fixed reference element with bounded degrees of freedom on the required Sobolev class. No arbitrary-input spectral rate is claimed.

The proof items preserve the law-specific and regularity hypotheses in Completeness and spectral rates require law-specific hypotheses.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Finite product-law polynomial-chaos input data

Definition

Fix an integer $d \geq 1$ , independent real random variables $Y_{1}, \dots, Y_{d}$ with marginal Borel probability laws $ρ_{1}, \dots, ρ_{d}$ , and a degree cutoff vector $p = (p_{1}, \dots, p_{d}) \in N_{0}^{d}$ . Their joint law is $ρ = ρ_{1} \otimes \dots \otimes ρ_{d}$ on $R^{d}$ , as in Independent random elements have product joint law. For a Borel probability law $ρ_{j}$ on $R$ , define $supp (ρ_{j}) : = {x \in R : ρ_{j} (U) > 0 for every open U ∋ x} .$ The complement is a union of zero-measure members of the countable rational interval basis, so countable additivity gives $ρ_{j} (supp (ρ_{j})) = 1$ . Set $Γ = \prod_{j = 1}^{d} supp (ρ_{j})$ and give it the trace Borel sigma-algebra from $R^{d}$ . The product rectangle formula gives $ρ (Γ) = 1$ . Whenever the measure space is written as $(Γ, B (Γ), ρ)$ , $ρ$ denotes the restriction of the product law on $R^{d}$ to $B (Γ)$ ; it is therefore a probability measure. The product rectangle formula also identifies this restriction with the finite product of the marginal laws restricted to their supports. The cutoff vector $p$ is admissible when, for every $j = 1, \dots, d$ , $\int ∣ y ∣^{2 p_{j}} d ρ_{j} (y) < \infty$ and the monomial Gram matrix $G_{j} = (\int y^{r + s} d ρ_{j} (y))_{0 \leq r, s \leq p_{j}}$ is positive definite, meaning $c^{T} G_{j} c > 0$ for each nonzero $c \in R^{p_{j} + 1}$ . Throughout the pair, assume the selected cutoff vector is admissible. The admissible degree box is ${α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j} for every j}$ . Admissibility gives a finite inner product on each stated polynomial space. It is only a finite-degree condition; it does not assert that polynomials are dense in any $L^{2} (ρ_{j})$ .

RemarkRemark: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Completeness and spectral rates require law-specific hypotheses

Remarks

The finite moment and Gram hypotheses in Finite product-law polynomial-chaos input data construct only the declared finite polynomial family. Completeness in $L^{2} (ρ)$ requires a separate density proof for that law; no such conclusion is inferred from moments alone. Likewise, an exponential or algebraic spectral rate requires quantified mixed-derivative or analyticity bounds. The weighted least-squares procedure in this page is a separately defined collocation method. No general equality with a spectral Galerkin method is asserted; exact agreement is limited to the polynomial degree conditions stated in the Gaussian-quadrature corollary.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Total-degree and hyperbolic-cross chaos index sets

Definition

For $d$ inputs with admissible cutoff vector $p = (p_{1}, \dots, p_{d})$ from Finite product-law polynomial-chaos input data, the cutoff total-degree set of order $s \geq 0$ and cutoff hyperbolic-cross set of threshold $N \geq 1$ are $Λ_{s, p}^{T D} = {α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j}, \sum_{j = 1}^{d} α_{j} \leq s}, Λ_{N, p}^{H C} = {α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j}, \prod_{j = 1}^{d} (α_{j} + 1) \leq N},$ The cutoff vector prevents either set from requesting polynomials above the moment degree declared in Finite product-law polynomial-chaos input data. Both are finite and contain zero. An arbitrary finite index set used below is denoted by $Λ$ ; whenever a downward-closed set is required, this means $α \in Λ$ and $0 \leq β_{j} \leq α_{j}$ for every $j$ imply $β \in Λ$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Bochner L2 space for a finite product-law input

Definition

Let $(Γ, B (Γ), ρ)$ be the finite product-law probability space of Finite product-law polynomial-chaos input data, and let $V$ be a separable real Hilbert space. Scalar integrals involving strongly measurable maps are taken using measurable versions: if simple maps converge to $U$ outside a measurable null set $N$ , set $\tilde{U} = U$ off $N$ and $\tilde{U} = 0$ on $N$ , and set the approximants to zero on $N$ . They then converge everywhere to $\tilde{U}$ . For every open set $O \subseteq V$ , convergence expresses ${\tilde{U}}^{- 1} (O)$ through countably many eventual bounds on distances of the simple maps to $V ∖ O$ , so this version is Borel measurable. Any two such versions agree outside a measurable null set. In particular, norm-square and pairing integrals below mean the integrals of $∥ \tilde{U} ∥_{V}^{2}$ and $(\tilde{U}, \tilde{W})_{V}$ ; their values are independent of the versions. This convention preserves arbitrary null-set values of strongly measurable maps and does not complete the underlying probability space.

Define $L_{ρ}^{2} (V)$ to be the space of strongly measurable maps $U : Γ \to V$ for which $\int_{Γ} ∥ U (y) ∥_{V}^{2} d ρ (y) < \infty$ , modulo equality $ρ$ -almost everywhere. Its inner product is $(U, W)_{L_{ρ}^{2} (V)} = \int_{Γ} (U (y), W (y))_{V} d ρ (y) .$

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Positive moment Gram matrices produce orthonormal polynomial families

Statement

Let $ρ_{j}$ be one factor law with admissible degree $p_{j}$ as in Finite product-law polynomial-chaos input data. The monomial classes $1, y, \dots, y^{p_{j}}$ are linearly independent in $L^{2} (ρ_{j})$ . Applying the finite Gram–Schmidt recursion to this ordered list gives real orthonormal polynomials $ϕ_{j, 0}, \dots, ϕ_{j, p_{j}}$ , where $ϕ_{j, k}$ has degree $k$ , positive leading coefficient, and $ϕ_{j, 0} = 1$ .

Facts & Assumptions

Given: A probability law $ρ_{j}$ with finite $2 p_{j}$ moment and positive-definite monomial Gram matrix.

[F1]

Finite product-law polynomial-chaos input data states that for every nonzero coefficient vector $c$ , the quadratic form $c^{T} G_{j} c$ is strictly positive.

[F2]

The finite $2 p_{j}$ moment makes every monomial of degree at most $2 p_{j}$ integrable: for $0 \leq q \leq 2 p_{j}$ , $∣ y ∣^{q} \leq 1 + ∣ y ∣^{2 p_{j}}$ .

[F3]

On a real inner-product space, Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans constructs the finite orthonormal list by recursive subtraction and normalization, preserving each successive span.

[F4]

The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product defines normalization by the positive norm of each nonzero residual.

Proof

1.1givenF1F2algebra

Let $P_{p_{j}}$ be the real vector space spanned by the formal monomials $1, y, \dots, y^{p_{j}}$ . For $q, r \in P_{p_{j}}$ define $⟨ q, r ⟩_{ρ_{j}} : = \int q (y) r (y) d ρ_{j} (y)$ . Every product $q r$ has degree at most $2 p_{j}$ and is integrable by [F2], so the pairing is finite; linearity of the integral makes it bilinear and symmetric. If $q (y) = \sum_{r = 0}^{p_{j}} c_{r} y^{r}$ , then $⟨ q, q ⟩_{ρ_{j}} = c^{T} G_{j} c$ , which is positive for each nonzero formal polynomial by [F1]. Thus this is an inner product on $P_{p_{j}}$ .

2.1step 1.1F1F2F3algebra

Every $q \in P_{p_{j}}$ has finite squared integral by [F2], so it defines an $L^{2} (ρ_{j})$ class. If that class is zero, then $q = 0$ almost everywhere and $\int q^{2} d ρ_{j} = 0$ ; by [F1] its coefficient vector must be zero. Thus the class map is injective and the monomial classes are linearly independent. The formal monomials are also independent by uniqueness of their coefficient lists, so [F3] applies to this list in $P_{p_{j}}$ . Its residual at stage $k$ is $y^{k}$ minus a polynomial of degree at most $k - 1$ , so it has degree $k$ and leading coefficient $1$ .

