Randomized Reduced Basis Methods for Parameterized Fractional Elliptic PDEs

TL;DR

This paper develops efficient reduced order models for fractional elliptic PDEs with parameters, using randomized compression and shifted linear solvers, enabling fast repeated simulations in applications like Gaussian processes.

Contribution

It introduces a novel randomized reduced basis method combined with shifted linear solvers for fractional PDEs, improving computational efficiency and memory usage.

Findings

Significant reduction in computational time for multiple parameter queries.

Effective memory compression of solution snapshots.

Successful application to Gaussian process simulations.

Abstract

This paper is interested in developing reduced order models (ROMs) for repeated simulation of fractional elliptic partial differential equations (PDEs) for multiple values of the parameters (e.g., diffusion coefficients or fractional exponent) governing these models. These problems arise in many applications including simulating Gaussian processes, and geophysical electromagnetics. The approach uses the Kato integral formula to express the solution as an integral involving the solution of a parametrized elliptic PDE, which is discretized using finite elements in space and sinc quadrature for the fractional part. The offline stage of the ROM is accelerated using a solver for shifted linear systems, MPGMRES-Sh, and using a randomized approach for compressing the snapshot matrix. Our approach is both computational and memory efficient. Numerical experiments on a range of model problems, including an application to Gaussian processes, show the benefits of our approach.