Hearing the Triangles: A Numerical Perspective

Abstract

We introduce a two-step numerical scheme for reconstructing the shape of a triangle by its Dirichlet spectrum. With the help of the asymptotic behavior of the heat trace, the first step is to determine the area, the perimeter, and the sum of the reciprocals of the angles of the triangle. The shape is then reconstructed, in the second step, by an application of the Newton's iterative method or the Levenberg-Marquardt algorithm for solving a nonlinear system of equations on the angles. Numerically, we have used only finitely many eigenvalues to reconstruct the triangles. To our best knowledge, this is the first numerical simulation for the classical inverse spectrum problem in the plane. In addition, we give a counter example to show that, even if we have infinitely many eigenvalues, the shape of a quadrilateral may not be heard.

Keywords

Inverse spectral problems / Newton iteration / Vandermonde matrix / ill-posedness / triangles

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