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An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation
Authors:Z Sabeh  Mostafa Shamsi  Ionel Michael Navon
Institution:1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran;2. Department of Scientific Computing, Florida State University, Tallahassee, FL 32306‐4120, USA
Abstract:This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.
Keywords:discrete empirical interpolation method (DEIM)  finite element method  indirect shooting method  necessary optimality conditions  optimal control of wave equation  proper orthogonal decomposition (POD)
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