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Register a new userVirtual Transmission Method, A New Distributed Algorithm to Solve Sparse Linear System
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Added by
Fei Wei
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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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 Kirchhoffs Current Law from circuit theory. We proved that GNBT is feasible to partition any SPD linear system.
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In this paper, we propose a new distributed algorithm, called Directed Transmission Method (DTM). DTM is a fully asynchronous and continuous-time iterative algorithm to solve SPD sparse linear system. As an architecture-aware algorithm, DTM could be freely running on all kinds of heterogeneous parallel computer. We proved that DTM is convergent by making use of the final-value theorem of Laplacian Transformation. Numerical experiments show that DTM is stable and efficient.
Waveform Relaxation method (WR) is a beautiful algorithm to solve Ordinary Differential Equations (ODEs). However, because of its poor convergence capability, it was rarely used. In this paper, we propose a new distributed algorithm, named Waveform Transmission Method (WTM), by virtually inserting waveform transmission lines into the dynamical system to achieve distributed computing of extremely large ODEs. WTM has better convergence capability than the traditional WR algorithms.
As known, physical circuits, e.g. integrated circuits or power system, work in a distributed manner, but these circuits could not be easily simulated in a distributed way. This is mainly because that the dynamical system of physical circuits is nonlinear and the linearized system of physical circuits is nonsymmetrical. This paper proposes a simple and natural strategy to mimic the distributed behavior of the physical circuit by mimicking the distributed behavior of the internal wires inside this circuit. Mimic Transmission Method (MTM) is a new distributed algorithm to solve the nonlinear ordinary differential equations extracted from physical circuits. It maps the transmission delay of interconnects between subcircuits to the communication delay of digital data link between processors. MTM is a black-box algorithm. By mimicking the transmission lines, MTM seals the nonlinear dynamical system within the subcircuit. As the result, we do not need to pay attention on how to solve the nonlinear dynamic system or nonsymmetrical linear system in parallel. MTM is a global direct algorithm, and it does only one distributed computation at each time window to obtain accurate result, so unconvergence issues do not need to be worried about.
We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-to-communication ratio of the parallel algorithm is approximately the volume-to-surface-area ratio of the subdomain owned by every processor. We present various numerical results to demonstrate the versatility and scalability of the parallel algorithm.
In this paper, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is to introduce an entropic regularization of the Kantorovich formulation of the Optimal Transport problem. The regularized problem then corresponds to the projection of a vector on the intersection of the constraints with respect to the Kullback-Leibler distance. Iterative Bregman projections on each marginal constraint are explicit which enables us to approximate the optimal transport plan. We validate the numerical method against analytical test cases.
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