Spectral preconditioners for the efficient numerical solution of a continuous branched transport model

Abstract

We consider the efficient solution of sequences of linear systems arising in the numerical solution of a branched transport model whose long time solution for specific parameter settings is equivalent to the solution of the Monge–Kantorovich equations of optimal transport. Galerkin Finite Element discretization combined with explicit Euler time stepping yield a linear system to be solved at each time step, characterized by a large sparse very ill conditioned symmetric positive definite (SPD) matrix A . Extreme cases even prevent the convergence of Preconditioned Conjugate Gradient (PCG) with standard preconditioners such as an Incomplete Cholesky (IC) factorization of A , which cannot always be computed. We investigate several preconditioning strategies that incorporate partial approximated spectral information. We present numerical evidence that the proposed techniques are efficient in reducing the condition number of the preconditioned systems, thus decreasing the number of PCG iterations and the overall CPU time.