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

Sergey Dolgov, Dante Kalise, Karl K. Kunisch

Research output: Contribution to journalArticlepeer-review

27 Citations (SciVal)


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
Issue number3
Publication statusPublished - 31 May 2021


  • 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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