An Efficient Monte Carlo Method for Optimal Control Problems with Uncertainty

A general framework is proposed for what we call the sensitivity derivative Monte Carlo (SDMC) solution of optimal control problems with a stochastic parameter. This method employs the residual in the first-order Taylor series expansion of the cost functional in terms of the stochastic parameter rather than the cost functional itself. A rigorous estimate is derived for the variance of the residual, and it is verified by numerical experiments involving the generalized steady-state Burgers equation with a stochastic coefficient of viscosity. Specifically, the numerical results show that for a given number of samples, the present method yields an order of magnitude higher accuracy than a conventional Monte Carlo method. In other words, the proposed variance reduction method based on sensitivity derivatives is shown to accelerate convergence of the Monte Carlo method. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the proposed method is a fraction of the total time of the Monte Carlo method.

[1]  Wei Chen,et al.  Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design , 2000 .

[2]  Dominique Pelletier,et al.  Parametric uncertainty analysis for thermal fluid calculations , 2001 .

[3]  Carl D. Sorensen,et al.  A general approach for robust optimal design , 1993 .

[4]  Robert W. Walters,et al.  Random field solutions including boundary condition uncertainty for the steady-state generalized Burgers equation , 2001 .

[5]  Max Gunzburger,et al.  SENSITIVITIES, ADJOINTS AND FLOW OPTIMIZATION , 1999 .

[6]  David H. Sharp,et al.  Prediction and the quantification of uncertainty , 1999 .

[7]  P. A. Newman,et al.  Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives , 2001 .

[8]  S. Kodiyalam,et al.  Structural optimization using probabilistic constraints , 1992 .

[9]  R. Nicolaides,et al.  A SELF-CONTAINED, AUTOMATED METHODOLOGY FOR OPTIMAL FLOW CONTROL VALIDATED FOR TRANSITION DELAY , 1995 .

[10]  Luc Huyse,et al.  Solving Problems of Optimization Under Uncertainty as Statistical Decision Problems , 2001 .

[11]  Hussaini M. Yousuff,et al.  A Self-Contained, Automated Methodology for Optimal Flow Control , 1997 .

[12]  David M. Gay,et al.  Algorithm 611: Subroutines for Unconstrained Minimization Using a Model/Trust-Region Approach , 1983, TOMS.

[13]  Yanzhao Cao,et al.  Shape optimization for noise radiation problems , 2002 .

[14]  Yoon-ha Lee,et al.  Uncertainty Quantification for Multiscale Simulations , 2002 .

[15]  M GayDavid,et al.  Algorithm 611: Subroutines for Unconstrained Minimization Using a Model/Trust-Region Approach , 1983 .