Simulated Annealing for Optimal Pivot Selection in Jacobian Accumulation

Abstract

We report on new results in logarithmic simulated annealing applied to the optimal Jacobian accumulation problem. This is a continuation of work that was presented at the SAGA’01 conference in Berlin, Germany [16]. We discuss the optimal edge elimination problem in linearized computational graphs [15] in the context of linear algebra. We introduce row and column pivoting on the extended Jacobian as analogs to front and back edge elimination in linearized computational graphs. Neighborhood relations for simulated annealing are defined on a metagraph that is derived from the computational graph. All prerequisites for logarithmic simulated annealing are fulfilled for dyadic pivoting, which is equivalent to vertex elimination in linearized computational graphs [7]. For row and column pivoting we cannot yet give a proof that the corresponding elimination sequences are polynomial in size. In practice, however, the likelihood for an exponential elimination sequence to occur is negligible. Numerical results are presented for algorithms based on both homogeneous and inhomogeneous Markov chains for all pivoting techniques. The superiority of row and column pivoting over dyadic pivoting can be observed when applying these techniques to Roe’s numerical flux [17].