3.1givenF3F4step 2.1algebra∎

Each residual is nonzero in $P_{p_{j}}$ by its degree and leading coefficient, so [F4] makes its norm strictly positive. The normalized residual is orthogonal to the preceding polynomials and has positive leading coefficient. At $k = 0$ , the residual is $1$ and its norm is $\sqrt{\int 1 d ρ_{j}} = 1$ because $ρ_{j}$ is a probability law; hence $ϕ_{j, 0} = 1$ . This proves the stated finite orthonormal family.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Finite orthonormal polynomial tensor basis

Definition

Let $ϕ_{j, k}$ be the Gram–Schmidt polynomials from Positive moment Gram matrices produce orthonormal polynomial families, and let $Λ$ be a finite subset of the admissible degree box in Finite product-law polynomial-chaos input data. For $α = (α_{1}, \dots, α_{d}) \in Λ$ , define $ψ_{α} (y_{1}, \dots, y_{d}) = \prod_{j = 1}^{d} ϕ_{j, α_{j}} (y_{j}) .$ Here $L^{2} (ρ)$ uses the full-mass restriction to $Γ$ specified in Finite product-law polynomial-chaos input data, which the product rectangle formula identifies with the product of the restricted marginal laws. The finite polynomial-chaos space is $P_{Λ} = span {ψ_{α} : α \in Λ}$ inside $L^{2} (ρ)$ . Whenever $0 \in Λ$ (in particular, for the stated total-degree and hyperbolic-cross sets), the family contains the constant mode $ψ_{0} = 1$ . No density of $P_{Λ}$ in $L^{2} (ρ)$ is part of this definition.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Uniformly elliptic affine random diffusion model

Definition

Let $D \subset R^{m}$ be a bounded polyhedral open set and $V = H_{0}^{1} (D; R)$ with norm $∥ v ∥_{V} = ∥ \nabla v ∥_{L^{2} (D)}$ . The zero-boundary Sobolev energy space is a separable Hilbert space supplies the separable Hilbert structure under Countable Choice; the gradient norm is equivalent to its Sobolev norm there.

Let $Y = (Y_{1}, \dots, Y_{d})$ have the product law $ρ$ of Finite product-law polynomial-chaos input data, and assume its support $Γ$ is bounded, so $∣ y_{j} ∣ \leq R_{j} < \infty$ for every $y \in Γ$ . For real coefficient functions $a_{0}, \dots, a_{d} \in L^{\infty} (D)$ define $a (x, y) = a_{0} (x) + \sum_{j = 1}^{d} y_{j} a_{j} (x) .$ Assume constants $0 < a_{*} \leq a^{*} < \infty$ satisfy, for every $y \in Γ$ , $a_{*} \leq a (x, y) \leq a^{*}$ for almost every $x \in D$ . This explicit all-parameter condition ensures the pointwise variational problem is well-posed for every stated parameter, not merely almost every one. Let $F (y) = f_{0} + \sum_{j = 1}^{d} y_{j} f_{j}$ with $f_{j} \in V^{*}$ .

The pointwise weak form is $a_{y} (u, v) = \int_{D} a (x, y) \nabla u (x) \cdot \nabla v (x) d x = F (y) (v), v \in V .$ The Bochner weak form uses the bilinear form $A (U, W) = \int_{Γ} a_{y} (U (y), W (y)) d ρ (y)$ and the functional $F (W) = \int_{Γ} F (y) (W (y)) d ρ (y)$ on $L_{ρ}^{2} (V)$ from Bochner L2 space for a finite product-law input.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

The stated chaos index sets are finite and contain zero

Statement

For every admissible cutoff vector $p \in N_{0}^{d}$ , order $s \geq 0$ , and threshold $N \geq 1$ , the sets $Λ_{s, p}^{T D}$ and $Λ_{N, p}^{H C}$ of Total-degree and hyperbolic-cross chaos index sets are finite and contain $0$ .

Facts & Assumptions

Given: $d \geq 1$ , an admissible cutoff vector $p \in N_{0}^{d}$ , $s \geq 0$ , and $N \geq 1$ , with the two index-set definitions above.

[F1]

Total-degree and hyperbolic-cross chaos index sets defines the sets by coordinate sums and products of positive integers.

Proof

1.1givenF1algebra

If $α \in Λ_{s, p}^{T D}$ , then every coordinate satisfies $0 \leq α_{j} \leq p_{j}$ by the cutoff condition in [F1]. Hence this set is a subset of the finite box $\prod_{j} {0, \dots, p_{j}}$ .

2.1givenF1step 1.1algebra∎

The same cutoff condition places $Λ_{N, p}^{H C}$ in the finite box $\prod_{j} {0, \dots, p_{j}}$ . Since $0 \leq s$ and $1 \leq N$ , zero satisfies both defining inequalities. Thus both sets are finite and contain zero.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Finite-input Bochner L2 of a separable Hilbert space is Hilbert

Statement

Assume Countable Choice. For the space $L_{ρ}^{2} (V)$ of Bochner L2 space for a finite product-law input, the displayed integral pairing is a well-defined real inner product, its norm is $∥ U ∥_{L_{ρ}^{2} (V)} = {(\int_{Γ} ∥ U (y) ∥_{V}^{2} d ρ (y))}^{1 / 2},$ and this norm is complete. Moreover, finite-valued strongly measurable functions are dense in this space.

Facts & Assumptions

Given: Countable Choice, a finite product Borel probability space $(Γ, B (Γ), ρ)$ , and a separable real Hilbert space $V$ .

[F1]

Bochner L2 space for a finite product-law input defines the elements as strongly measurable $V$ -valued classes with square-integrable norm, modulo equality almost everywhere.

[F2]

Hilbert space gives completeness of $V$ and the positive-definite inner product $(\cdot, \cdot)_{V}$ .

[F3]

Cauchy-Schwarz inequality for $L^{2}$ gives the scalar integral estimate $\int f g d ρ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for nonnegative $f, g \in L^{2} (ρ)$ .

[F4]

Minkowski's inequality and monotone convergence for nonnegative integrals are Minkowski's inequality for integrals, including $p = \infty$ and Monotone convergence for the integral.

[F5]

A Borel map into a separable Banach space on a probability space is strongly measurable (A Borel map into a separable Banach space is strongly measurable).

[F6]

Dominated convergence applies to integrable scalar majorants (Dominated convergence).

[A1]

Countable Choice permits selecting a subsequence of a Cauchy sequence with prescribed summable successive distances (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

Countable Choice also permits selecting strongly measurable representatives and their simple-approximation witnesses for a countable sequence of equivalence classes; replacing each representative on its measurable null exceptional set by zero gives a Borel version.

Proof

1.1F1F2F3given

For $U, W \in L_{ρ}^{2} (V)$ , use the Borel representatives obtained by the null-set modification in [F1], and denote them again by $U, W$ . Their norm functions and pairings are measurable. Pointwise Cauchy–Schwarz gives $∣ (U (y), W (y))_{V} ∣ \leq ∥ U (y) ∥_{V} ∥ W (y) ∥_{V}$ . Integrating and applying [F3] shows that the product is integrable and the pairing is finite. Changing either representative on a null set does not change the integral. The pairing is bilinear and symmetric. If its value on $(U, U)$ is zero, then for each integer $n \geq 1$ the set $E_{n} = {y : ∥ U (y) ∥_{V} \geq 1 / n}$ has measure zero, since $0 = \int ∥ U ∥^{2} \geq ρ (E_{n}) / n^{2}$ ; their countable union contains every point where $U \neq 0$ . Thus the class of $U$ is zero, proving positive definiteness.

1.2F1F4A1A2given

Let $(U_{n})$ be Cauchy in the displayed norm. By [A1] select a subsequence $(U_{n_{k}})$ with $∥ U_{n_{k + 1}} - U_{n_{k}} ∥_{L_{ρ}^{2} (V)} \leq 2^{- k}$ . By [A2], choose strongly measurable representatives for these countably many classes and simple-approximation witnesses for each representative; replace each representative by zero on its measurable null exceptional set, obtaining Borel versions of the same classes. Put $d_{k} (y) = ∥ U_{n_{k + 1}} (y) - U_{n_{k}} (y) ∥_{V}$ and $s_{m} = \sum_{k = 1}^{m} d_{k}$ . Minkowski [F4] gives $∥ s_{m} ∥_{2} \leq \sum_{k = 1}^{m} 2^{- k} \leq 1$ . The functions $s_{m}^{2}$ increase, so monotone convergence [F4] yields $\int (\lim_{m} s_{m})^{2} d ρ \leq 1$ . Therefore $\sum_{k} d_{k} (y) < \infty$ outside a measurable null set; otherwise its square integral would be infinite on a set of positive measure.

