A parallel iterative algorithm for differential linear complementarity problems

Abstract

We propose a parallel iterative algorithm for solving the differential linear complementarity problems consisting of two systems, a linear ODE system and a linear complementarity system (LCS). At each iteration we proceed in a system decoupling way: by using a rough approximation of the state variable obtained from the previous iteration, we solve the LCS; then we solve the ODE system and update the state variable for preparing for the next iteration, by using the obtained constraint variable as a known source term. The algorithm is highly parallelizable, because at each iteration the computations of both the LCS and the ODE system at all the time points of interest can start simultaneously. The parallelism for solving the LCS is natural and for the ODE system it is achieved by using the Laplace inversion technique. For the P-matrix LCS, we prove that the algorithm converges superlinearly with arbitrarily chosen initial iterate and for the Z-matrix LCS the algorithm still converges superlinearly if we use the initial value as the initial iterate. We show that this algorithm is superior to the widely used time-stepping method, with respect to robustness, flexibility, and computation time.