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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 language | English |
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Pages (from-to) | A1625-A1650 |
Number of pages | 26 |
Journal | SIAM Journal on Scientific Computing |
Volume | 43 |
Issue number | 3 |
DOIs | |
Publication status | Published - 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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Dive into the research topics of 'Tensor decomposition methods for high-dimensional Hamilton-Jacobi-Bellman equations'. Together they form a unique fingerprint.Projects
- 3 Finished
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Overcoming the curse of dimensionality in dynamic programming by tensor decompositions
Dolgov, S. (PI)
Engineering and Physical Sciences Research Council
10/05/21 → 9/05/23
Project: Research council
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Tensor product numerical methods for high-dimensional problems in probability and quantum calculations
Dolgov, S. (PI)
1/01/16 → 31/12/18
Project: Research council
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Sergey Dolgov Fellowship - Tensor Product Numerical Methods for High-Dimensional Problems in Probablility and Quantum Calculations
Scheichl, R. (PI)
Engineering and Physical Sciences Research Council
1/01/16 → 31/12/18
Project: Research council