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

Fei Wei

Huazhong Yang

Department of Electronic Engineering, Tsinghua University, Beijing, China

Technical Report

Preprinted at arXiv.org

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 background is the lossless transmission line and microwave network. The basic idea of VTM is to insert the virtual transmission lines into the 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 Electric Vertex Splitting (EVS). It is based on Kirchhoff’s Current Law from circuit theory. We proved that EVS is feasible to partition any SPD linear system.

Key words: Distributed Algorithm, Sparse Linear System, Partitioning, Convergence, Performance Modeling, Preconditioning, Transmission Line, Interconnect, Wire New words: Virtual Transmission Method (VTM), Electric Vertex Splitting (EVS), Transmission Delay Equations (TDE), Virtual Transmission Line (VTL), Interconnect, Wire, Electric Graph, Neighbor-To-Neighbor (N2N), Conformal Splitting Existence

2 Theory, Impedance Matching, Lines Coupling, Wire Tearing.

1. Introduction The linear system, Ax = b, is widely encountered in scientific computing. When the coefficient matrix A is symmetric-positive-definite (SPD), the linear system is called SPD system, which is extremely common in engineering applications [1, 2]. For example, most of the linear systems generated by the finite element method are SPD systems. Therefore, in many scientific disciplines, solving SPD systems is an inevitable task and the efficiency will be the dominant factor in those fields. To solve the SPD system, there are two basic approaches, direct methods and iterative methods.
   The direct methods are mainly based on the Sparse Cholesky Factorization. In order to efficiently compute the dense submatrices inside the sparse matrix, supernodal method and multifrontal method are used [3].
   The representatives of the iterative methods are Conjugate Gradient method (CG) and Multigrid method (MG). CG is based on the Krylov subspace projection. If the preconditioner is properly chosen, the convergence of CG will be fast. MG is efficient for the linear systems generated from the elliptic partial differential equations [4]. All the algorithms mentioned above work well on the traditional single-processor computers, but they would get into trouble on parallel computers [5, 6]. The parallel version of Sparse Cholesky Factorization suffers from the limited concurrency which depends on the distribution of the nonzero elements in the sparse matrix. For the parallel CG, it is difficult to choose a proper preconditioner in a parallel way [4]. Another well known parallel method for large sparse linear system is the Domain Decomposition Method (DDM). DDM refers to a collection of techniques which revolve around the principle of divide and conquer [4]. Schur Complement method, Additive Schwarz method and the Dual-Prime Finite Element Tearing and Interconnection (FETI-DP) method are three commonly-adopted parallel methods of DDM [7]. The Schur Complement method makes use of the master-slave model [8]. This method first partitions the large linear system into a number of subsystems. Then these subsystems are simplified and solved by the slave processors in parallel. After that the simplified results are merged into a new linear system, which is much smaller than the original one. At last this new system is solved by the master processor. This model suffers from the heavy communication overheads imposed on the master processor, especially when the number of slave processors is large. Consequently, the scalability and concurrency of the Schur Complement method is limited. The Additive Schwarz method is similar to the block Jacobi iteration. For a SPD system, it needs two assumptions to be convergent, and the convergence speed depends on these two assumptions [4].

3 The FETI-DP method is a scalable method to solve large problems [7, 9]. FETI-DP has to solve a coarse problem. This procedure needs global communication of the residual errors and the concurrency is difficult to explore. Consequently, the parallel efficiency of FETI-DP is affected. VTM is a new parallel algorithm for large-scale sparse SPD systems. It is inspired by the behavior of transmission lines in the electrical engineering. Although VTM is a distributed iterative algorithm, it is sure to be convergent because of its physical back

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