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Improving the Convergence Rates for the Kinetic Fokker-Planck Equation by Optimal Control

Tobias Breiten, Karl Kunisch

TL;DR

This work investigates speeding up the long-time convergence of the kinetic Fokker-Planck equation for Langevin dynamics by a bilinear, potential-based control that preserves the invariant measure. It develops a rigorous operator-theoretic framework, proves well-posedness of the controlled system, and decouples the invariant measure to reduce analysis to a mean-zero subspace with a spectral gap. An infinite-horizon infinite-dimensional LQ control problem is formulated on $Y_0$, with a Riccati-based feasible feedback and an existence result for an optimal control under admissibility. Numerical experiments with a Sinc spatial discretization and Kleinman iterations illustrate improved exponential convergence and the feasibility of the Riccati-based approach. The results provide a principled method to accelerate convergence in hypocoercive kinetic equations, with potential implications for sampling and stochastic optimization methods that rely on Langevin dynamics.

Abstract

The long time behavior and detailed convergence analysis of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings.

Improving the Convergence Rates for the Kinetic Fokker-Planck Equation by Optimal Control

TL;DR

This work investigates speeding up the long-time convergence of the kinetic Fokker-Planck equation for Langevin dynamics by a bilinear, potential-based control that preserves the invariant measure. It develops a rigorous operator-theoretic framework, proves well-posedness of the controlled system, and decouples the invariant measure to reduce analysis to a mean-zero subspace with a spectral gap. An infinite-horizon infinite-dimensional LQ control problem is formulated on , with a Riccati-based feasible feedback and an existence result for an optimal control under admissibility. Numerical experiments with a Sinc spatial discretization and Kleinman iterations illustrate improved exponential convergence and the feasibility of the Riccati-based approach. The results provide a principled method to accelerate convergence in hypocoercive kinetic equations, with potential implications for sampling and stochastic optimization methods that rely on Langevin dynamics.

Abstract

The long time behavior and detailed convergence analysis of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings.
Paper Structure (11 sections, 15 theorems, 137 equations, 4 figures)

This paper contains 11 sections, 15 theorems, 137 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Operator form of the kinetic Fokker-Planck equation
  3. The controlled kinetic Fokker-Planck equation
  4. Decoupling of the invariant measure
  5. An infinite-horizon bilinear optimal control problem
  6. Existence of a feasible control
  7. Existence of an optimal control
  8. Numerical experiments
  9. Spatial discretization and numerical realization
  10. Examples
  11. Proofs

Key Result

Proposition 1

Let Assumption ass:A1 hold. Then the operators $\overline{\mathcal{L}}$ and $\overline{\mathcal{L}^*}$ are maximally dissipative in $Y$ and they generate contraction semigroups $e^{\overline{\mathcal{L}} t}$ and $e^{\overline{\mathcal{L}^*} t}$ on $Y$. Moreover ${\overline{\mathcal{L}^*}} = \mathcal

Figures (4)

  • Figure 1: Left. Stationary distribution for a confining triple well potential. Center/Right. Two different initial configurations.
  • Figure 2: Exponential decay of the (un)controlled trajectories for perturbed (left) and rotated (right) initial configurations.
  • Figure 3: Left. Entire spectrum of the (un)controlled system. Right. Zoom around the $\delta$-unstable part.
  • Figure 4: Left. Gradients of the control shape functions $\alpha_i$. Center. Feedback control laws for the perturbed initial configuration. Right. Feedback control laws for the rotated initial configuration.

Theorems & Definitions (33)

  • Proposition 1
  • proof
  • Remark 2
  • Lemma 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • Proposition 6
  • proof
  • ...and 23 more