The inverse wave problem in strongly scattering media is highly non-linear. In this seminar talk we show how ideas from reduced-order modeling can be used to mitigate this nonlinearity.
First we will study the one-dimensional dissipative wave equation and show that the wave impedance and losses can be directly recovered from measurements in the frequency-domain. This is an extension of results from Mark Krein's studies of strings with point masses and their connection to Stieltjes continued fractions.
Using the intuition and insights from this one-dimensional problem we will tackle the time-domain inverse problem in higher dimensions. From the measured data, we construct a small dynamical system, a so-called reduced-order model, which explains the observed data and has the structure of a discretized wave operator. Rather than minimizing a data misfit (as is done in classical full waveform inversion), we propose to minimize the misfit in these reduced-order models. To be more specific, we try to find a medium that generates the same reduced-order model as the measured data, rather than a medium that generates that same data. We show that this mitigates the nonlinearity of this problem. Speaker(s): Jörn Zimmerling (University of Michigan)