Application of optimal transport to the computation of Wasserstein gradient flows and an inverse problem in seismic imaging

Optimal transport is nowadays an established mathematical tool for many applications. In this talk I will discuss two specific instances. In the first part I will review the work I have done during my thesis on the computation of Wasserstein gradient flows. Many parabolic PDEs can be cast as steepest descent curves in the space of probability measures endowed with the Wasserstein distance, an optimal transport metric. Taking advantage of this structure may be extremely helpful both at the theoretical and the numerical level. I will show how we used it to construct finite volume schemes to approximate these problems, preserving the underlying variational structure, that could be at the same time robust, precise and efficient. In the second part I will present the subject of my current postdoc, an application of optimal transport to seismic imaging. The objective is to use optimal transport to measure the misfit between multi-component seismograms, namely signed oscillating signals. Multi-component data can be lifted into positive semi-definite matrices using for example Pauli's transformation. Therefore, we designed an optimal transport problem between positive semi-definite matrix valued measures. We achieved this by extending in a natural way to this setting the Beckmann's formulation of the L1 transport. I will provide some good preliminary results of the application of this misfit function to the inverse problem of Full Waveform Inversion, showing the effectiveness of the approach.