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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. [ABSTRACT FROM AUTHOR]

This article deals with the design and analysis of reliable and efficient numerical methods for the solution of inverse problems in diffuse optical tomography. The proposed fully discrete algorithms are based on iterative regularization methods, derived and analysed on the continuous level, and their careful discretization by finite element methods. This guarantees convergence of the fully discrete algorithms under the same conditions as required on the continuous level, and allows to establish mesh-independent reliability and performance. The derivatives and adjoints can be shown to be exact on the discrete level, which implies that the Gauss-Newton systems used in the reconstructions are real symmetric and positive definite, and the conjugate gradient method can be used for their efficient solutions. We also present complexity estimates of the algorithms, and discuss a posteriori error estimators for assessing the discretization error. The efficiency and robustness of the proposed methods are demonstrated in numerical tests. [ABSTRACT FROM AUTHOR]

Fluorescence optical tomography is a noninvasive imaging modality that employs the absorption and reemission of light by fluorescent dyes. The aim is to reconstruct the fluorophore distribution in a body from measurements of light intensities at the boundary. Due to the diffusive nature of light propagation in tissue, fluorescence tomography is a nonlinear and severely ill-posed problem, and some sort of regularization is required for a stable solution. In this paper, we investigate reconstruction methods based on Tikhonov regularization with nonlinear penalty terms, namely total-variation regularization and a levelset-type method using a nonlinear parameterization of the unknown function. Moreover, we use the full 3-D nonlinear forward model, which arises from the governing system of partial differential equations. We discuss the numerical realization of the regularization schemes by Newton-type iterations, present some details of the discretization by finite-element methods, and outline the efficient implementation of sensitivity systems via adjoint methods. As we will demonstrate in numerical tests, the proposed nonlinear methods provide better reconstructions than standard methods based on linearized forward models and linear penalty terms. We will additionally illustrate that the careful discretization of the methods derived on the continuous level allows us to obtain reliable, mesh-independent reconstruction algorithms. [ABSTRACT FROM PUBLISHER]

Many inverse problems are ill-posed in the sense that their solution is unstable with respect to data perturbations, and hence regularization methods have to be used for their stable solution. Two possible drawbacks of standard regularization methods are (a) saturation, i.e., only suboptimal approximations can be found for smooth solutions—this is the case, e.g., for Tikhonov regularization; (b) numerical effort, e.g., the large number of iterations of methods like Landweber iteration. A framework that allows us to overcome both effects at least for certain classes of inverse problems is regularization in Hilbert scales, where a solution is searched in a scale of spaces (Hilbert scale), but convergence is achieved in the original space. Regularization methods in Hilbert scales can be viewed as modified (preconditioned) versions of standard methods. In order to make the advantages of the Hilbert scale approach applicable to a new class of problems, we propose to use a scale of spaces over the image space instead. This results in a new family of methods which we call γ-scale regularization, and whose (optimal) convergence properties are analyzed. One of the key steps in the analysis is the formulation of an adequate a posteriori stopping rule and the proof of optimal convergence rates. The theoretical results are illustrated in several numerical examples. [ABSTRACT FROM AUTHOR]