2.1F2F5F6step 1.2

Let $E = {y : \sum_{k} d_{k} (y) < \infty}$ , a Borel set of full measure by step 1.2. On $E$ , $(U_{n_{k}} (y))$ is Cauchy in $V$ and hence has a limit by [F2]. Define $U (y)$ to be this limit on $E$ and zero on $E^{c}$ . The Borel maps equal to $U_{n_{k}}$ on $E$ and zero on $E^{c}$ converge pointwise everywhere to $U$ ; their pointwise limit is Borel measurable (equivalently, distances to a countable dense subset of $V$ are Borel limits). By [F5], $U$ is strongly measurable. If $g_{m} = \sum_{k \geq m} d_{k}$ , then $g_{m} ↓ 0$ almost everywhere, $g_{m} \leq \lim_{k} s_{k}$ , and the latter is in $L^{2}$ by step 1.2. Dominated convergence [F6] gives $∥ g_{m} ∥_{2} \to 0$ , while $∥ U - U_{n_{m}} ∥_{V} \leq g_{m}$ almost everywhere on $E$ . Thus $U_{n_{m}} \to U$ in $L_{ρ}^{2} (V)$ .

3.1step 1.2step 2.1algebra

Since $(U_{n})$ is Cauchy and its subsequence converges to $U$ , the full sequence converges to $U$ : for $ε > 0$ , choose $N$ so $∥ U_{n} - U_{m} ∥ < ε / 2$ for $m, n \geq N$ , then choose $n_{k} \geq N$ with $∥ U_{n_{k}} - U ∥ < ε / 2$ ; the triangle inequality gives $∥ U_{n} - U ∥ < ε$ for $n \geq N$ . This proves completeness.

4.1F1F6∎

Let $U \in L_{ρ}^{2} (V)$ and choose a strongly measurable representative with simple approximants and measurable null exceptional set as in the definition. Change the representative and approximants to zero on that set; the resulting representative is measurable and the finite-valued simple approximants converge to it everywhere. Truncate each approximant to zero where its norm exceeds $2 ∥ U (y) ∥_{V}$ . The truncation set is measurable; the truncated functions remain finite-valued simple functions, converge to $U$ almost everywhere, and their squared errors are bounded by $9 ∥ U (y) ∥_{V}^{2}$ . Dominated convergence [F6] gives convergence in $L_{ρ}^{2} (V)$ , proving density.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

Christoffel-weighted least-squares polynomial collocation

Definition

Let $Λ$ be finite, contain $0$ , and let $ψ = (ψ_{α})_{α \in Λ}$ be the real orthonormal basis of Finite orthonormal polynomial tensor basis. Set $N = ∣ Λ ∣$ and $K_{Λ} (y) = \sum_{α \in Λ} ∣ ψ_{α} (y) ∣^{2} .$ The algorithm takes an evaluable real function $u$ , a positive integer sample count $M$ , and independent samples $X_{1}, \dots, X_{M}$ from the probability law $d ν = (K_{Λ} / N) d ρ$ . It forms $G = \frac{1}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} ψ (X_{i}) ψ (X_{i})^{T}, b = \frac{1}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} u (X_{i}) ψ (X_{i}) .$ If every eigenvalue of $G$ lies in $[1 / 2, 3 / 2]$ , return the unique coefficient vector $c = G^{- 1} b$ and polynomial $\hat{u} = \sum_{α \in Λ} c_{α} ψ_{α}$ . Otherwise return unstable-sample and no coefficients. The weighted empirical objective minimized on the accepted branch is $\frac{1}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} ∣ u (X_{i}) - p (X_{i}) ∣^{2},$ over $p \in P_{Λ}$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Finite polynomial-chaos orthogonal projection

Definition

For real $u \in L^{2} (ρ)$ and a finite subset $Λ$ of the admissible degree box, with $(ψ_{α})_{α \in Λ}$ as in Finite orthonormal polynomial tensor basis, define $c_{α} = ⟨ u, ψ_{α} ⟩_{L^{2} (ρ)} = \int u (y) ψ_{α} (y) d ρ (y), P_{Λ} u = \sum_{α \in Λ} c_{α} ψ_{α} .$ The scalar integral pairing is choice-free: for representatives $f, g$ of classes in $L^{2} (ρ)$ , Cauchy–Schwarz makes $f g$ integrable, and the integral is unchanged when either representative is changed on a null set. Its bilinearity follows from linearity of the integral and its symmetry from commutativity of multiplication; moreover $\int f^{2} d ρ = ∥ [f] ∥_{2}^{2}$ , so it is positive definite on the quotient. These facts use only the integral pairing, not completeness of $L^{2} (ρ)$ . The displayed finite sum is the orthogonal projection onto $P_{Λ}$ of Finite orthonormal polynomial tensor basis.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Affine diffusion forms are well-defined on Bochner L2

Statement

For the model of Uniformly elliptic affine random diffusion model, let $H = L_{ρ}^{2} (V)$ . For $U, W \in H$ , the scalar functions $y \mapsto a_{y} (U (y), W (y))$ and $y \mapsto F (y) (W (y))$ have measurable integrable versions in the convention of Bochner L2 space for a finite product-law input, so $A (U, W)$ and $F (W)$ are well-defined on equivalence classes. They satisfy $∣ A (U, W) ∣ \leq a^{*} ∥ U ∥_{H} ∥ W ∥_{H}, A (U, U) \geq a_{*} ∥ U ∥_{H}^{2},$ and $∣ F (W) ∣ \leq ∥ F ∥_{L_{ρ}^{2} (V^{*})} ∥ W ∥_{H} < \infty .$

Facts & Assumptions

Given: A bounded-support product law, the affine coefficient and forcing, and the uniform coefficient bounds in Uniformly elliptic affine random diffusion model.

[F1]

For every $y \in Γ$ , the pointwise form satisfies $∣ a_{y} (v, w) ∣ \leq a^{*} ∥ v ∥_{V} ∥ w ∥_{V}$ and $a_{y} (v, v) \geq a_{*} ∥ v ∥_{V}^{2}$ ; $F (y) = f_{0} + \sum_{j} y_{j} f_{j}$ is bounded and affine (Uniformly elliptic affine random diffusion model).

[F2]

Strongly measurable maps are almost-everywhere limits of measurable finite-valued simple maps (Strongly measurable Banach-valued function).

[F3]

The scalar integral Cauchy--Schwarz inequality is Cauchy-Schwarz inequality for $L^{2}$ .

[F4]

Bochner L2 space for a finite product-law input identifies $H$ with strongly measurable square-integrable maps modulo almost-everywhere equality.

Proof

1.1F1F2givenalgebra

Choose finite-valued measurable simple maps $U_{n}, W_{n}$ converging respectively to $U, W$ almost everywhere, using [F2]. On every intersection of their finite partition sets, $y \mapsto a_{y} (U_{n} (y), W_{n} (y))$ is an affine continuous scalar function of $y$ by the displayed coefficient formula in [F1]. It is therefore measurable. On the common full-measure set where both approximations converge, continuity of the bounded bilinear form gives convergence to $a_{y} (U (y), W (y))$ . Setting the scalar function to zero on the common measurable exceptional null set gives a measurable version; an arbitrary modification on that set need not be measurable. The integral uses this version, as in [F4].

1.2F1F2given

For a finite-valued simple $W_{n}$ converging almost everywhere to $W$ , the scalar function $F (y) (W_{n} (y))$ is measurable on each partition piece because $F (y)$ is affine continuous into the finite-dimensional span of $f_{0}, \dots, f_{d}$ . Its pointwise limit on the full-measure convergence set is $F (y) (W (y))$ , so setting the limit to zero on the measurable exceptional null set gives a measurable version in the convention [F4]. Since $∣ y_{j} ∣ \leq R_{j}$ , $∥ F (y) ∥_{V^{*}} \leq ∥ f_{0} ∥_{V^{*}} + \sum_{j} R_{j} ∥ f_{j} ∥_{V^{*}}$ ; hence $F \in L_{ρ}^{2} (V^{*})$ .

2.1F1F3F4step 1.1

The pointwise bound [F1] gives $∣ a_{y} (U (y), W (y)) ∣ \leq a^{*} ∥ U (y) ∥_{V} ∥ W (y) ∥_{V}$ . Both norm functions are square-integrable, so [F3] makes this majorant integrable. The form integral is finite and depends only on the almost-everywhere classes by [F4].

2.2F3step 1.2

Pointwise duality gives $∣ F (y) (W (y)) ∣ \leq ∥ F (y) ∥_{V^{*}} ∥ W (y) ∥_{V}$ . Apply [F3] to obtain the stated integrability and functional bound.

3.1F1F3F4step 2.1∎

Integrating the pointwise coercivity bound [F1] gives $A (U, U) \geq a_{*} \int ∥ U (y) ∥_{V}^{2} d ρ = a_{*} ∥ U ∥_{H}^{2}$ . Integrating the pointwise continuity bound and applying [F3] as in step 2.1 gives $∣ A (U, W) ∣ \leq a^{*} ∥ U ∥_{H} ∥ W ∥_{H}$ . These estimates prove the remaining assertions.

