|Prof. Dr. Liliana Borcea
Peter Field Collegiate Professor of Mathematics, University of Michigan, USA
Working Areas: Inverse scattering in random media, Electro-magnetic inverse problems, Effective properties of composite materials, transport in high contrast, heterogeneous media
Title of the talk: Quantitative inverse scattering via reduced order modeling.
Abstract: I will discuss an inverse problem for the wave equation, where an array of sensors probes an unknown heterogeneous medium with waves and measures the echoes.The goal is to determine scattering structures in the medium modeled by a reflectivity function.
Much of the existing imaging methodology is based on a linear least squares data fit approach. However, the mapping between the reflectivity and the wave measured at the array is nonlinear and the resulting images have artifacts. I will show how to use a reduced order model (ROM) approach to solve the inverse scattering problem. The ROM is data driven i.e., it is constructed from the data, with no knowledge of the medium. It approximates the wave propagator, which is the operator that maps the wave from one time step to the next. I will show how to use the ROM to: (1) Remove the multiple scattering (nonlinear) effects from the data, which can then be used with any linearized inversion algorithm. (2) Obtain a well conditioned quantitative inversion algorithm for estimating the reflectivity.
|Prof. Dr. Bernd Hofmann
Head of Professorship Inverse Problems, Faculty of Mathematics Chemnitz University of Technology, Germany
Working Areas: Regularization of inverse and ill-posed problems
Title of the talk: Ill-posedness concepts and the distinguished role of smoothness in regularization for linear and nonlinear inverse problems. Abstract: In a first part, we consider different concepts of ill-posedness for mathematical models of inverse problems expressed by linear and non-linear operator equations in infinite dimensional Hilbert and Banach spaces. The concepts of Hadamard and Nashed are recalled, which are appropriate for linear operator equations. Compact linear forward operators in Hilbert spaces allow for a pre-selection of the degree of ill-posedness by verifying the decay rate of the singular values indicating the operator smoothness. The interplay of operator smoothness and solution smoothness is responsible for opportunities and limitations of stable approximate solutions to inverse problems by regularization. For non-linear operator equations, the nature of ill-posedness may vary with the solution point. The presented concept of local ill-posedness is adapted to that feature.
In a second part, the impact of different varieties of smoothness on variants of variational regularization is under consideration. In this context, variational source conditions represent a sophisticated tool for expressing the solution smoothness with respect to the character of the forward operator, for non-linear problems even in combination with the occurring structure of non-linearity. Also the role of conditional stability estimates in combination with Tikhonov regularization is outlined. Several example situations for applying regularization approaches are presented, including sparsity-promoting versions. We also discuss the specific difficulties and some recent results for the Tikhonov regularization with oversmoothing penalties.
Research is supported by the German Research Foundation (DFG) under grant HO 1454/12-1 and embedded in the Austrian/German joint research project “Novel Error Measures and Source Conditions of Regularization Methods for Inverse Problems (SCIP)” with Prof. Otmar Scherzer (Vienna) based on the D-A-CH Lead-Agency Agreement.
|Prof. Dr. John C Schotland
Director, Michigan Center for Applied and Interdisciplinary Mathematics University of Michigan, USA
Working Areas: Inverse problems with applications to imaging, Scattering theory, Waves in random media, Nano-scale optics, Coherence theory and quantum optics
Title of the talk: Superresolution and inverse problems with internal sources
Abstract: I will discuss a method to reconstruct the optical properties of a scattering medium with subwavelength resolution. The method is based on the solution to the inverse scattering problem with internal sources. Applications to photoactivated localization microscopy are described.
|Prof. Dr. Erkki Somersalo
Professor of Mathematics, Case Western Reserve University, USA
Working Areas: Computational and statistical inverse problems, Probabilistic methods for uncertainty quantification, Modeling of complex systems, Biomedical applications
Title of the talk: Hypermodels, sparsity and approximate Bayesian computing.
Abstract: In numerous applications involving an underdetermined large scale inverse problem, sparsity of the solution is a desired property. In the Bayesian framework of inverse problems, sparsity requirement of the solution may be implemented by properly defining the prior distribution. While Gaussian priors are not well suited for promoting sparsity, certain hierarchical, conditionally Gaussian models have been demonstrated to be efficient. In this talk, we review a general class of conditionally Gaussian hypermodels that provide a flexible framework for promoting sparsity of the solution, and discuss approximate iterative methods that can be used both for finding single sparse solutions as well as for approximate sampling of the posterior distributions. The motivation for this work comes from an ongoing work on brain imaging using magnetoencephalography data.
|Prof. Dr. Gunter Uhlmann
Walker Family Endowed Professor of Mathematics, University of Washington, USA IAS Si Yuan Professor and Chair Professor of Mathematics, Institute for Advanced Study, HKUST, Hong Kong
Working Areas: Inverse problems and imaging, Partial differential equations, Microlocal analysis, Scattering theory
Title of the talk: Inverse Problems for Nonlinear Equations.
Abstract: We will survey recent developments in the solution of inverse problems for nonlinear equations. The nonlinearity helps to solve several inverse problems that cannot be solved for the linearized equations.