DATA-DRIVEN TENSOR TRAIN GRADIENT CROSS APPROXIMATION FOR HAMILTON-JACOBI-BELLMAN EQUATIONS

Sergey Dolgov; Dante Kalise; Luca Saluzzi

doi:10.1137/22M1498401

DATA-DRIVEN TENSOR TRAIN GRADIENT CROSS APPROXIMATION FOR HAMILTON-JACOBI-BELLMAN EQUATIONS

Sergey Dolgov, Dante Kalise, Luca Saluzzi

Imperial College London

Research output: Contribution to journal › Article › peer-review

9 Citations (SciVal)

Abstract

A gradient-enhanced functional tensor train cross approximation method for the resolution of the Hamilton-Jacobi-Bellman (HJB) equations associated with optimal feedback control of nonlinear dynamics is presented. The procedure uses samples of both the solution of the HJB equation and its gradient to obtain a tensor train approximation of the value function. The collection of the data for the algorithm is based on two possible techniques: Pontryagin Maximum Principle and State-Dependent Riccati Equations. Several numerical tests are presented in low and high dimension showing the effectiveness of the proposed method and its robustness with respect to inexact data evaluations, provided by the gradient information. The resulting tensor train approximation paves the way towards fast synthesis of the control signal in real-time applications.

Original language	English
Pages (from-to)	A2153-A2184
Journal	SIAM Journal on Scientific Computing
Volume	45
Issue number	5
Early online date	13 Sept 2023
DOIs	https://doi.org/10.1137/22M1498401
Publication status	Published - 31 Oct 2023

Bibliographical note

Funding Information:
\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section May 25, 2022; accepted for publication (in revised form) February 22, 2023; published electronically September 13, 2023. https://doi.org/10.1137/22M1498401

Funding: This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/V04771X/1 and EP/T024429/1. \dagger Department of Mathematical Sciences, University of Bath, North Road, BA2 7AY Bath, United Kingdom ([email protected]). \ddagger Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom ([email protected], [email protected]).

Data access. Matlab codes implementing the gradient cross and numerical examples are available at https://github.com/saluzzi/TT-Gradient-Cross

Funding

\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section May 25, 2022; accepted for publication (in revised form) February 22, 2023; published electronically September 13, 2023. https://doi.org/10.1137/22M1498401 Funding: This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/V04771X/1 and EP/T024429/1. \dagger Department of Mathematical Sciences, University of Bath, North Road, BA2 7AY Bath, United Kingdom ([email protected]). \ddagger Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom ([email protected], [email protected]).

Keywords

dynamic programming
Hamilton-Jacobi-Bellman equations
high-dimensional approximation
optimal feedback control
tensor decomposition

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

Access to Document

10.1137/22M1498401

https://arxiv.org/pdf/2205.05109.pdfLicence: CC BY

Cite this

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title = "DATA-DRIVEN TENSOR TRAIN GRADIENT CROSS APPROXIMATION FOR HAMILTON-JACOBI-BELLMAN EQUATIONS",

abstract = "A gradient-enhanced functional tensor train cross approximation method for the resolution of the Hamilton-Jacobi-Bellman (HJB) equations associated with optimal feedback control of nonlinear dynamics is presented. The procedure uses samples of both the solution of the HJB equation and its gradient to obtain a tensor train approximation of the value function. The collection of the data for the algorithm is based on two possible techniques: Pontryagin Maximum Principle and State-Dependent Riccati Equations. Several numerical tests are presented in low and high dimension showing the effectiveness of the proposed method and its robustness with respect to inexact data evaluations, provided by the gradient information. The resulting tensor train approximation paves the way towards fast synthesis of the control signal in real-time applications.",

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author = "Sergey Dolgov and Dante Kalise and Luca Saluzzi",

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N1 - Funding Information: \ast Submitted to the journal's Methods and Algorithms for Scientific Computing section May 25, 2022; accepted for publication (in revised form) February 22, 2023; published electronically September 13, 2023. https://doi.org/10.1137/22M1498401 Funding: This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grants EP/V04771X/1 and EP/T024429/1. \dagger Department of Mathematical Sciences, University of Bath, North Road, BA2 7AY Bath, United Kingdom ([email protected]). \ddagger Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, United Kingdom ([email protected], [email protected]). Data access. Matlab codes implementing the gradient cross and numerical examples are available at https://github.com/saluzzi/TT-Gradient-Cross

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N2 - A gradient-enhanced functional tensor train cross approximation method for the resolution of the Hamilton-Jacobi-Bellman (HJB) equations associated with optimal feedback control of nonlinear dynamics is presented. The procedure uses samples of both the solution of the HJB equation and its gradient to obtain a tensor train approximation of the value function. The collection of the data for the algorithm is based on two possible techniques: Pontryagin Maximum Principle and State-Dependent Riccati Equations. Several numerical tests are presented in low and high dimension showing the effectiveness of the proposed method and its robustness with respect to inexact data evaluations, provided by the gradient information. The resulting tensor train approximation paves the way towards fast synthesis of the control signal in real-time applications.

AB - A gradient-enhanced functional tensor train cross approximation method for the resolution of the Hamilton-Jacobi-Bellman (HJB) equations associated with optimal feedback control of nonlinear dynamics is presented. The procedure uses samples of both the solution of the HJB equation and its gradient to obtain a tensor train approximation of the value function. The collection of the data for the algorithm is based on two possible techniques: Pontryagin Maximum Principle and State-Dependent Riccati Equations. Several numerical tests are presented in low and high dimension showing the effectiveness of the proposed method and its robustness with respect to inexact data evaluations, provided by the gradient information. The resulting tensor train approximation paves the way towards fast synthesis of the control signal in real-time applications.

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DATA-DRIVEN TENSOR TRAIN GRADIENT CROSS APPROXIMATION FOR HAMILTON-JACOBI-BELLMAN EQUATIONS

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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Overcoming the curse of dimensionality in dynamic programming by tensor decompositions

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DATA-DRIVEN TENSOR TRAIN GRADIENT CROSS APPROXIMATION FOR HAMILTON-JACOBI-BELLMAN EQUATIONS

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Overcoming the curse of dimensionality in dynamic programming by tensor decompositions

Cite this