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Independent polynomial-chaos tensor products are orthonormal

Statement

Let $ρ = \otimes_{j = 1}^{d} ρ_{j}$ and form $ψ_{α}$ from Finite orthonormal polynomial tensor basis for indices in the admissible degree box. Then each $ψ_{α}$ belongs to $L^{2} (ρ)$ and $\int_{Γ} ψ_{α} (y) ψ_{β} (y) d ρ (y) = {\begin{cases} 1, & α = β, \\ 0, & α \neq β . \end{cases}$ In particular, the family indexed by any finite subset $Λ$ of the box is orthonormal; when $0 \in Λ$ , it includes the constant mode $ψ_{0} = 1$ .

Facts & Assumptions

Given: Independent real inputs with product law $ρ$ , admissible one-dimensional moment data, and indices $α, β$ in the finite degree box.

[F1]

Positive moment Gram matrices produce orthonormal polynomial families gives $\int ϕ_{j, k} ϕ_{j, l} d ρ_{j} = δ_{k l}$ and $ϕ_{j, 0} = 1$ .

[F2]

Independent random elements have product joint law identifies the joint law with $\otimes_{j} ρ_{j}$ .

[F3]

Tonelli and Fubini hold for the uncompleted product of sigma-finite measure spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product); finite iteration gives factorization for a finite product.

[F4]

The product law on $Γ$ is the full-mass restriction specified in Finite orthonormal polynomial tensor basis, so Tonelli and Fubini apply to the product of the restricted marginal laws.

Proof

1.1F1F2F3F4given

For $α, β$ in the admissible box set $g_{j} (y_{j}) = ϕ_{j, α_{j}} (y_{j}) ϕ_{j, β_{j}} (y_{j})$ . Each $g_{j}$ is integrable because it is a polynomial of degree at most $2 p_{j}$ and the $2 p_{j}$ moment is finite. By [F4], the restricted measure on $Γ$ is the product of the restricted marginal laws. Repeated Tonelli [F3] first gives $\int_{Γ} \prod_{j} ∣ g_{j} (y_{j}) ∣ d ρ = \prod_{j = 1}^{d} \int ∣ g_{j} ∣ d ρ_{j} < \infty$ ; repeated Fubini [F3] then gives $\int_{Γ} \prod_{j} g_{j} (y_{j}) d ρ = \prod_{j = 1}^{d} \int g_{j} d ρ_{j}$ . The product-law statement [F2] identifies the ambient finite product measure with the joint law, and [F4] restricts it to $Γ$ .

2.1F1step 1.1algebra∎

The factorization in step 1.1 and univariate orthonormality [F1] give $\int_{Γ} ψ_{α} ψ_{β} d ρ = \prod_{j = 1}^{d} δ_{α_{j} β_{j}}$ . This product is one exactly when every coordinate agrees, and zero otherwise; with $β = α$ it also shows $ψ_{α} \in L^{2} (ρ)$ . Multiplying $ϕ_{j, 0} = 1$ over the finitely many coordinates gives $ψ_{0} = 1$ , so whenever $0 \in Λ$ the finite family includes this constant mode.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Quadrature estimates of polynomial-chaos coefficients

Definition

Let $Λ$ be a finite subset of the admissible degree box, with $(ψ_{α})_{α \in Λ}$ as in Finite orthonormal polynomial tensor basis, and let $Q$ be any declared finite cubature functional $Q (g) = \sum_{q = 1}^{J} w_{q} g (y^{(q)})$ whose nodes lie in the domain of evaluation of $g$ . For an evaluable real function $u$ , define the quadrature coefficient and the resulting polynomial by ${\hat{c}}_{α} : = Q (u ψ_{α}), {\hat{P}}_{Λ} u : = \sum_{α \in Λ} {\hat{c}}_{α} ψ_{α} .$ When $u$ represents an element of $L^{2} (ρ)$ , its exact coefficients $c_{α}$ and projection $P_{Λ} u$ are those of Finite polynomial-chaos orthogonal projection. The quadrature quantities use the specified node values of $u$ and need not be invariant under changes on null sets. This definition gives no accuracy claim for a general cubature rule.

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Christoffel sampling is normalized and its weighted Gram has mean identity

Statement

Let $Λ$ and $K_{Λ}$ be as in Christoffel-weighted least-squares polynomial collocation, with $N = ∣ Λ ∣ \geq 1$ . Then $K_{Λ} \geq 1$ pointwise, the measure $d ν = (K_{Λ} / N) d ρ$ is a probability measure, and the random matrix $A (X) = \frac{N}{K_{Λ} (X)} ψ (X) ψ (X)^{T}, X \sim ν,$ satisfies $A ⪰ 0$ , $E_{ν} A = I_{N}$ , and $∥ A ∥_{o p} \leq N$ almost surely.

Facts & Assumptions

Given: A finite real orthonormal polynomial family indexed by $Λ$ , with $0 \in Λ$ , and its Christoffel kernel.

[F1]

Christoffel-weighted least-squares polynomial collocation defines $K_{Λ} = \sum_{α \in Λ} ∣ ψ_{α} ∣^{2}$ and the rank-one matrix $A$ .

[F2]

Independent polynomial-chaos tensor products are orthonormal gives $\int ψ_{α} ψ_{β} d ρ = δ_{α β}$ and $ψ_{0} = 1$ .

[F3]

Integration is linear on integrable functions (The Lebesgue integral is linear on $L^{1} (μ)$ ); the finite sums defining $K_{Λ}$ can therefore be integrated term by term.

[F4]

Cauchy-Schwarz inequality for $L^{2}$ makes each product $ψ_{α} ψ_{β}$ integrable. Since $d ν = (K_{Λ} / N) d ρ$ , integration against $ν$ is integration against that density for nonnegative measurable functions, and for signed functions once absolute integrability is established.

Proof

1.1F1F2F3algebra

Since $0 \in Λ$ and $ψ_{0} = 1$ , $K_{Λ} (y) \geq 1$ . By [F2]–[F3], $\int K_{Λ} d ρ = \sum_{α \in Λ} \int ∣ ψ_{α} ∣^{2} d ρ = N$ . Therefore $d ν = (K_{Λ} / N) d ρ$ is a probability measure and the denominator in [F1] is positive everywhere.

2.1F1F2F4step 1.1algebra∎

For each pair $α, β$ , [F4] and $\int ∣ ψ_{α} ψ_{β} ∣ d ρ \leq 1$ show $\int ∣ A_{α β} ∣ d ν = \int ∣ ψ_{α} ψ_{β} ∣ d ρ \leq 1$ , so the entry is integrable under $ν$ . Using the signed density identity in [F4], its expectation is $\int_{Γ} ψ_{α} (y) ψ_{β} (y) d ρ (y) = δ_{α β}$ by [F2], hence $E_{ν} A = I_{N}$ . The rank-one outer product is positive semidefinite and has operator norm $N ∥ ψ (y) ∥_{2}^{2} / K_{Λ} (y) = N$ , proving the remaining assertions.

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Affine random diffusion has a measurable Bochner solution

Statement

Under the hypotheses of Uniformly elliptic affine random diffusion model, for each $y \in Γ$ there is a unique $u (y) \in V$ satisfying the pointwise weak problem. The map $y \mapsto u (y)$ is Lipschitz on the bounded parameter set, strongly measurable, and belongs to $L_{ρ}^{2} (V)$ . It is also the unique solution of the Bochner weak equation $A (u, W) = F (W) (W \in L_{ρ}^{2} (V)) .$

Facts & Assumptions

Given: A bounded product parameter set, affine coefficient and forcing, and constants $0 < a_{*} \leq a^{*}$ as in Uniformly elliptic affine random diffusion model.

[F1]

The zero-boundary Sobolev energy space is a separable Hilbert space supplies the separable real Hilbert structure on $V = H_{0}^{1} (D)$ under Countable Choice.

[F2]

Pointwise $a_{y}$ is a bounded real bilinear form, with $∣ a_{y} (v, w) ∣ \leq a^{*} ∥ v ∥_{V} ∥ w ∥_{V}$ and $a_{y} (v, v) \geq a_{*} ∥ v ∥_{V}^{2}$ , by the coefficient bounds in Uniformly elliptic affine random diffusion model.

[F3]

The real bilinear Lax–Milgram theorem under Countable Choice gives a unique solution and $∥ u (y) ∥_{V} \leq ∥ F (y) ∥_{V^{*}} / a_{*}$ (The Lax--Milgram theorem, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F4]

A Borel map into the separable Banach space $V$ is strongly measurable (A Borel map into a separable Banach space is strongly measurable).

