Alexander Mamonov, mamonov@gmail.com
Co-authors: Vladimir Druskin, Mikhail Zaslavsky
Krein–Gelfand–Levitan algorithm for inverse hyperbolic problems via spectrally matched finite-difference
grids
We present a method for the numerical solution of inverse problems for coefficients of hyperbolic PDEs based
on the spectrally matched finite-difference grids (a.k.a. Gaussian quadrature rules or optimal grids). The method
is built around an algorithm for interpolation of the measured time domain data. Once an interpolant is obtained,
it can be expressed in terms of Stieltjes continued fraction or its matrix generalization. The use of S-fraction
coefficients in inversion is twofold. First, they can be used to reformulate the traditional optimization-based
approaches to drastically improve the objective functional, which addresses issues such as the abundance of
local minima and slow convergence. Second, the coefficients provide a way to obtain direct, non-iterative reconstructions on the spectrally matched grids. We supplement the theoretical considerations with the numerical
results.