@Article{Aruliah:2015:APP,
  author =       "Dhavide A. Aruliah and Lennaert van Veen and Alex Dubitski",
  title =        "Algorithm 956: {PAMPAC}: A Parallel Adaptive Method
                  for Pseudo-Arclength Continuation",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "42",
  number =       "1",
  accepted =     "2 January 2015",
  upcoming =     "true", 
  abstract =     "
                  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 correction sequences that have not
                  converged 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.",
}