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Error bounds for model reduction of feedback-controlled linear stochastic dynamics on Hilbert spaces

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  • Becker, Simon
  • Hartmann, Carsten
  • Redmann, Martin
  • Richter, Lorenz

Abstract

We analyze structure-preserving model order reduction methods for Ornstein–Uhlenbeck processes and linear S(P)DEs with multiplicative noise based on balanced truncation. For the first time, we include in this study the analysis of non-zero initial conditions. We moreover allow for feedback-controlled dynamics for solving stochastic optimal control problems with reduced-order models and prove novel error bounds for a class of linear quadratic regulator problems. We provide numerical evidence for the bounds and discuss the application of our approach to enhanced sampling methods from non-equilibrium statistical mechanics.

Suggested Citation

  • Becker, Simon & Hartmann, Carsten & Redmann, Martin & Richter, Lorenz, 2022. "Error bounds for model reduction of feedback-controlled linear stochastic dynamics on Hilbert spaces," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 107-141.
    • Handle: RePEc:eee:spapps:v:149:y:2022:i:c:p:107-141
    DOI: 10.1016/j.spa.2022.03.009
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    References listed on IDEAS

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    1. Daraghmeh, Adnan & Hartmann, Carsten & Qatanani, Naji, 2019. "Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 295-307.
    2. S. Endres & J. Stübinger, 2019. "Optimal trading strategies for Lévy-driven Ornstein–Uhlenbeck processes," Applied Economics, Taylor & Francis Journals, vol. 51(29), pages 3153-3169, June.
    3. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    4. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    5. Nikolas Nüsken & Lorenz Richter, 2021. "Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space," Partial Differential Equations and Applications, Springer, vol. 2(4), pages 1-48, August.
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