Verified Numerical Computation for an Inverse Elliptic Eigenvalue Problem

Yoshitaka Watanabe, Nobito Yamamoto and Mitsuhiro T. Nakao
9th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2000) jointly with International Conference on Interval Methods in Science and Engineering (Interval 2000), September 18-22, 2000, Karlsruhe, Germany.

In this talk, it is considered that the reconstruction, from the finite input data of eigenvalues, of an unkown potential in the two-dimensional inverse elliptic eigenvalue problem with Dirichlet boundary conditions. This problem is transformed into a system of finite-dimensional nonlinear equations and, using an interval Newton method described by [1], an algorithm enclosing a solution can be constructed.

In the reconstruction procedure, for some given potential functions, guaranteed enclosures of eigenvalues and normalized eigenfunctions for elliptic eigenvalue problem are needed. And the interval Newton method also requires the numerically verified Jacobian of the finite-dimensional nonlinear map. [1] proposed the reconstruction method in the inverse Sturm-Lioville problem which computes guaranteed enclosures of eigenvalues and eigenfunctions by a modification of the shooting method. It is, however, difficult to directly apply this method for the two-dimensional problems.

This presentation gives an extended method to compute with guaranteed accuracy a set including the exact potential of the inverse elliptic problems incorporating with the numerical verification tequniques proposed in [2]. Some verified numerical examples will be shown.

References:
[1] Neher, M.; Enclosing solutions of an inverse Sturm-Liuoville problem with finite data, Computing 53 (1994), 379--395.
[2] Nagatou, K.; A numerical method to verify the elliptic eigenvalue problems including a uniqueness property, Computing 63 (1999), 109--130.