[F5]

Affine diffusion forms are well-defined on Bochner L2 supplies measurability, integrability, continuity, and coercivity of $A$ and boundedness of $F$ on $L_{ρ}^{2} (V)$ .

[F6]

$L_{ρ}^{2} (V)$ is a Hilbert space and finite-valued simple functions are dense (Finite-input Bochner L2 of a separable Hilbert space is Hilbert).

Proof

1.1F1F2F3given

For each $y \in Γ$ , [F2] verifies the boundedness and coercivity hypotheses of [F3], while $F (y) = f_{0} + \sum_{j} y_{j} f_{j}$ is a bounded functional. Lax–Milgram gives a unique $u (y) \in V$ and $∥ u (y) ∥_{V} \leq F_{\max} / a_{*}$ , where $F_{\max} : = \sup_{y \in Γ} ∥ F (y) ∥_{V^{*}} < \infty$ because $Γ$ is bounded.

2.1F2step 1.1algebra

For $y, z \in Γ$ , put $d = u (y) - u (z)$ and subtract the weak equations: $a_{y} (d, v) = (F (y) - F (z)) (v) + (a_{z} - a_{y}) (u (z), v) .$ The affine formulas give $∥ F (y) - F (z) ∥_{V^{*}} \leq C_{F} ∥ y - z ∥_{1}$ and $∣ a_{z} (v, w) - a_{y} (v, w) ∣ \leq C_{a} ∥ y - z ∥_{1} ∥ v ∥_{V} ∥ w ∥_{V}$ , where $C_{F} = \sum_{j} ∥ f_{j} ∥_{V^{*}}$ and $C_{a} = \sum_{j} ∥ a_{j} ∥_{L^{\infty} (D)}$ . Set $v = d$ and use [F2] and the bound $∥ u (z) ∥_{V} \leq F_{\max} / a_{*}$ from step 1.1. If $d \neq 0$ , divide the resulting coercive inequality by $∥ d ∥_{V}$ ; if $d = 0$ the conclusion is immediate. This proves $∥ u (y) - u (z) ∥_{V} \leq a_{*}^{- 1} (C_{F} + C_{a} F_{\max} / a_{*}) ∥ y - z ∥_{1}$ . Thus $u$ is Lipschitz and Borel measurable.

2.2F1F4step 1.1

Since $V$ is separable, [F4] makes $u$ strongly measurable. The uniform bound in step 1.1 and $ρ (Γ) = 1$ give $\int_{Γ} ∥ u (y) ∥_{V}^{2} d ρ (y) \leq (F_{\max} / a_{*})^{2} < \infty$ , so $u \in L_{ρ}^{2} (V)$ .

3.1F5step 2.2

For a finite-valued simple test $W (y) = \sum_{i = 1}^{m} 1_{E_{i}} (y) v_{i}$ , integrate the pointwise weak identity for each $v_{i}$ over $E_{i}$ and sum; [F5] ensures the scalar integrands are measurable and integrable. This gives $A (u, W) = F (W)$ .

4.1F5F6step 3.1algebra∎

The bounds in [F5] make both sides continuous on $L_{ρ}^{2} (V)$ . Finite-valued simple tests are dense by [F6], so step 3.1 extends to every $W$ . If two Bochner solutions existed, testing their difference against itself and using the coercivity bound in [F5] would force their difference to be zero.

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Finite chaos projection, Parseval identity, mean, and variance

Statement

Let $Λ$ be finite, contain $0$ , and let the real basis $(ψ_{α})_{α \in Λ}$ be as in Finite orthonormal polynomial tensor basis. For real $u \in L^{2} (ρ)$ set $c_{α} = ⟨ u, ψ_{α} ⟩$ and $P_{Λ} u = \sum_{α \in Λ} c_{α} ψ_{α}$ as in Finite polynomial-chaos orthogonal projection. Then $∥ u ∥_{L^{2} (ρ)}^{2} = \sum_{α \in Λ} c_{α}^{2} + ∥ u - P_{Λ} u ∥_{L^{2} (ρ)}^{2} .$ The residual is orthogonal to $P_{Λ}$ , $E [u] = c_{0}$ , and $Var (u) = \sum_{α \in Λ ∖ {0}} c_{α}^{2} + ∥ u - P_{Λ} u ∥_{L^{2} (ρ)}^{2} .$ Here $Var (u) = E [(u - E u)^{2}]$ ; no completeness of the polynomial family is assumed.

Facts & Assumptions

Given: A finite real orthonormal polynomial family containing $ψ_{0} = 1$ and $u \in L^{2} (ρ)$ .

[F1]

Finite polynomial-chaos orthogonal projection defines $c_{α}$ and $P_{Λ} u$ by the displayed finite sum.

[F2]

Independent polynomial-chaos tensor products are orthonormal gives $⟨ ψ_{α}, ψ_{β} ⟩ = δ_{α β}$ and $ψ_{0} = 1$ .

[F3]

Finite polynomial-chaos orthogonal projection establishes the choice-free integral inner product and its norm-square identity; bilinearity and symmetry give the finite orthogonal norm identities by expansion, without completeness.

Proof

1.1F1F2algebra

For each $β \in Λ$ , bilinearity and [F2] give $⟨ u - P_{Λ} u, ψ_{β} ⟩ = c_{β} - \sum_{α \in Λ} c_{α} δ_{α β} = 0$ . Thus the residual is orthogonal to every basis vector and hence to their finite span.

2.1F2F3step 1.1algebra

Write $u = P_{Λ} u + (u - P_{Λ} u)$ . By step 1.1 the two terms are orthogonal, so bilinearity and symmetry from [F3] give $∥ u ∥_{2}^{2} = ∥ P_{Λ} u ∥_{2}^{2} + ∥ u - P_{Λ} u ∥_{2}^{2}$ . Expanding the finite sum with [F2] gives $∥ P_{Λ} u ∥_{2}^{2} = \sum_{α, β \in Λ} c_{α} c_{β} δ_{α β} = \sum_{α \in Λ} c_{α}^{2}$ . This proves the norm identity directly from the integral pairing.

3.1F1F2F3step 1.1algebra∎

Since $ψ_{0} = 1$ , $c_{0} = ⟨ u, 1 ⟩ = E [u]$ . The centered variable decomposes as $u - E [u] = \sum_{α \in Λ ∖ {0}} c_{α} ψ_{α} + (u - P_{Λ} u)$ . The two summands are orthogonal by step 1.1, and expansion of the integral norm square as in step 2.1, followed by [F2], gives the displayed variance identity.

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Tensor products are dense in Bochner L2 under scalar and spatial density

Statement

Let $S$ be the scalar linear span of a declared polynomial-chaos family in $L^{2} (ρ)$ and let $V_{0}$ be a linear subspace dense in the separable real Hilbert space $V$ . Assume separately that $S$ is dense in $L^{2} (ρ)$ ; this law-specific completeness hypothesis is not inferred from finite moments. Then the algebraic tensor span $span {s (\cdot) v : s \in S, v \in V_{0}}$ is dense in $L_{ρ}^{2} (V)$ .

Facts & Assumptions

Given: A finite product probability space, the stated scalar density hypothesis, and a dense linear spatial subspace $V_{0} \subseteq V$ .

[F1]

Bochner L2 space for a finite product-law input defines $L_{ρ}^{2} (V)$ using strongly measurable square-integrable maps.

[F2]

Dominated convergence gives convergence of scalar integrals for pointwise almost-everywhere convergence under an integrable measurable majorant.

[F3]

Scalar density of $S$ in $L^{2} (ρ)$ is an explicit hypothesis in Given, not a conclusion of the finite-basis orthonormality theorem.

[F4]

Density of $V_{0}$ in $V$ is an explicit hypothesis in Given.

[F5]

Strongly measurable Banach-valued function characterizes strong measurability through simple-map approximation; a scalar measurable function times a fixed vector is strongly measurable.

Proof

1.1F1F2F5givenchoose

Fix $U \in L_{ρ}^{2} (V)$ and $ε > 0$ . By [F1] and [F5], choose a simple approximation sequence and replace it and $U$ by zero on their measurable null exceptional set. Write $S_{n}$ for the resulting simple maps, which converge everywhere to the Borel representative $U$ . Set $T_{n} = S_{n} 1_{{∥ S_{n} ∥_{V} \leq 2 ∥ U ∥_{V}}}$ . The truncation sets are measurable and $T_{n}$ is finite-valued simple. At $U = 0$ it is zero; at every other point it eventually equals $S_{n}$ , so $T_{n} \to U$ . Since $∥ T_{n} - U ∥_{V}^{2} \leq 9 ∥ U ∥_{V}^{2}$ , [F2] gives $∥ T_{n} - U ∥_{L_{ρ}^{2} (V)} \to 0$ . Choose one such map $S = \sum_{k = 1}^{m} 1_{E_{k}} v_{k}$ with error below $ε / 2$ , taking $m \geq 1$ by allowing a zero value. It remains to approximate its finitely many terms with total error below $ε / 2$ .

