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

In this paper, we propose a new distributed numerical algorithm which could work naturally on the parallel computer with arbitrary number of processors. This algorithm is named virtual transmission method (VTM), which is inspired by the behavior of transmission line from the electrical engineering. VTM is a scalable, distributed and global-iterative-local-direct algorithm to solve the large sparse linear system whose coefficient matrix is symmetric-positive-definite (SPD). As a distributedly-iterative algorithm, VTM is proved to be convergent. VTM requires simple hardware and could be easily implemented on any kind of parallel computer, including the manycore microprocessor, and all the traditional serial solvers could be assembled into VTM without any change. Numerical experiments show that VTM is an efficient and accurate algorithm. Accompanied with VTM, we bring in a new technique to partition the symmetric linear system, which is named electric vertex splitting technique (EVS). We present the conformal splitting existence theorem to assure that this splitting technique is feasible to partition any SPD linear system.