Parallel Methods for Solving Nonlinear Block Bordered Systems of Equations

A group of parallel algorithms, and their implementation for solving a special class of nonlinear equations, are discussed. The type of sparsity occurring in these problems, which arise in VLSI design, structural engineering, and many other areas, is called a block bordered structure. The explicit method and several implicit methods are described, and the new corrected implicit method for solving block bordered nonlinear problems is presented. The relationship between the two types of methods is analyzed, and some computational comparisons are performed. Several variations and globally convergent modifications of the implicit method are also described. Parallel implementations of these algorithms for solving block bordered nonlinear equations are described, and experimental results on the Intel hypercube that show the effectiveness of the parallel implicit algorithms are presented. These experiments include a fairly large circuit simulation that leads to a multilevel block bordered system of nonlinear equations.