Parallel Randomized and Matrix-Free Direct Solvers for Large Structured Dense Linear Systems

Abstract

We design efficient and distributed-memory parallel randomized direct solvers for large structured dense linear systems, including a fully matrix-free version based on matrix-vector multiplications and a partially matrix-free one. The dense coefficient matrix $A$ has an off-diagonal low-rank structure, as often encountered in practical applications such as Toeplitz systems and discretized integral and partial differential equations. A distributed-memory parallel framework for randomized structured solution is shown. Scalable adaptive randomized sampling and hierarchical compression algorithms are designed to approximate $A$ by hierarchically semiseparable (HSS) matrices. Systematic process grid storage schemes are given for different HSS forms. Parallel hierarchical algorithms are proposed for the resulting HSS forms. As compared with existing work on parallel HSS methods, our algorithms have several remarkable advantages, including the matrix-free schemes that avoid directly using dense $A$, a synchroniz...