Optimization of Approximate Maps for Linear Systems Arising in Discretized PDEs

TL;DR

Several sparsity patterns for computing the sparse approximate map (SAM) update are examined to characterize optimal or near‐optimal sparsity patterns for linear systems arising from discretized PDEs.

Abstract

Generally, discretization of partial differential equations (PDEs) creates a sequence of linear systems $A_k x_k = b_k, k = 0, 1, 2, ..., N$ with well-known and structured sparsity patterns. Preconditioners are often necessary to achieve fast convergence When solving these linear systems using iterative solvers. We can use preconditioner updates for closely related systems instead of computing a preconditioner for each system from scratch. One such preconditioner update is the sparse approximate map (SAM), which is based on the sparse approximate inverse preconditioner using a least squares approximation. A SAM then acts as a map from one matrix in the sequence to another nearby one for which we have an effective preconditioner. To efficiently compute an effective SAM update (i.e., one that facilitates fast convergence of the iterative solver), we seek to compute an optimal sparsity pattern. In this paper, we examine several sparsity patterns for computing the SAM update to characterize optimal or near-optimal sparsity patterns for linear systems arising from discretized PDEs.

Figures (11)

• Figure 1: Adjacency graph of target matrix, $G(A_0)$
• Figure 2: Adjacency graph of source matrix, $G(A_1)$
• Figure 3: Transitive closure of source matrix, $G^*(A_1)$
• Figure 4: Adjacency graph of exact map, $G(\widehat{N_1})$.
• Figure 5: Sparsity patterns for THT matrices and global sparsification applied to the exact map,$\widehat{N_k}$, between selected matrices in the sequence. $\widehat{N_j} > 1.e-p$ denotes the dropping of elements (in absolute value) from the exact map that is less than $1.e-p$, for $p=2,4$.