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MS56 1: Inverse Problems of Transport Equations and Related Topics
Time:
Monday, 04/Sept/2023:
1:30pm - 3:30pm
Session Chair: Ru-Yu Lai Session Chair: Hanming Zhou
Location:VG2.106
Presentations
Multiscale Parameter Identification: mesoscopic kernel reconstruction from macroscopic data
Kathrin Hellmuth
University of Würzburg, Germany
Motivated from a biological application, we study mesoscopic velocity jump (run-and-tumble) models for particle motion in the phase space. The motion is characterized by a sudden change of direction which is governed by the turning rate. The inverse problem is to determining this mesoscopic turning rate from macroscopic, i.e. directionally averaged data. The lack of directional information in the measurements poses problems in the reconstruction of the mesoscopic quantity. These problems can be leveraged by the use of time dependent interior domain data, as theoretical results on the reconstructability suggest. We then investigate the macroscopic limit behaviour for the inverse problem on the macroscopic regime and present first results on the numerical inversion.
This is joint work with Christian Klingenberg (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).
A 1-Wasserstein framework for forward-peaked diffusive transport
Guillaume Bal1, Benjamin Palacios2
1University of Chicago, USA; 2Pontificia Universidad Catolica, Chile
In this talk, I will present a framework to study inverse problems involving forward-peaked diffusive transport. More specifically, we will study Fokker-Planck approximations to highly forward-peaked scattering and the accuracy of its respective approximations via Fermi pencil-beams, in a metric based on the 1-Wasserstein distance. We argue that metrics of this kind are suitable for analyzing the stability of related inverse problems, for instance, in inverse transport, microscopy, and off-axis laser detection. This is joint work with Guillaume Bal.
Recovery of coefficients in nonlinear transport equations
Hanming Zhou
University of California Santa Barbara, United States of America
In this talk, we will discuss the determination of coefficients in time-dependent nonlinear transport equations. We consider both cases of time-independent and time-dependent coefficients. The talk is based on joint work with Ru-Yu Lai and Gunther Uhlmann.
Mapping properties and functional relations for the hyperbolic X-ray transform
1University of California Santa Cruz, United States of America; 2Leibniz University Hannover; 3Northwestern University
I will discuss recent results on the range characterization of the X-ray transform on the hyperbolic disk, along with functional relations with distinguished differential operators, and mapping properties in adapted scales of Sobolev spaces.
This is based on joint work with Nikolaos Eptaminitakis (Leibniz University Hannover) and Yuzhou Zou (Northwestern).
