We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography.
Variational Regularization of Inverse Problems for Manifold-Valued Data
Image Analysis and Learning Group, Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Germany（图像分析与学习组，跨学科科学计算中心，海德堡大学，德国）；Department of Mathematics and Natural Sciences, Hochschule Darmstadt, and Institute of Computational Biology, Helmholtz Zentrum München, Germany（数学与自然科学系，达姆施塔特应用科学大学，以及计算生物学研究所，海德堡研究中心，德国）
Inverse regression for ridge recovery: A data-driven approach for parameter reduction in computer experiments
Department of Computer Science University of Colorado Boulder（科罗拉多大学博尔德分校计算机科学系）；Department of Applied Statistics University of Minnesota（明尼苏达大学应用统计学系）
Towards a machine learning pipeline in reduced order modelling for inverse problems: neural networks for boundary parametrization, dimensionality reduction and solution manifold approximation
