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

TL;DR

This paper develops advanced sparse-grid interpolation and quadrature methods for high-dimensional PDE solution manifolds, improving convergence rates and applicability to infinite-dimensional problems with applications to elliptic equations and holomorphic maps.

Contribution

It generalizes sparse-grid approximation techniques to infinite-variate functions in Bochner spaces, with improved convergence rates and applicability to complex PDEs and holomorphic maps.

Findings

Achieved convergence rates for sparse-grid tensor-product approximations.

Verified assumptions in elliptic PDE and holomorphic map applications.

Improved quadrature convergence free from curse-of-dimension effects.

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.