Virtual Transmission Method, A New Distributed Algorithm to Solve Sparse Linear System

TL;DR

This paper introduces the Virtual Transmission Method (VTM), a novel parallel algorithm inspired by electrical transmission lines, designed to efficiently solve sparse symmetric positive definite linear systems on distributed computing architectures.

Contribution

The paper presents VTM, a new distributed algorithm for sparse linear systems, along with a novel partitioning technique called GNBT, both grounded in circuit theory.

Findings

VTM is proven to converge for SPD systems.

Numerical experiments demonstrate VTM's efficiency and stability.

GNBT effectively partitions SPD linear systems.

Abstract

In this paper, we propose a new parallel algorithm which could work naturally on the parallel computer with arbitrary number of processors. This algorithm is named Virtual Transmission Method (VTM). Its physical backgroud is the lossless transmission line and microwave network. The basic idea of VTM is to insert lossless transmission lines into the sparse linear system to achieve distributed computing. VTM is proved to be convergent to solve SPD linear system. Preconditioning method and performance model are presented. Numerical experiments show that VTM is efficient, accurate and stable. Accompanied with VTM, we bring in a new technique to partition the symmetric linear system, which is named Generalized Node & Branch Tearing (GNBT). It is based on Kirchhoff's Current Law from circuit theory. We proved that GNBT is feasible to partition any SPD linear system.