Two-level method for a time-independent Fokker–Planck control problem

Abstract

A time-independent Fokker–Planck (FP) control problem and a two-level numerical method are presented. We aim to formulate a control problem that controls the drift of the stochastic process so that the probability density function (PDF) attains a specific steady-state configuration. First-order optimality conditions, which characterize the solution of the control problem, are discretized by the Chang-Cooper (CC) scheme. For positivity and conservativeness of a PDF in the stationary FP control formulation and discretization, we take advantage of CC-scheme. We investigate a two-grid method with coarsening by a factor-of-three strategy. It is found that the coarsening by a factor-of-three strategy simplifies the inter-grid transfer operators and hence the computations. We present several numerical experiments to show the effectiveness of the proposed two-level framework to solve Fokker–Planck or stochastic models control problems with and without control-constrained.

KEYWORDS:

• Fokker–Planck equation
• multidimensional stochastic process
• PDE-constrained optimization
• Chang-Cooper scheme
• multigrid
• finite difference

MSC (2010) CLASSIFICATIONS:

• 35Q84
• 35Q93
• 49K20
• 65N55

Acknowledgements

The author is thankful to the anonymous referees for their careful reading and useful comments that help to improve the paper significantly.

Disclosure statement

No potential conflict of interest was reported by the author(s).