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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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Sources

Albert Cohen and Giovanni Migliorati, Optimal Weighted Least-Squares Methods, §§2–3 (standard reference, not scraped)
Bruno Sudret, Polynomial Chaos Expansions and Stochastic Finite Element Methods, §3.4.3 (standard reference, not scraped)