A Parallel Adaptive Method for Pseudo-arclength Continuation

Pseudo-arclength continuation is a well-established method for constructing a numerical curve comprising solutions of a system of nonlinear equations. In many complicated high-dimensional systems, the corrector steps within pseudo-arclength continuation are extremely costly to compute; as a result, the step-length of the preceding prediction step must be adapted carefully to avoid prohibitively many failed steps. We describe the essence of a parallel method for adapting the step-length of pseudo-arclength continuation. Our method employs several predictor-corrector sequences with differing step-lengths running concurrently on distinct processors. Our parallel framework permits intermediate results of correction sequences that have not yet converged to seed new predictor-corrector sequences with various step-lengths; the goal is to amortize the cost of corrector steps to make further progress along the underlying numerical curve. Results from numerical experiments suggest a three-fold speedup is attainable when the continuation curve sought has great topological complexity and the corrector steps require significant processor time.