Di., 29.05.2018 um 16:30 Uhr in G.15.20
Vortrag im Mathematischen Kolloquium
Inverse problems are omnipresent in many scientific fields such as systems biology, engineering, medical imaging, and geophysics. The main challenges toward obtaining meaningful real-time solutions to large, data-intensive inverse problems are ill-posedness of the problem, large parameter dimensions, and/or complex model constraints. This talk discusses computational challenges of inverse problems by exploiting a combination of tools from applied linear algebra, parameter estimation and optimization, and statistics. For instance, for large scale ill-posed inverse
problems, approximate solutions are computed using a regularization method that solves a nearby well-posed problem. Oftentimes, the selection of a proper regularization parameter is the most critical and computationally intensive task and
may hinder real-time computations of the solution. We present a new framework for solving ill-posed inverse problems by computing optimal regularized inverse matrices. We further discuss randomized Newton and randomized quasi-Newton
approaches to efficiently solve large linear least-squares problems, where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In this framework, randomness is introduced as a means to overcome computational limitations, and probability distributions that can exploit structure and/or sparsity are considered. We will present numerical examples, from deblurring, tomography, and machine learning to illustrate the challenges and our proposed methods.