High-performance VLSI model elliptic solvers

Abstract

VLSI parallel algorithms for a solution of fundamental elliptic problems with Laplace operators (Dirichlet and first boundary value problem for Poisson and biharmonic equation respectively) on a rectangular N×N grid are proposed. A standard multigrid algorithm is adopted for Poisson equation which allows a parallel solution of this problem in T=O(logN) parallel steps. A special network consisting of N×N processor elements and of O(NlogN) interconnection lines in each direction results in a design the area of which is A=O(N ² log ² N). AT ² estimation for a complexity of this Poisson solver is O(N ² log ⁴ N) which improves the best result known until now by a factor of O(N/logN). This VLSI multigrid Poisson solver is applied to the semidirect method for solving the biharmonic equation. The parallel time of the algorithm is O(√Nlog ² N) and the area needed is A=O(N ³ logN). The total complexity for such VLSI semidirect solver is AT ²=O(N ⁴ log ⁵ N).

The work presented in this paper was carried out as part of the research project Innovative Tools for Parallel Programming funded by the Austrian Ministry for Science and Research (BMWF) and the project Algorithms for Massively Parallel Computer Systems of the Slovak Grant Agency (GAV).