[Paper Review] Linear collective collocation and Galerkin approximations for parametric and stochastic elliptic PDEs

Ðinh Dũng|arXiv (Cornell University)|Nov 11, 2015

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper introduces linear collective approximation methods—Taylor, collocation, and Galerkin—for high-dimensional parametric and stochastic elliptic PDEs with affine parametric dependence. Under mild regularity conditions, these methods achieve the same convergence rate as the underlying nonparametric finite element approximations, effectively breaking the curse of dimensionality while maintaining optimal error bounds with explicit constant estimates.

ABSTRACT

Consider the parametric elliptic problem \begin{equation} - \operatorname{dv} \big(a(y)(x) abla u(y)(x)\big) \ = \ f(x) \quad x \in D, \ y \in [-1,1]^\infty, \quad u|_{\partial D} \ = \ 0, \end{equation} where $D \subset {\mathbb R}^m$ is a bounded Lipschitz domain, $[-1,1]^\infty$, $f \in L_2(D)$, and the diffusions $a$ satisfy the uniform ellipticity assumption and are affinely dependent with respect to $y$. The parametric variable $y$ may be deterministic or random. In the present paper, a central question to be studied is as follows. Assume that we have an approximation property that there is a sequence of finite element approximations with a certain error convergence rate in energy norm of the space $V:=H^1_0(D)$ for the nonparametric problem $- \operatorname{dv} \big(a(y_0)(x) abla u(y_0)(x)\big) = f(x)$ at every point $y_0 \in [-1,1]^\infty$. Then under what assumptions does this sequence induce a sequence of finite element approximations with the same error convergence rate for the parametric elliptic problem in the norm of the Bochner spaces $L_\infty([-1,1]^\infty,V)$ or $L_2([-1,1]^\infty,V)$? We solved this question by linear collective Taylor, collocation and Galerkin methods, based on Taylor expansions, Lagrange polynomial interpolations and Legendre polynomials expansions, respectively, on the parametric domain $[-1,1]^\infty$. Under very light conditions, we show that all these approximation methods give the same error convergence rate as that by the sequence of finite element approximations for the nonparametric elliptic problem. The parametric infinite-variate part completely disappears from the convergence rate and influences only the constant. Hence the curse of dimensionality is broken by linear methods.

Motivation & Objective

• To determine under what conditions a sequence of finite element approximations with a given convergence rate for a nonparametric elliptic PDE induces the same convergence rate for the corresponding parametric problem.
• To address the curse of dimensionality in high-dimensional parametric PDEs where the number of parameters may be countably infinite.
• To develop linear collective approximation schemes that treat spatial and parametric variables jointly, avoiding separable decomposition.
• To establish error convergence rates in Bochner spaces $L_\infty(\mathbb{I}^\infty, V)$ and $L_2(\mathbb{I}^\infty, V)$ that match the nonparametric rate.
• To demonstrate that the parametric dependence, while affecting constants, does not degrade the convergence rate when affine and under $\ell_p$-summability conditions.

Proposed method

• Constructs linear collective Taylor, collocation, and Galerkin approximations using Taylor expansions, Lagrange interpolation, and Legendre polynomial expansions on the infinite-dimensional parametric domain $\mathbb{I}^\infty$.
• Uses a sequence of finite element approximations for the nonparametric problem as a base, inducing a collective approximation framework for the parametric case.
• Applies a Legendre expansion-based method with a sequence $\sigma_s$ defined via $\sigma_s^{-1} = \frac{|s|!}{s!} b^s$, where $b$ controls the decay of parametric derivatives.
• Imposes the $\ell_p$-summability condition $\|\psi_j\|_{W^{1}_\infty(D)} \in \ell_p(\mathbb{N})$ for $0 < p < 1$ to control parametric regularity.
• Derives error bounds in Bochner norms by estimating the growth of parametric derivatives via the sequence $\sigma_s$ and applying interpolation and approximation theory.
• Establishes that the convergence rate in $L_\infty(\mathbb{I}^\infty, V)$ or $L_2(\mathbb{I}^\infty, V)$ matches the nonparametric rate under mild assumptions on the diffusion coefficients and domain regularity.

Experimental results

Research questions

• RQ1 Under what conditions does a finite element approximation sequence with a given convergence rate for a nonparametric elliptic PDE induce the same rate for the parametric version in Bochner space norms?
• RQ2 Can linear collective methods break the curse of dimensionality in parametric elliptic PDEs with infinite-dimensional parameter spaces?
• RQ3 How do affine parametric dependence and $\ell_p$-summability of derivative coefficients affect the convergence rate of collective approximations?
• RQ4 To what extent can the parametric regularity be exploited to achieve optimal convergence without splitting spatial and parametric variables?
• RQ5 Can the same convergence rate be preserved in $L_\infty(\mathbb{I}^\infty, V)$ and $L_2(\mathbb{I}^\infty, V)$ using linear, non-intrusive methods?

Key findings

• The linear collective Galerkin and collocation methods achieve the same convergence rate $n^{-\min(1/p - 1, \alpha)}$ as the base nonparametric finite element approximations, with explicit constant $C$ depending on $\|\sigma^{-1}\|_{\ell_p(\mathbb{F})}$.
• When $p = \frac{1}{1 + \alpha}$, the convergence rate simplifies to $n^{-\alpha}$, matching the nonparametric rate exactly.
• The method achieves optimal convergence without requiring separability of spatial and parametric variables, treating them collectively via tensorized approximation spaces.
• The curse of dimensionality is broken: the convergence rate is independent of the parametric dimension, even when $d = \infty$, under $\ell_p$-summability of the parametric derivative coefficients.
• The constant $C$ in the error bound is explicitly estimated and depends only on the $\ell_p$-norm of the coefficient sequence and problem data, not on the dimension.
• The results extend to cases with arbitrary $\alpha > 0$ if sufficient regularity of $a(y)$, $f$, and $D$ is assumed, suggesting broader applicability beyond the current regularity constraints.