Subspace Acceleration for a Sequence of Linear Systems and Application to Plasma Simulation

TL;DR

This paper introduces a novel acceleration method for solving sequences of large-scale linear systems, combining reduced-order projection with randomized linear algebra, significantly reducing iteration counts and computational time in plasma simulations.

Contribution

It proposes a new approach that integrates reduced-order projection and randomized linear algebra to efficiently generate initial guesses for iterative solvers in time-dependent PDEs.

Findings

Reduced number of iterations for convergence.

Theoretical analysis of initial guess accuracy.

Application to plasma turbulence simulation shows significant speedup.

Abstract

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.