In several initial value problems with particularly expensive right-hand side
evaluation or implicit step computation, there is a tradeoff between accuracy and
computational effort. We consider inexact spectral deferred correction (SDC)
methods for solving such initial value problems. SDC methods are interpreted as
fixed-point iterations and, due to their corrective iterative nature, allow one to
exploit the accuracy-work tradeoff for a reduction of the total computational
effort. First we derive error models bounding the total error in terms of the
evaluation errors. Then we define work models describing the computational
effort in terms of the evaluation accuracy. Combining both, a theoretically
optimal local tolerance selection is worked out by minimizing the total work
subject to achieving the requested tolerance. The properties of optimal local
tolerances and the predicted efficiency gain compared to simpler heuristics,
and reasonable practical performance, are illustrated with simple numerical
examples.
Keywords
spectral deferred corrections, initial value problems,
error propagation, adaptive control of tolerances, inexact,
work models, accuracy models
