# Time-parallel solutions to differential equations via functional optimization

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## Abstract

We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate.

## Keywords

Time-parallel computing Differential equations Functional optimization## Mathematics Subject Classification

65Y05 Numerical analysis-Computer aspects of numerical algorithms-Parallel computation 34-04 Ordinary differential equations-Explicit machine computation and programs## Notes

### Acknowledgments

The authors wish to acknowledge the support of the Air Force Office of Scientific Research (AFOSR), Grant No. 12RZ06COR (PM: Dr. F. Fahroo) for this work. We would also like to thank Dr. Justin Koo of AFRL at Edwards AFB for many fruitful discussions.

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