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Approximation of PDE solution manifolds: Sparse-grid interpolation and quadrature

Dinh Dũng, Van Kien Nguyen, Duong Thanh Pham, Christoph Schwab

TL;DR

This work develops fully discrete sparse-grid interpolation and quadrature methods for infinite-variate, $X$-valued functions under a countable tensor-product Jacobi measure, leveraging double-weighted summability of Jacobi GPC coefficients. The approach combines stable spatial discretizations with sparse-grid tensor-product projectors built from univariate Chebyshev nodes, and exploits symmetry-induced cancellations in ultra-spherical settings to achieve accelerated convergence and mitigate the curse of dimensionality. The authors establish rigorous error bounds in ${\mathcal L}_p(X^1)$ and for ${\mathcal L}_p(V)$-norms, provide thresholded index set constructions with explicit dimension growth, and demonstrate the framework in two practical applications: affine-parametric linear elliptic PDEs and holomorphic maps with affine-parametric encoding. The results are largely self-contained and pave the way for extensions to neural-network surrogates for unbounded parameter ranges, while offering high-order, dimension-controlled approximations for a broad class of infinite-dimensional parametric problems.

Abstract

We study fully-discrete approximations and quadratures of infinite-variate functions in abstract Bochner spaces associated with a Hilbert space $X$ and an infinite-tensor-product Jacobi measure. For target infinite-variate functions taking values in $X$ which admit absolutely convergent Jacobi generalized polynomial chaos expansions, with suitable weighted summability conditions for the coefficient sequences, we generalize and improve prior results on construction of sequences of finite sparse-grid tensor-product polynomial interpolation approximations and quadratures, based on the univariate Chebyshev points. For a generic stable discretization of $X$ in terms of a dense sequence $(V_m)_{m \in \mathbb{N}_0}$ of finite-dimensional subspaces, we obtain fully-discrete, linear approximations in terms of so-called sparse-grid tensor-product projectors, with convergence rates of approximations as well as of sparse-grid tensor-product quadratures of the target functions. We verify the abstract assumptions in two fundamental application settings: first, a linear elliptic diffusion equation with affine-parametric coefficients and second, abstract holomorphic maps between separable Hilbert spaces with affine-parametric input data encoding. For these settings, as in [37,20], cancellation of anti-symmetric terms in ultra-spherical Jacobi generalized polynomial chaos expansion coefficients implies crucially improved convergence rates of sparse-grid tensor-product quadrature with respect to the infinite-tensor-product Jacobi weight, free from the ``curse-of-dimension". Largely self-contained proofs of all results are developed. Approximation convergence rate results in the present setting which are based on construction of neural network surrogates, for unbounded parameter ranges with Gaussian measures, will be developed in extensions of the present work.

Approximation of PDE solution manifolds: Sparse-grid interpolation and quadrature

TL;DR

This work develops fully discrete sparse-grid interpolation and quadrature methods for infinite-variate, -valued functions under a countable tensor-product Jacobi measure, leveraging double-weighted summability of Jacobi GPC coefficients. The approach combines stable spatial discretizations with sparse-grid tensor-product projectors built from univariate Chebyshev nodes, and exploits symmetry-induced cancellations in ultra-spherical settings to achieve accelerated convergence and mitigate the curse of dimensionality. The authors establish rigorous error bounds in and for -norms, provide thresholded index set constructions with explicit dimension growth, and demonstrate the framework in two practical applications: affine-parametric linear elliptic PDEs and holomorphic maps with affine-parametric encoding. The results are largely self-contained and pave the way for extensions to neural-network surrogates for unbounded parameter ranges, while offering high-order, dimension-controlled approximations for a broad class of infinite-dimensional parametric problems.

Abstract

We study fully-discrete approximations and quadratures of infinite-variate functions in abstract Bochner spaces associated with a Hilbert space and an infinite-tensor-product Jacobi measure. For target infinite-variate functions taking values in which admit absolutely convergent Jacobi generalized polynomial chaos expansions, with suitable weighted summability conditions for the coefficient sequences, we generalize and improve prior results on construction of sequences of finite sparse-grid tensor-product polynomial interpolation approximations and quadratures, based on the univariate Chebyshev points. For a generic stable discretization of in terms of a dense sequence of finite-dimensional subspaces, we obtain fully-discrete, linear approximations in terms of so-called sparse-grid tensor-product projectors, with convergence rates of approximations as well as of sparse-grid tensor-product quadratures of the target functions. We verify the abstract assumptions in two fundamental application settings: first, a linear elliptic diffusion equation with affine-parametric coefficients and second, abstract holomorphic maps between separable Hilbert spaces with affine-parametric input data encoding. For these settings, as in [37,20], cancellation of anti-symmetric terms in ultra-spherical Jacobi generalized polynomial chaos expansion coefficients implies crucially improved convergence rates of sparse-grid tensor-product quadrature with respect to the infinite-tensor-product Jacobi weight, free from the ``curse-of-dimension". Largely self-contained proofs of all results are developed. Approximation convergence rate results in the present setting which are based on construction of neural network surrogates, for unbounded parameter ranges with Gaussian measures, will be developed in extensions of the present work.
Paper Structure (9 sections, 13 theorems, 139 equations)

This paper contains 9 sections, 13 theorems, 139 equations.

Table of Contents

  1. Notation
  2. Fully discrete approximations in Bochner spaces
  3. Jacobi polynomials
  4. Jacobi chaos
  5. Sparse-grid tensor-product polynomial interpolation
  6. Sparse-grid quadratures
  7. Applications
  8. Affine-parametric, linear elliptic PDEs
  9. Holomorphic maps with affine-parametric encoding

Key Result

Lemma 2.2

Let $\varepsilon$ be a fixed positive number and $C_\varepsilon$ such that with $\theta_0, \lambda_0$ given in lambda_0 and theta_0. Let $v\in {\mathcal{L}}_2(X)$ and satisfy Assumption assum1 with Then the function $v$ can be identified with an element in $C({\mathbb I}^\infty, X)$. Additionally, for every ${\boldsymbol{y}}\in {\mathbb I}^\infty$ we can represent $v({\boldsymbol{y}})$ by the se

Theorems & Definitions (30)

  • Lemma 2.2
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • proof
  • Remark 2.1
  • ...and 20 more