TY - JOUR
T1 - Hearing the Triangles: A Numerical Perspective
AU - Gong , Wei
AU - Liu , Xiaodong
AU - Wang , Jing
JO - CSIAM Transactions on Applied Mathematics
VL - 1
SP - 58
EP - 72
PY - 2024
DA - 2024/02
SN - 5
DO - http://doi.org/10.4208/csiam-am.SO-2023-0027
UR - https://global-sci.org/intro/article_detail/csiam-am/22920.html
KW - Inverse spectral problems, Newton iteration, Vandermonde matrix, ill-posedness, triangles.
AB - <p style="text-align: justify;">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.</p>