Model order reduction in contour integral methods for parametric PDEs

TL;DR

This paper introduces a projection model order reduction method for parametric linear evolution PDEs using Laplace transform, enabling direct solution computation at specific times and reducing the computational complexity compared to traditional time stepping methods.

Contribution

The paper presents a novel MOR approach based on contour integral approximation via Laplace transform, improving reduction efficiency and avoiding slow singular value decay issues.

Findings

Effective for parabolic PDEs in finance

Reduces number of vectors needed in MOR

Avoids slow decay of singular values in advection problems

Abstract

In this paper we discuss a projection model order reduction (MOR) method for a class of parametric linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of some classical MOR methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a linear advection equation, does not suffer the problem of slow decay of singular values which instead affects time stepping methods for the numerical approximation of the Cauchy problem arising from space discretization.