2.1F1F3F4F5step 1.1algebra

For each $k$ , use [F3] and [F4] to choose $s_{k} \in S$ and $w_{k} \in V_{0}$ with $∥ 1_{E_{k}} - s_{k} ∥_{L^{2} (ρ)} ∥ v_{k} ∥_{V} + ∥ s_{k} ∥_{L^{2} (ρ)} ∥ v_{k} - w_{k} ∥_{V} < ε / (2 m)$ . Such choices are possible by first making the scalar error arbitrarily small, which bounds $∥ s_{k} ∥_{2} \leq ∥ 1_{E_{k}} ∥_{2} + ∥ 1_{E_{k}} - s_{k} ∥_{2}$ , and then making the vector error small. The product norm estimate $∥ 1_{E_{k}} v_{k} - s_{k} w_{k} ∥_{L_{ρ}^{2} (V)} \leq ∥ 1_{E_{k}} - s_{k} ∥_{2} ∥ v_{k} ∥_{V} + ∥ s_{k} ∥_{2} ∥ v_{k} - w_{k} ∥_{V}$ follows pointwise from the triangle inequality and then from scalar square-integral estimation. Each product is strongly measurable by [F5], so these are legitimate Bochner-space norms.

3.1step 1.1step 2.1algebra∎

The finite sum $\sum_{k = 1}^{m} s_{k} w_{k}$ belongs to the displayed algebraic tensor span, and the triangle inequality with step 2.1 gives distance less than $ε / 2$ from $S$ . Together with step 1.1 its distance from $U$ is less than $ε$ . Since $U$ and $ε$ were arbitrary, the span is dense.

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Projection and quadrature errors split in L2

Statement

Let $u \in L^{2} (ρ)$ be real, choose a representative $\tilde{u}$ defined at every node of the finite cubature rule used below, let $Λ$ be finite, and use the exact and quadrature coefficients from Finite polynomial-chaos orthogonal projection and Quadrature estimates of polynomial-chaos coefficients, with $\tilde{u}$ as the evaluable input to that estimator. Then $∥ u - {\hat{P}}_{Λ} u ∥_{L^{2} (ρ)}^{2} = ∥ u - P_{Λ} u ∥_{L^{2} (ρ)}^{2} + \sum_{α \in Λ} ∣ {\hat{c}}_{α} - c_{α} ∣^{2} .$ Since the finite sum equals $∥ {\hat{P}}_{Λ} u - P_{Λ} u ∥_{L^{2} (ρ)}^{2}$ , the triangle inequality also gives $∥ u - {\hat{P}}_{Λ} u ∥_{L^{2} (ρ)} \leq ∥ u - P_{Λ} u ∥_{L^{2} (ρ)} + ∥ {\hat{P}}_{Λ} u - P_{Λ} u ∥_{L^{2} (ρ)} .$ Thus coefficient quadrature error and finite-space projection error remain separate even when no completeness statement is available.

Facts & Assumptions

Given: A finite real orthonormal family, $u \in L^{2} (ρ)$ , and finite quadrature values ${\hat{c}}_{α}$ .

[F1]

Quadrature estimates of polynomial-chaos coefficients defines ${\hat{P}}_{Λ} u$ using the finite basis sum.

[F2]

Finite polynomial-chaos orthogonal projection defines $c_{α} = ⟨ u, ψ_{α} ⟩$ and $P_{Λ} u = \sum_{α \in Λ} c_{α} ψ_{α}$ .

[F3]

Independent polynomial-chaos tensor products are orthonormal gives the orthonormality of the finite basis.

[F4]

Finite polynomial-chaos orthogonal projection gives the choice-free bilinear integral pairing inducing the $L^{2}$ norm; the finite orthogonality identities follow by expansion.

[F5]

Cauchy-Schwarz inequality for $L^{2}$ bounds $\int ∣ v w ∣ d ρ$ , and The modulus of an integral is bounded by the integral of the modulus bounds $∣ \int v w d ρ ∣$ by that integral. Expanding $∥ v + w ∥_{2}^{2}$ then gives the norm triangle inequality.

Proof

1.1F1F2F3F4algebra

For each $β \in Λ$ , the coefficient formula [F2], orthonormality [F3], and bilinearity [F4] give $⟨ u - P_{Λ} u, ψ_{β} ⟩ = c_{β} - \sum_{α \in Λ} c_{α} δ_{α β} = 0$ . Thus $u - P_{Λ} u$ is orthogonal to the finite span $P_{Λ}$ , including when $Λ$ is empty. Subtracting the two finite expansions gives ${\hat{P}}_{Λ} u - P_{Λ} u = \sum_{α \in Λ} ({\hat{c}}_{α} - c_{α}) ψ_{α} \in P_{Λ}$ , so this difference is orthogonal to the residual.

2.1F3F4F5step 1.1algebra∎

The identity $u - {\hat{P}}_{Λ} u = (u - P_{Λ} u) - ({\hat{P}}_{Λ} u - P_{Λ} u)$ is an orthogonal sum by step 1.1. Expanding its squared norm using [F4] gives $∥ u - {\hat{P}}_{Λ} u ∥^{2} = ∥ u - P_{Λ} u ∥^{2} + ∥ {\hat{P}}_{Λ} u - P_{Λ} u ∥^{2}$ ; orthonormality [F3] identifies the second term with $\sum_{α \in Λ} ∣ {\hat{c}}_{α} - c_{α} ∣^{2}$ . To obtain the final inequality, use [F5] to expand $∥ v + w ∥_{2}^{2}$ and deduce $∥ v + w ∥_{2} \leq ∥ v ∥_{2} + ∥ w ∥_{2}$ for $v = u - P_{Λ} u$ and $w = P_{Λ} u - {\hat{P}}_{Λ} u$ .

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Bounded-Christoffel sampling stabilizes the empirical Gram matrix

Statement

Let $Λ$ , $N = ∣ Λ ∣ \geq 1$ , and the independent Christoffel samples $X_{1}, \dots, X_{M}$ be as in Christoffel-weighted least-squares polynomial collocation. Let $G$ be its weighted empirical Gram matrix. For $0 < δ < 1$ , if $M \geq 12 N \log (2 N / δ),$ then $P {spec (G) \subseteq [1 / 2, 3 / 2]} \geq 1 - δ .$

Facts & Assumptions

Given: A finite real orthonormal basis of size $N \geq 1$ , independent Christoffel samples, and $0 < δ < 1$ .

[F1]

Christoffel sampling is normalized and its weighted Gram has mean identity gives rank-one matrices $A_{i} ⪰ 0$ with $E A_{i} = I_{N}$ and $∥ A_{i} ∥_{o p} \leq N$ .

[F2]

Christoffel-weighted least-squares polynomial collocation specifies independent samples, hence the matrices formed from them are independent.

[F3]

Bounded independent matrix Bernstein bound from the trace exponential gives, for centered independent Hermitian summands bounded by $R$ with variance bound $v$ , the two-sided tail $P {∥ \sum_{i} B_{i} ∥_{o p} \geq t} \leq 2 N \exp (- t^{2} / (2 (M v + R t / 3)))$ .

[F4]

Christoffel-weighted least-squares polynomial collocation defines $G = M^{- 1} \sum_{i} A_{i}$ .

Proof

1.1F1F2algebra

Put $B_{i} = A_{i} - I_{N}$ . Then $E B_{i} = 0$ . Since $0 ⪯ A_{i} ⪯ N I_{N}$ , spectral calculus gives $A_{i}^{2} ⪯ N A_{i}$ , and hence $0 ⪯ E B_{i}^{2} = E A_{i}^{2} - I_{N} ⪯ (N - 1) I_{N} ⪯ N I_{N}$ . Also every eigenvalue of $A_{i}$ lies in $[0, N]$ , so $∥ B_{i} ∥_{o p} \leq N$ . These are the hypotheses of [F2] with $R = v = N$ .

2.1F2F3F4step 1.1algebra

Apply [F3] to $Y = \sum_{i = 1}^{M} B_{i}$ with $t = M / 2$ . Direct substitution gives $P {∥ G - I_{N} ∥_{o p} > 1 / 2} \leq 2 N \exp (- 3 M / (28 N))$ , because $G - I_{N} = Y / M$ and $(M / 2)^{2} / [2 (M N + N (M / 2) / 3)] = 3 M / (28 N)$ .

