Algorithms And Analysis For Parameter Identification Of Partial Differential Equations

Inverse problems of differential equations have extensive applications in science and engi-neering fields. The lack of well-posedness makes inverse problems much more challenging than forward problems in common cases. Therefore, finding algorithms for the inverse problems is of substantial and growing interest in the field of scientific computation.Parameter identification of differential equations is a key branch of inverse problems. The paper presents the contemporary state of the theory, some applications and numerical methods of inverse problems in partial differential equations.An inverse problem of differential equations is usually recast into the optimization of cor-responding output least-square functional with respect to the unknown coefficient. We focus on the numerical solution of Robin inverse problem of partial differential equations, and obtain the following new achievements:1. To reconstruct the piecewise constant Robin coefficient, a technique based on determining the discontinuous points of the unknown coefficient is suggested. We investigate the first and twice differentiability of the solution and the objective functional with respect to the discontinuous points. Then we apply the Gauss-Newton method to reconstruct the shape of the unknown Robin coefficient. Numerical examples illustrate its efficiency and stability.2. In general, we proposed both complex and real coupled boundary method, combining both Neumann and Dirichlet measurements into a single Robin condition on the inaccessible part of the boundary, and reconstruct the unknown Robin coefficient by optimizing the corresponding objective functionals.For the complex case, we recast the origin real-space inverse problem into a complex one, then we optimize the imaginary part of the solution in the domain to determine the unknown piecewise Robin coefficient.For the real case, two different boundary value problems, depending on two different positive numbers α1 and α2, are introduced such that the two types of boundary measurements are coupled into a single Robin condition. By applying the Kohn-Vogelius approach with Tikhonov regulariza-tion, a feasible reconstruction could be obtained even for very small regularization parameter.