报告题目：A neural network approach to learning solutions of a class of elliptic variational inequalities

报 告 人：Prof. Dr. Michael Hintermüller (Weierstrass Insitute (WIAS) and Humboldt University of Berlin)

报告时间：2024年12月9日16:50-

报告地点：格物楼数学研究中心528报告厅

报告摘要：We develop a weak adversarial approach to solving obstacle problems using neural networks. By using (generalized) regularized gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to a learning approach. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity or lack of strict complementarity, a situation that poses difficulty for traditional numerical methods.

报告人简介：Michael Hintermüller is currently the director of Weierstrass Institute for Applied Analysis and Stochastics and the Chair Professor of Applied Mathematics in Humboldt University of Berlin. Among a couple of prominent positions, he is also the Chair of Berlin Mathematics Research Center named MATH+, which is one of the three Excellence Clusters in Mathematics funded by German Science Foundation (DFG).  Michael Hintermüller received his Ph.D. degree in technical mathematics from the University of Linz, Austria, in 1997. He had a Postdoctoral position and an Assistant Professor position at the University of Graz, where he got his Habilitation in Mathematics in 2003. From 2003 to 2004, he held a visiting Associate Professorship at Rice University, Houston, and an Associate Professorship at the University of Graz from 2004–2007. From 2007 to 2008, he held a Chair in Applied Mathematics at the University of Sussex, U.K. In 2008, he became a MATHEON Research Professor and a W3-Professor in Applied Mathematics at the Department of Mathematics of Humboldt-University of Berlin, Germany. He has won several important prizes and honours, e.g., in 2005, he received the prestigious START-prize from the Austrian Ministry of Science and Education; in 2006, he got a SIAM Outstanding Paper Prize; he was a Young Member of Austrian Academy of Sciences from 2009 - 2017, and was elected as SIAM Fellow in 2016. He is currently or was on the editorial board of many important journals in applied and computational mathematics, including SIOPT, Math. Program., SINUM, SISC for example.His main research areas are optimization with partial differential equation constraints, mathematical image processing, inverse problems and problems in shape and topology optimization。