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Numerical solution and convergence speed of variational formulation for linear schrodinger equation

This paper presents a numerical solution of an optimal control problem for linear parabolic equations. The estimates for the error of the difference scheme and the speed of convergence have been established. Numerical results are reported on test problems. | Turk J Math 26 (2002) , 397 – 420. ¨ ITAK ˙ c TUB Numerical Solution and Convergence Speed of Variational Formulation for Linear Schr¨ odinger Equation Murat Suba¸sı, B¨ unyamin Yıldız Abstract This paper presents a numerical solution of an optimal control problem for linear parabolic equations. The estimates for the error of the difference scheme and the speed of convergence have been established. Numerical results are reported on test problems. Key Words: Optimal control problem, Finite difference method, Inverse problem 1. Introduction and Statement of the Problem Many theoretical phenomena which are governed by linear and nonlinear parabolic equations have been investigated in the field of optimal control. Many problems in theoretical physics, for example [1], [2], [3], [4], can be expanded to these types of problems. Also, determining the quantum-mechanical potential is one of the basic problems of the quantum mechanics. Given simplifying assumptions, this potential is determined on the basis of intuitive concepts [1]. Problems of determining interaction potentials have stimulated the development of scattering theory [1]. Different approaches for solving optimal control problems of parabolic systems and inverse problems were proposed in [8], [10], AMS Mathematics Subject Classification: 49J20, 65L12, 81U40 397 SUBAS ¸ I, YILDIZ [11], [12] and numerical solutions of Schr¨ odinger equations were investigated in [9], [16]. In many applications as in the quantum-mechanical, heat equation and hydrology there is a need to recover control coefficient v from boundary measurements of solutions of a parabolic equations. The inverse problem of determining the quantum-mechanical potential is not well-posed, and this also holds in the variational formulation, so it is quite difficult to obtain numerical solution. In the variational formulation, an optimal control of the coefficient, that is, of the potential in the Schrodinger equation, is produced [13]. The .