3.1givenstep 2.1algebra∎

The sample-size condition makes the last bound at most $2 N \exp [- (9 / 7) \log (2 N / δ)] = δ (δ / (2 N))^{2 / 7} \leq δ$ , since $0 < δ / (2 N) < 1$ . On the complementary event, for every unit vector $x$ , $∣ x^{T} (G - I_{N}) x ∣ \leq ∥ G - I_{N} ∥_{o p} \leq 1 / 2$ ; the Rayleigh quotient characterization therefore puts every eigenvalue of $G$ in $[1 / 2, 3 / 2]$ .

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Stochastic Galerkin coercivity and Cea quasi-optimality

Statement

Assume Countable Choice and the affine diffusion hypotheses of Uniformly elliptic affine random diffusion model. Let $p$ be the selected admissible cutoff vector of the finite product-law input, and let $Λ \subseteq {α \in N_{0}^{d} : 0 \leq α_{j} \leq p_{j} for all j}$ be finite. Let $V_{h} \subset V$ be any finite-dimensional conforming spatial subspace. Define the stochastic Galerkin space $V_{Λ, h} : = span {ψ_{α} (\cdot) v : α \in Λ, v \in V_{h}} \subset L_{ρ}^{2} (V) .$ The Bochner form is continuous with constant $a^{*}$ and coercive with constant $a_{*}$ . There is a unique $U_{Λ, h} \in V_{Λ, h}$ satisfying $A (U_{Λ, h}, W_{h}) = F (W_{h}) (W_{h} \in V_{Λ, h}),$ and, for the exact solution $u$ of Affine random diffusion has a measurable Bochner solution, $∥ u - U_{Λ, h} ∥_{L_{ρ}^{2} (V)} \leq \frac{a^{*}}{a_{*}} \inf_{W_{h} \in V_{Λ, h}} ∥ u - W_{h} ∥_{L_{ρ}^{2} (V)} .$ No stochastic-basis completeness or spectral-rate claim is part of this finite-dimensional estimate.

Facts & Assumptions

Given: The real affine diffusion model, Countable Choice, a finite chaos index set contained in the selected admissible cutoff box, and a finite-dimensional conforming spatial subspace.

[F1]

Affine diffusion forms are well-defined on Bochner L2 gives a well-defined bounded coercive real bilinear form with constants $a^{*}$ and $a_{*}$ and a bounded linear functional $F$ .

[F2]

Affine random diffusion has a measurable Bochner solution gives the exact $u \in L_{ρ}^{2} (V)$ satisfying $A (u, W) = F (W)$ for all $W$ .

[F3]

Finite-input Bochner L2 of a separable Hilbert space is Hilbert makes $L_{ρ}^{2} (V)$ a real Hilbert space.

[F4]

Independent polynomial-chaos tensor products are orthonormal gives $∥ ψ_{α} ∥_{L^{2} (ρ)} = 1$ , so $ψ_{α} v \in L_{ρ}^{2} (V)$ for every $v \in V$ .

[F5]

Strongly measurable Banach-valued function characterizes strong measurability by almost-everywhere limits of finite-valued measurable simple maps.

[F6]

Céa quasi-optimality for coercive conforming Galerkin problems gives existence, uniqueness, and the $M / α$ error bound for finite-dimensional conforming subspaces under Countable Choice.

Proof

1.1F3F4F5given

Each $ψ_{α}$ is a continuous scalar polynomial. Approximate it pointwise by finite-valued measurable simple real functions; multiplying those functions by a fixed $v \in V$ gives simple $V$ -valued maps converging to $ψ_{α} v$ . Thus $ψ_{α} v$ is strongly measurable by [F5] and has Bochner norm $∥ ψ_{α} ∥_{L^{2} (ρ)} ∥ v ∥_{V} = ∥ v ∥_{V}$ by [F4]. A finite basis of $V_{h}$ and finite $Λ$ therefore generate $V_{Λ, h}$ from finitely many members of $L_{ρ}^{2} (V)$ , making it finite-dimensional and conforming.

2.1F1F2F3F6step 1.1given∎

The model form and load satisfy the boundedness and coercivity hypotheses in [F6] by [F1], and the ambient space is Hilbert by [F3]. The finite-dimensional subspace condition is step 1.1. Apply [F6] with $M = a^{*}$ and $α = a_{*}$ ; it gives the unique discrete solution and $∥ u - U_{Λ, h} ∥ \leq (a^{*} / a_{*}) \inf_{W_{h} \in V_{Λ, h}} ∥ u - W_{h} ∥$ . This is the claimed estimate.

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Smolyak mixed-regularity bounds control chaos coefficient error

Statement

Assume $ρ$ is uniform product probability on $[0, 1]^{d}$ , so its integral is $I_{d}$ in Explicit work and error rate for dyadic Simpson Smolyak cubature. Let $p$ be an admissible cutoff vector for this law and let $Λ$ be a finite subset of its admissible degree box, with basis from Finite orthonormal polynomial tensor basis. Let $Q_{Λ_{n}}$ be that theorem's dyadic Simpson Smolyak rule, with $n \geq 2 d$ . Let $u \in L^{2} (ρ)$ be real and fix a representative $\tilde{u}$ defined at every node of $Q_{Λ_{n}}$ . For every $α \in Λ$ , suppose the coefficient integrand $g_{α} = \tilde{u} ψ_{α}$ has the continuous mixed fourth derivatives and finite bounds $M_{α, S}$ required by Mixed-derivative bounds for dyadic Simpson details. Define $E_{α, n} : = C_{d} ∥ g_{α} ∥_{m i x, d} W_{n, d}^{- 4} (\log (2 + W_{n, d}))^{5 (d - 1)},$ with the constants and mixed norm of the sparse-grid theorem. Then ${(\sum_{α \in Λ} ∣ Q_{Λ_{n}} (u ψ_{α}) - ⟨ u, ψ_{α} ⟩ ∣^{2})}^{1 / 2} \leq {(\sum_{α \in Λ} E_{α, n}^{2})}^{1 / 2} .$ The error-split theorem consequently bounds the L2 coefficient contribution by the right-hand side; this claim is restricted to the stated cube law and mixed regularity.

Facts & Assumptions

Given: Uniform product measure on $[0, 1]^{d}$ , a finite chaos basis, and the stated mixed-derivative hypotheses for each coefficient integrand.

[F1]

Explicit work and error rate for dyadic Simpson Smolyak cubature bounds each scalar Smolyak integration error by $E_{α, n}$ under its displayed hypotheses.

[F2]

Mixed-derivative bounds for dyadic Simpson details supplies the exact mixed-derivative and endpoint convention required by [F1].

[F3]

Smolyak cubature on a finite downward-closed index set defines $Q_{Λ_{n}}$ as a finite cubature functional.

[F4]

Projection and quadrature errors split in L2 identifies the Euclidean coefficient-error norm as the L2 quadrature contribution.

Proof

1.1F1F2F3given

For each $α$ , apply [F1] to $g_{α} = \tilde{u} ψ_{α}$ . Its assumptions are exactly the mixed-derivative bounds and cube domain in Given, with [F2] supplying the theorem's specified mixed norm. Thus $∣ Q_{Λ_{n}} (g_{α}) - \int g_{α} d ρ ∣ \leq E_{α, n}$ . Since $\tilde{u}$ represents $u$ , the integral equals $⟨ u, ψ_{α} ⟩$ by the uniform product law.

2.1step 1.1algebra

Square the finitely many nonnegative inequalities from step 1.1, sum over $α$ , and take square roots; monotonicity of the square root gives the stated Euclidean bound.

3.1F4step 2.1∎

Apply [F4] with these coefficient errors. Its exact orthogonal error decomposition leaves the projection residual unchanged and adds the bounded Euclidean coefficient term, which proves the final assertion.

