Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Manifold learning and an inverse problem for a wave equation
Speaker: Matti Lassas, University of Helsinki
Date: November 30, 2020
Time: 16:00:00 Hours
Venue: AG-77
Abstract: We consider invariant manifold learning and its applications in wave imaging. The invariant manifold learning problem, also known as the geometric Whitney problem, means the construction of a manifold $M$ and its Riemannian metric $g$ using a discrete metric space $(X,d_X)$ that approximates the manifold in the Gromov-Hausdorff sense. This problem is closely related to manifold interpolation where a smooth $n$-dimensional surface $S\subset \mathbb R^m$, $m>n$ needs to be constructed to approximate a point cloud in $\mathbb R^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. As an example, we consider an inverse problem for a wave equation $(\partial_t^2-\Delta_g)u(x,t)=F(x,t)$ on a Riemannian manifold $(M,g)$. We assume that we are given an open subset $V$ of $M$ and the source-to-solution map that maps a source supported in $V\times \mathbb R_+$ to the restriction of the solution $u$ in the set $V\times \mathbb R_+$. This map corresponds to the measurements made on the set $V$. The results on the first problem are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan, and the results on the second problem with R. Bosi and Y. Kurylev.
Title of Talk: Manifold learning and an inverse problem for a wave equation
Speaker: Matti Lassas, University of Helsinki
Date: November 30, 2020
Time: 16:00:00 Hours
Venue: AG-77
Abstract: We consider invariant manifold learning and its applications in wave imaging. The invariant manifold learning problem, also known as the geometric Whitney problem, means the construction of a manifold $M$ and its Riemannian metric $g$ using a discrete metric space $(X,d_X)$ that approximates the manifold in the Gromov-Hausdorff sense. This problem is closely related to manifold interpolation where a smooth $n$-dimensional surface $S\subset \mathbb R^m$, $m>n$ needs to be constructed to approximate a point cloud in $\mathbb R^m$. These questions are encountered in differential geometry, machine learning, and in many inverse problems encountered in applications. As an example, we consider an inverse problem for a wave equation $(\partial_t^2-\Delta_g)u(x,t)=F(x,t)$ on a Riemannian manifold $(M,g)$. We assume that we are given an open subset $V$ of $M$ and the source-to-solution map that maps a source supported in $V\times \mathbb R_+$ to the restriction of the solution $u$ in the set $V\times \mathbb R_+$. This map corresponds to the measurements made on the set $V$. The results on the first problem are done in collaboration with C. Fefferman, S. Ivanov, Y. Kurylev, and H. Narayanan, and the results on the second problem with R. Bosi and Y. Kurylev.