Inverse Problems for Graphs and Discrete Spaces

Abstract

The speaker will give a review on the recent results on inverse problems on finite graphs that are analogous to extensively studied inverse spectral problems and inverse problems for heat equations studied for the partial differential equations. He and his research group study the inverse problem of determining a finite weighted graph $(X,E)$ from the source-to-solution map on a vertex subset $B\subset X$ for heat equations on graphs, where the time variable can be either discrete or continuous. They prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on $B$. In particular, they consider inverse problems for discrete-time random walks on finite graphs. They show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on $B$, or from the observation on $B$ of one realization of the random walk. In addition, they will consider the problem of approximating discrete metric spaces or graphs by smooth manifolds.

About the Speaker

Prof. Matti LASSAS obtained his MS and PhD, both in Mathematics, from the University of Helsinki in 1992 and 1996 respectively. He continued his research as a Postdoctoral Fellow and then a Research Fellow at the Academy of Finland in 1998-2004. He joined the Helsinki University of Technology as a Professor in 2004 and then moved to the University of Helsinki in 2009. He is currently a Professor of Applied Mathematics in the Department of Mathematics and Statistics in the University of Helsinki. He is also the Director of the Centre of Excellence of Inverse Modelling and Imaging, funded by the Academy of Finland.

Prof. Lassas’ research focuses on the theoretical aspects in geometric, analytic and statistical inverse problems. He is the editor of journals, SIAM Journal on Applied Mathematics, Research in the Mathematical Sciences, Inverse Problems and Imaging and Abstract and Applied Analysis.

Prof. Lassas is the recipient of the 2020 E. J. Nyström Prize of the Finnish Society of Sciences and Letters, the 2007 Calderón Prize of the International Inverse Problems Association and the 2004 Väisälä Prize of the Finnish Academy of Science and Letters. In 2012 and 2020, he was elected as a Member of the Finnish Academy of Science and Letters and of the Finnish Society of Sciences and Letters respectively. He is also the Member of the American Mathematical Society, the Society of Industrial and Applied Mathematics and the European Mathematical Society.

For Attendees' Attention

This talk will be held online via Zoom. To attend, please join the Zoom meeting at https://hkust.zoom.us/j/93180011402 (Meeting ID: 931 8001 1402 / Passcode: iasip2023).

About the Program

For more information, please refer to the program website at https://iasprogram.hkust.edu.hk/inverseproblems/.