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PAMPAC: A Parallel Adaptive Method for Pseudo-Arclength Continuation

Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined. Convergence of the iterative corrections is guaranteed only for sufficiently small prediction steps. In high-dimensional systems, corrector steps are extremely costly to compute and the prediction step-length must be adapted carefully to avoid failed steps or unnecessarily slow progress. We describe a parallel method for adapting the step-length employing several predictor-corrector sequences of different step lengths computed concurrently. In addition, the algorithm permits intermediate results of unconverged correction sequences to seed new predictions. This strategy results in an aggressive optimization of the step length at the cost of redundancy in the concurrent computation. We present two examples of convoluted solution curves of high-dimensional systems showing that speed-up by a factor of two can be attained on a multi-core CPU while a factor of three is attainable on a small cluster.