Theoretical and practical aspects of parallel numerical algorithms for initial value problems, with applications

Theoretical and practical aspects of parallel numerical methods for solving initial value problems are investigated, with particular attention to two applications from electrical engineering. First, algorithms for massively parallel circuit-level simulation of the grid-based analog signal processing arrays currently being developed for robotic vision applications are described, and simulation results presented. The trapezoidal rule is used to discretize the differential equations that describe the analog array behavior, Newton's method is used to solve the nonlinear equations generated at each time-step, and a block preconditioned conjugate-gradient squared algorithm is used to solve the linear equations generated by Newton's method. Excellent parallel performance of the algorithm is achieved through the use of a novel, but very natural, mapping of the circuit data onto a massively parallel architecture. The mapping takes advantage of the underlying computer architecture and the structure of the analog array problem. Experimental results demonstrate that a fullsize Connection Machine can provide a 650 times speedup over a SUN-4/490 workstation. Next, a new conjugate direction algorithm for accelerating waveform relaxation applied to the semiconductor device transient simulation problem is developed. A Galerkin method is applied to solving the system of second-kind Volterra integral equations which characterize the classical dynamic iteration methods for the linear time-varying initial value problem. It is shown that the Galerkin approximations can be computed iteratively using conjugate-direction algorithms. The resulting iterative methods are combined with an operator Newton method and applied to solving the nonlinear differential-algebraic system generated by spatial discretization of the time-dependent semiconductor device equations. Experimental results ae included which demonstrate the conjugate-direction methods are significantly faster than classical dynamic iteration methods. The results from both applications are encouraging and demonstrate that for specific initial value problems, the largest performance gains can be achieved by using closely matched algorithms and architectures to exploit characteristic features of the particular problem to be solved.