Tensor decomposition methods for high-dimensional Hamilton-Jacobi-Bellman equations

Sergey Dolgov, Dante Kalise, Karl K. Kunisch

Research output: Contribution to journalArticlepeer-review

42 Citations (SciVal)

Abstract

A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables.

Original languageEnglish
Pages (from-to)A1625-A1650
Number of pages26
JournalSIAM Journal on Scientific Computing
Volume43
Issue number3
DOIs
Publication statusPublished - 31 May 2021

Bibliographical note

Funding Information:
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section December 6, 2019; accepted for publication (in revised form) February 11, 2021; published electronically May 10, 2021. https://doi.org/10.1137/19M1305136 Funding: This work was supported by the PGMO-PRMO program of the FMJH and by the Engineering and Physical Sciences Research Council (EPSRC) New Horizons grant EP/V04771X/1. The first author was also supported by EPSRC Fellowship EP/M019004/1. The second author was supported by a public grant as part of the Investissement d'avenir project ANR-11-LABX-0056-LMH, LabEx LMH, and by EPSRC grants EP/T024429/1 and EP/V025899/1. The third author was supported by the ERC advanced grant 668998 (OCLOC) under the EU's H2020 research program.

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics

Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

Keywords

  • Dynamic programming
  • Hamilton-Jacobi-Bellman equations
  • High-dimensional approximation
  • Optimal feedback control
  • Tensor calculus

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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