Time-Parallel Solutions to Ordinary Differential Equations on GPUs with a New Functional Optimization Approach Related to the Sobolev Gradient Method.

摘要

The problem of finding a solution to an ODE can by reformulated as a problem of finding the minimum of a specific energy functional. We present an efficient approach to finding this minimum and relate it to the Sobolev gradient method and Newton s method. The proposed approach requires only well studied parallel efficient algorithms in contrast to some other approaches that still require repeated serial solvers but at a lower resolutions. We present examples where, with a very good initial guess of the solution, convergence can be obtained in a single iteration. Even with a very poor guess, only a few iterations are required and convergence is faster than the pure sequential approach. We discussed how the speed of the method decreases rapidly with more parallelization and the method in the context of multi-scale modeling.