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Optimal control of stochastic cylinder flow using data-driven compressive sensing method

Liuhong Chen, Ju Ming, Max D. Gunzburger

Abstract

A stochastic optimal control problem for incompressible Newtonian channel flow past a circular cylinder is used as a prototype optimal control problem for the stochastic Navier-Stokes equations. The inlet flow and the rotation speed of the cylinder are allowed to have stochastic perturbations. The control acts on the cylinder via adjustment of the rotation speed. Possible objectives of the control include, among others, tracking a desired (given) velocity field or minimizing the kinetic energy, enstrophy, or the drag of the flow over a given body. Owing to the high computational requirements, the direct application of the classical Monte Carlo methods for our problem is limited. To overcome the difficulty, we use a multi-fidelity data-driven compressive sensing based polynomial chaos expansions (MDCS-PCE). An effective gradient-based optimization for the discrete optimality systems resulted from the MDCS-PCE discretization is developed. The strategy can be applied broadly to many stochastic flow control problems. Numerical tests are performed to validate our methodology.

Optimal control of stochastic cylinder flow using data-driven compressive sensing method

Abstract

A stochastic optimal control problem for incompressible Newtonian channel flow past a circular cylinder is used as a prototype optimal control problem for the stochastic Navier-Stokes equations. The inlet flow and the rotation speed of the cylinder are allowed to have stochastic perturbations. The control acts on the cylinder via adjustment of the rotation speed. Possible objectives of the control include, among others, tracking a desired (given) velocity field or minimizing the kinetic energy, enstrophy, or the drag of the flow over a given body. Owing to the high computational requirements, the direct application of the classical Monte Carlo methods for our problem is limited. To overcome the difficulty, we use a multi-fidelity data-driven compressive sensing based polynomial chaos expansions (MDCS-PCE). An effective gradient-based optimization for the discrete optimality systems resulted from the MDCS-PCE discretization is developed. The strategy can be applied broadly to many stochastic flow control problems. Numerical tests are performed to validate our methodology.
Paper Structure (15 sections, 1 theorem, 45 equations, 13 figures, 6 tables, 2 algorithms)

This paper contains 15 sections, 1 theorem, 45 equations, 13 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Stochastic cylinder flow
  3. Representation of white noise
  4. Polynomial chaos expansion
  5. Formulation of the cylinder flow problem
  6. Data-Driven compressive sensing approach for stochastic cylinder flow
  7. Data-Driven compressive sensing approach
  8. Simulation results using data-driven compressive sensing
  9. Deterministic cylinder flow
  10. Stochastic cylinder flow
  11. Optimal boundary control of stochastic cylinder flow
  12. Objective functionals
  13. Optimization algorithm
  14. Computational experiments
  15. Conclusion

Key Result

Theorem 2.1

[Cameron--Martin theorem cameron1947orthogonal] For any given point $(\bm x, s)\in \Omega\times [0,t)$, if $u(\bm x,s)$ is a stochastic functional of Brownian motion $\omega$ with $\mathbb{E}[u(\bm x,s)]^{2} < \infty$, then $u(\bm x,s)$ can be represented by the expansion where $\{\mathcal{H}_{\bm{\alpha}}\}_{\bm{\alpha}\in\mathcal{I}}$ is the set of polynomials defined by Wick and $\bm{\xi}=(\xi

Figures (13)

  • Figure 1: Channel flow past a circular cylinder.
  • Figure 2: Scaled speed of the deterministic cylinder flow (top) and the associated contours (bottom) at $t=10$.
  • Figure 3: Pressure contours at $t=10$.
  • Figure 4: Time history of the drag coefficient $C_{D}$ and lift coefficient $C_L$ for the deterministic problem.
  • Figure 5: Time history of a realization of the stochastic inlet velocity $u_{in}(0,y,t;\bm{\xi})$ given in \ref{['stin']} for $0$.
  • ...and 8 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Remark 2.1
  • Remark 3.1