A Formula for Time-to-Frequency Wave Boundary Data Conversion by the Boundary Control Method

TL;DR

This paper presents an explicit, stable method to convert time-domain boundary data into frequency-domain data for wave equations on manifolds, enabling better boundary control and reconstruction in inverse problems.

Contribution

It derives a new explicit formula for converting time-to-frequency boundary data using the boundary control method, applicable to general Riemannian manifolds.

Findings

The formula is explicit and stable with fixed regularization.

Numerical validation is demonstrated on 1D Euclidean and non-Euclidean examples.

The method requires only boundary data and is applicable at any non-eigenfrequency.

Abstract

Given the wave equation on a compact Riemannian manifold with boundary, we derive an explicit reconstruction procedure to represent the frequency-domain Neumann-to-Dirichlet map in terms of the time-domain Neumann-to-Dirichlet map at any non-eigenfrequency. If the wave equation is exactly controllable, we derive an explicit formula to compute the former from the latter. The derivation is based on the boundary control method and requires only knowledge on the boundary of the manifold. The formula is stable when the level of regularization is fixed. The numerical feasibility is validated using one-dimensional examples in both Euclidean and non-Euclidean geometries.