Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity

TL;DR

This paper analyzes the convergence rates of the Multi-index Stochastic Collocation method for solving PDEs with random coefficients, providing theoretical estimates and demonstrating its effectiveness through numerical experiments.

Contribution

It introduces a new error analysis for MISC, applying a greedy optimization for mixed differences, and derives algebraic convergence rates based on parametric regularity.

Findings

MISC achieves algebraic convergence rates depending on solution regularity.

The method outperforms Multi-index Monte Carlo in numerical experiments.

Convergence rates are validated against theoretical predictions.

Abstract

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDEs) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data and, naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDEs in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Mat\'ern model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite-dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.