Multilevel Optimization for Inverse Problems

Simon Weissmann, Ashia Wilson, Jakob Zech
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:5489-5524, 2022.

Abstract

Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationally expensive forward models. We introduce a unifying framework of multilevel optimization that can be applied to a wide range of optimization-based solvers. Our framework provably reduces the computational cost associated with evaluating the expensive forward maps stemming from various physical models. To demonstrate the versatility of our analysis, we discuss its implications for various methodologies including multilevel (accelerated, stochastic) gradient descent, a multilevel ensemble Kalman inversion and a multilevel Langevin sampler. We also provide numerical experiments to verify our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-weissmann22a, title = {Multilevel Optimization for Inverse Problems}, author = {Weissmann, Simon and Wilson, Ashia and Zech, Jakob}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {5489--5524}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/weissmann22a/weissmann22a.pdf}, url = {https://proceedings.mlr.press/v178/weissmann22a.html}, abstract = {Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationally expensive forward models. We introduce a unifying framework of multilevel optimization that can be applied to a wide range of optimization-based solvers. Our framework provably reduces the computational cost associated with evaluating the expensive forward maps stemming from various physical models. To demonstrate the versatility of our analysis, we discuss its implications for various methodologies including multilevel (accelerated, stochastic) gradient descent, a multilevel ensemble Kalman inversion and a multilevel Langevin sampler. We also provide numerical experiments to verify our theoretical findings.} }
Endnote
%0 Conference Paper %T Multilevel Optimization for Inverse Problems %A Simon Weissmann %A Ashia Wilson %A Jakob Zech %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-weissmann22a %I PMLR %P 5489--5524 %U https://proceedings.mlr.press/v178/weissmann22a.html %V 178 %X Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationally expensive forward models. We introduce a unifying framework of multilevel optimization that can be applied to a wide range of optimization-based solvers. Our framework provably reduces the computational cost associated with evaluating the expensive forward maps stemming from various physical models. To demonstrate the versatility of our analysis, we discuss its implications for various methodologies including multilevel (accelerated, stochastic) gradient descent, a multilevel ensemble Kalman inversion and a multilevel Langevin sampler. We also provide numerical experiments to verify our theoretical findings.
APA
Weissmann, S., Wilson, A. & Zech, J.. (2022). Multilevel Optimization for Inverse Problems. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:5489-5524 Available from https://proceedings.mlr.press/v178/weissmann22a.html.

Related Material