Title: Unique continuation for the wave equation: The stability landscape
Abstract
Inverse and ill-posed problems subject to the wave equation occur in a variety of applications ranging from seismology to medicine. Among these problems, the task of unique continuation is perhaps one of the simplest: Given measurements of the solution (of the wave equation) in a subset of the space-time cylinder but no initial data, we aim to extend the solution as far as possible beyond the data set – ideally to the entire cylinder. It is well-known that this maximal extension is indeed possible in a Lipschitz-stable manner, provided that additional measurements of the solution on the lateral boundary of the cylinder are available (and certain geometric conditions are fulfilled). In this setting, many innovative numerical methods have been proposed in recent years to compute the extension numerically.
However, the question what happens if no measurements on the boundary are available seems to be far less understood. In this talk we provide analytical and numerical results which show that in this case the solution can be extended in a Hölder-stable manner to a certain proper(!) subset of the space-time domain. Trying to extend further leads to degeneration of the stability until a logarithmic regime is reached and the accuracy of the corresponding extension becomes too poor to suit practical purposes. We present a possible remedy for this problem: If one can explicitly construct a small finite dimensional space in which the trace of the solution on the boundary can be approximated with high-accuracy, then the solution can be recovered with Lipschitz stability everywhere. This is an extremely powerful yet dangerous technique as its success hinges entirely on a proper choice of this finite dimensional space.
The talk is based on joint work with Erik Burman and Lauri Oksanen.
