Hearing the Triangles: A Numerical Perspective (CSIAM Transactions on Applied Mathematics, 2024)
We introduce a two-step numerical scheme for reconstructing the shape ofa triangle by its Dirichlet spectrum. With the help of the asymptotic behavior of theheat trace, the first step is to determine the area, the perimeter, and the sum of thereciprocals of the angles of the triangle. The shape is then reconstructed, in the secondstep, by an application of the Newton’s iterative method or the Levenberg-Marquardtalgorithm for solving a nonlinear system of equations on the angles. Numerically,we have used only finitely many eigenvalues to reconstruct the triangles. To our bestknowledge, this is the first numerical simulation for the classical inverse spectrumproblem in the plane. In addition, we give a counter example to show that, even if wehave infinitely many eigenvalues, the shape of a quadrilateral may not be heard.
Keywords: Inverse spectral problems; Newton iteration; Vandermonde matrix; ill-posedness; triangles.