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Conditional spatial finite-element rate for stochastic Galerkin

Statement

Assume the hypotheses of the stochastic Galerkin theorem with $D \subset R^{d}$ , $d \geq 2$ , and suppose the exact solution has the exact finite expansion $u (y) = \sum_{α \in Λ} u_{α} ψ_{α} (y), u_{α} \in H^{r} (D) \cap H_{0}^{1} (D),$ for integers $k \geq 0$ and $1 \leq ℓ \leq r \leq k + 1$ . Choose a fixed reference finite element whose polynomial space contains $P_{k}$ and whose unisolvent degrees of freedom are bounded on $W^{ℓ, 2}$ of the reference cell; let its reference interpolant reproduce $P_{k}$ and define local interpolants by affine pullback as in Local interpolation estimates on shape-regular affine meshes. Let ${T_{h}}$ be a shape-regular family of matching affine simplicial meshes as in Matching affine simplicial meshes and shape regularity, with maximum cell diameter $h$ , and assume for each mesh that a globally conforming spatial interpolant $I_{h}$ maps these coefficients into its finite-element space, restricts cellwise to those local interpolants, and preserves the zero boundary condition. These are the generic bounded-degree-of-freedom hypotheses of the local estimate. Then for the stochastic Galerkin solution $U_{Λ, h}$ , $∥ u - U_{Λ, h} ∥_{L_{ρ}^{2} (H_{0}^{1} (D))} \leq \frac{a^{*}}{a_{*}} C h^{r - 1} {(\sum_{α \in Λ} ∣ u_{α} ∣_{H^{r} (D)}^{2})}^{1 / 2} .$ The regularity and finite-expansion assumptions are explicit; this is not a general spatial rate for arbitrary random solutions.

Facts & Assumptions

Given: Uniform ellipticity, the exact finite expansion and coefficient regularity above, a shape-regular mesh, and a globally conforming interpolant with the stated cell restrictions.

[F1]

Stochastic Galerkin coercivity and Cea quasi-optimality gives the Galerkin error bounded by $(a^{*} / a_{*})$ times best approximation in the tensor space.

[F2]

Local interpolation estimates on shape-regular affine meshes with $p = 2$ , $m = 1$ , the fixed reference space containing $P_{k}$ , bounded degrees of freedom on $W^{ℓ, 2}$ , and $ℓ \leq r \leq k + 1$ gives $∣ v - I_{K} v ∣_{W^{1, 2} (K)} \leq C h_{K}^{r - 1} ∣ v ∣_{W^{r, 2} (K)}$ . The generic theorem uses the boundedness hypothesis directly; the corollary does not infer nodal boundedness from point evaluation.

[F3]

Independent polynomial-chaos tensor products are orthonormal gives the orthonormal finite random basis.

Proof

1.1F3given

Define $W_{h} (y) = \sum_{α \in Λ} ψ_{α} (y) I_{h} u_{α}$ . Each interpolated coefficient belongs to the spatial finite-element subspace, so $W_{h} \in V_{Λ, h}$ . By [F3], orthogonality in the random variable gives $∥ u - W_{h} ∥_{L_{ρ}^{2} (V)}^{2} = \sum_{α \in Λ} ∥ u_{α} - I_{h} u_{α} ∥_{V}^{2} .$

1.2F2givenalgebra

For each coefficient and cell $K$ , [F2] gives $∣ u_{α} - I_{K} u_{α} ∣_{W^{1, 2} (K)} \leq C h_{K}^{r - 1} ∣ u_{α} ∣_{W^{r, 2} (K)}$ . Square, sum over the mesh cells, and use $h_{K} \leq h$ to obtain $∥ u_{α} - I_{h} u_{α} ∥_{V} \leq C h^{r - 1} ∣ u_{α} ∣_{H^{r} (D)}$ . This uses the assumed cellwise conformity and disjoint cell interiors.

2.1F1step 1.1step 1.2algebra∎

Square the bounds from step 1.2, sum over the finite $Λ$ , and use step 1.1 to get $∥ u - W_{h} ∥_{L_{ρ}^{2} (V)} \leq C h^{r - 1} (\sum_{α} ∣ u_{α} ∣_{H^{r} (D)}^{2})^{1 / 2}$ . Then [F1] bounds the Galerkin error by $(a^{*} / a_{*})$ times this particular approximation error, proving the claim.

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Weighted least squares is stable on the good Gram event

Statement

Let real $u \in L^{2} (ρ)$ have a representative evaluable at the Christoffel samples, and let $\hat{u}$ be the accepted weighted least-squares fit from Christoffel-weighted least-squares polynomial collocation. On the event $spec (G) \subseteq [1 / 2, 3 / 2]$ , the normal equations have a unique solution, and $\hat{u}$ is the unique minimizer over $p \in P_{Λ}$ of the weighted empirical objective $J_{M} (p) : = \frac{1}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} ∣ u (X_{i}) - p (X_{i}) ∣^{2} .$ Moreover, $∥ \hat{u} - P_{Λ} u ∥_{L^{2} (ρ)} \leq \sqrt{2} {(\frac{1}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} ∣ u (X_{i}) - P_{Λ} u (X_{i}) ∣^{2})}^{1 / 2},$ so $∥ u - \hat{u} ∥_{L^{2} (ρ)}^{2} \leq ∥ u - P_{Λ} u ∥_{L^{2} (ρ)}^{2} + \frac{2}{M} \sum_{i = 1}^{M} \frac{N}{K_{Λ} (X_{i})} ∣ u (X_{i}) - P_{Λ} u (X_{i}) ∣^{2} .$ For $0 < δ < 1$ and $M \geq 12 N \log (2 N / δ)$ , the good event has probability at least $1 - δ$ by Bounded-Christoffel sampling stabilizes the empirical Gram matrix.

Facts & Assumptions

Given: A finite real orthonormal family, $u \in L^{2} (ρ)$ , samples and weights from the Christoffel least-squares algorithm, and the good Gram event.

[F1]

Christoffel-weighted least-squares polynomial collocation defines $G$ , the weighted normal-equation vector, and the accepted fit.

[F2]

Bounded-Christoffel sampling stabilizes the empirical Gram matrix gives probability at least $1 - δ$ for the good Gram event under the stated sample budget.

[F3]

Finite polynomial-chaos orthogonal projection defines $P_{Λ} u$ and its coefficient vector $c$ .

[F4]

Finite chaos projection, Parseval identity, mean, and variance gives $u - P_{Λ} u ⊥ P_{Λ}$ .

[F5]

Independent polynomial-chaos tensor products are orthonormal gives the real orthonormal basis used by the design matrix.

Proof

1.1F1givenalgebra

Put $w_{i} = N / K_{Λ} (X_{i}) > 0$ , and let the $i$ th row of the design matrix $A$ be $\sqrt{w_{i} / M} ψ (X_{i})^{T}$ . Then $A^{T} A = G$ . On the good event, $G$ is positive definite, so the normal equations have the unique solution $\hat{c} = G^{- 1} A^{T} z$ , where $z_{i} = \sqrt{w_{i} / M} u (X_{i})$ .

1.2F1step 1.1algebra

For any coefficient vector $γ$ , the objective at $p = \sum_{α \in Λ} γ_{α} ψ_{α}$ is $J_{M} (p) = ∥ A γ - z ∥_{2}^{2}$ . The normal equations give $A^{T} (A \hat{c} - z) = 0$ . With $d : = γ - \hat{c}$ , expansion yields $J_{M} (p) - J_{M} (\hat{u}) = 2 d^{T} A^{T} (A \hat{c} - z) + d^{T} A^{T} A d = d^{T} G d .$ Positive definiteness of $G$ makes this difference strictly positive whenever $γ \neq \hat{c}$ . Since the basis coefficients uniquely identify elements of $P_{Λ}$ , $\hat{u}$ is the unique minimizer.

2.1F1F3F5step 1.1algebra

Let $c$ be the exact coefficient vector of $P_{Λ} u$ and let $r_{i} = u (X_{i}) - P_{Λ} u (X_{i})$ . Then $z - A c$ has entries $\sqrt{w_{i} / M} r_{i}$ , and the normal equations give $\hat{c} - c = G^{- 1} A^{T} (z - A c)$ . Moreover, $(G^{- 1} A^{T}) (G^{- 1} A^{T})^{T} = G^{- 1} A^{T} A G^{- 1} = G^{- 1}$ . Thus $∥ G^{- 1} A^{T} ∥_{o p}^{2} = λ_{\max} (G^{- 1}) = 1 / λ_{\min} (G) \leq 2$ , yielding the first bound.

3.1F4F5step 2.1algebra

Orthonormality [F5] gives $∥ \hat{u} - P_{Λ} u ∥_{L^{2} (ρ)} = ∥ \hat{c} - c ∥_{2}$ . By [F4], $u - P_{Λ} u$ is orthogonal to this polynomial difference, so Pythagoras gives $∥ u - \hat{u} ∥^{2} = ∥ u - P_{Λ} u ∥^{2} + ∥ \hat{u} - P_{Λ} u ∥^{2}$ . Substitute the first bound and square it to obtain the second inequality.

4.1F2given∎

The probability assertion is exactly [F2] under its stated independent sampling and budget hypotheses; no probability is asserted outside that good-event regime.

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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.

5 · Examples, counterexamples and false statements

None yet.