Proper Orthogonal Decomposition:

Optimal control problems for nonlinear partial differential equations are often hard to tackle numerically so that the need for developing novel techniques emerges. One such technique is given by suboptimal control strategies as developed, for instance, in [1,2] in the context of optimal control of the Navier-Stokes equations. Another approach is given by reduced order methods. These techniques are based on projecting the dynamical system onto subspaces consisting of basis elements that contain characteristics of the expected solution. This is in contrast to e.g. finite element techniques, where the elements of the subspaces are uncorrelated to the physical properties of the system that they approximate. The reduced basis method as developed in [3] is one such reduced order method with the basis elements corresponding to the dynamics of expected control regimes. The proper orthogonal decomposition (POD) is a procedure to determine the basis elements for the approximating subspaces on which to solve the optimal control problems. POD has proved to be an efficient procedure in several areas of science including the modeling of turbulent flow (see e.g. [3,4]). In [5] POD is utilized to solve open and closed loop optimal control problems for the Burgers equation which is a simple model for non-linear convection diffusion phenomena. The relative simplicity of the equation allows comparison of POD-based algorithms with numerical results obtained from finite element discretization of the optimality system. For closed loop control suboptimal state feedback strategies are presented. The work [6] is devoted to study POD as a minimization problem in a general Hilbert space setting. The POD-basis functions are given by the solution to the first-order necessary optimality conditions. In this work POD is utilized to solve optimal control problems for a phase-field model. The numerical results are compared with finite element solutions. In [7] the close connection between the singular value decomposition and POD is studied in complex Hilbert spaces. For the practical calculation of the POD basis functions perturbation error bounds are presented. By a numerical example the theoretical results are illustrated. In [8] error estimates for Galerkin POD based methods for parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. First, linear evolution problems are studied. For the time integration the backward Euler, Crank-Nicolson as well as the forward Euler methods are analyzed. Secondly, the analysis is extended to certain non-linear problems: to semi-linear problems with Lipschitz non-linearity and to the Burgers equation.

[1] Choi, H., Hinze, M., and Kunisch, K.: Suboptimal control of backward-facing-step-flow. Preprint No. 571/97, Technical University of Berlin, 1997.

[2] Choi, H., Temam, R., Moin, P., and Kim, J.: Feedback control for unsteady flow and its application to the stochastic Burgers equation. Journal of Fluid Mechanics, 1993.

[3] Ito, K., and Ravindran, S. S.: A reduced basis method for control problems governed by PDEs. Control and Esimation of Distributed Parameter Systems, International Series of Numerical Mathematics, Vol. 126, 1998.

[4] Broomhead, D. S., and King, G. P.: Extracting qualitative dynamics from experimantal data. Physica, Vol. 20D, pp. 217-236, 1986.

[5] Kunisch, K., and Volkwein, S.: Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition. To appear in JOTA

[6] Volkwein, S.: Optimal control of a phase-field model using the proper orthogonal decomposition. To appear in ZAMM.

[7] Volkwein, S.: Proper orthogonal decomposition and singular value decomposition. Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Juni 1999.

[8] Kunisch, K. and Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Spezialforschungsbereich F 003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 171, September 1999.

last changed: 6/23/2003