Regularizing Inverse Problems

An inverse problem reconstructs the unknown internal parameters of a subject based on collected data derived synthetically or from real measurements. Inverse problems often lack the well-posedness defined by J. Hadamard; in other words, solutions of inverse problems, namely the reconstructions of the parameters, may not exist, may not be unique or may be unstable. Regularization is a technique that deals with such situations. The well-known Tikhonov regularization method translates the original inverse problem to optimization problems of minimizing the norm of the data misfit plus a weighted regularization functional that incorporates the a priori information we may have about the original problem. The choices of the regularization functional r(q) include ‖q‖■(2@L^(2 ) )┤‖q‖■(2@H^(1) ), |q|BV and |q|TV. However, each of these has its limitations. In this work, we develop a novel H^(s) seminorm regularization method and present numerical results for model problems. This method relies on the evaluation of the seminorms of an intermediary Hilbert space, namely H^(s) space, that stays between L^(2) and H^(1). The H^(s) seminorm regularization is designed to minimize the undesirable aspects of the existing L^(2) and H^(1) regularization functionals. The H^(s) seminorm regularization also allows discontinuities and stabilizes the perturbations. We study the H^(s) seminorm regularization method both theoretically and numerically. We consider the theoretical analysis of this new regularization method based on a model problem. We show that a stable solution can be achieved with some conditions. In addition, we prove the convergence and guarantee a convergence rate provided additional conditions for the model problem when the considered domain is 1D. Numerically, we produce an approximated discretization of the H^(s) seminorm regularization that can be applied to 1D, 2D or 3D examples. We also provide reconstructions of both continuous and discontinuous parameters from synthetic data and a comparison of these solutions to the ones based on existing L^(2) and H^(1) regularization methods. Furthermore, we also apply the H^(s) seminorm regularization method to a fluorescence optical tomography problem. In summary, we study and implement the H^(s) seminorm regularization method for inverse problems, which can provide a stable solution to the model problem. The numerical results indicate the robustness of the new method and suggests that the H^(s) seminorm regularization method produces the closest approximation of the exact solution than the L^(2) norm and H^(1) seminorm regularization methods for the model problem

Regularization Methods in Banach Spaces applied to Inverse Medium Scattering Problems

This work handles inverse scattering problems for both acoustic and electromagnetic waves. That is to reconstruct the irradiated media from measurements of the scattered felds by regularization methods. As a particular feature, the contrasts of the scattering objects are assumed to be supported within a small region, hence called sparse. To apply sparsity regularization schemes it becomes crucial to model the problems in Banach spaces. Traditionally, they are given in a Hilbert space setting, such that reformulation in an L p-sense becomes a key point. Contrasts are linked to the data by forward operators, basing on beforehand stated solution operators and their continuity properties. Thereby, appropriate regularization techniques providing sparsity are given. As the case of scalar-valued contrast functions is already covered in the literature, mainly inverse scattering problems for anisotropic media are shown. In the case where electromagnetic waves are considered, a distinction is made between magnetic and non-magnetic media, since the latter is less complex. Finally, the case of inverse acoustic backscattering is handled, which is rarely seen in literature

Regularized Linear Inversion with Randomized Singular Value Decomposition

In this work, we develop efficient solvers for linear inverse problems based on randomized singular value decomposition (RSVD). This is achieved by combining RSVD with classical regularization methods, e.g., truncated singular value decomposition, Tikhonov regularization, and general Tikhonov regularization with a smoothness penalty. One distinct feature of the proposed approach is that it explicitly preserves the structure of the regularized solution in the sense that it always lies in the range of a certain adjoint operator. We provide error estimates between the approximation and the exact solution under canonical source condition, and interpret the approach in the lens of convex duality. Extensive numerical experiments are provided to illustrate the efficiency and accuracy of the approach.Comment: 20 pages, 4 figure

Quanta of Maths

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics

Regularization techniques based on Krylov subspace methods for ill-posed linear systems

This thesis is focussed on the regularization of large-scale linear discrete ill-posed problems. Problems of this kind arise in a variety of applications, and, in a continuous setting, they are often formulated as Fredholm integral equations of the first kind, with smooth kernel, modeling an inverse problem (i.e., the unknown of these equations is the cause of an observed effect). Upon discretization, linear systems whose coefficient matrix is ill-conditioned and whose right-hand side vector is affected by some perturbations (noise) must be solved. %Because of the ill-conditioning of the system matrix and the errors in the data, In this setting, a straightforward solution of the available linear system is meaningless because the computed solution would be dominated by errors; moreover, for large-scale problems, solving directly the available system could be computationally infeasible. Therefore, in order to recover a meaningful approximation of the original solution, some regularization must be employed, i.e., the original linear system must be replaced by a nearby problem having better numerical properties. The first part of this thesis (Chapter 1) gives an overview on inverse problems and briefly describes their properties in the continuous setting; then, in a discrete setting, the most well-known regularization techniques relying on some factorization of the system matrix are surveyed. The remaining part of the thesis is concerned with iterative regularization strategies based on some Krylov subspaces methods, which are well-suited for large-scale problems. More precisely, in Chapter 2, an extensive overview of the Krylov subspace methods most successfully employed with regularizing purposes is presented: historically, the first methods to be used were related to the normal equations and many issues linked to the analysis of their behavior have already been addressed. The situation is different for the methods based on the Arnoldi algorithm, whose regularizing properties are not well understood or widely accepted, yet. Therefore, still in Chapter 2, a novel analysis of the approximation properties of the Arnoldi algorithm when employed to solve linear discrete ill-posed problems is presented, in order to provide some insight on the use of Arnoldi-based methods for regularization purposes. The core results of this thesis are related to class of the Arnoldi-Tikhonov methods, first introduced about ten years ago, and described in Chapter 3. The Arnoldi-Tikhonov approach to regularization consists in solving a Tikhonov-regularized problem by means of an iterative strategy based on the Arnoldi algorithm. With respect to a purely iterative approach to regularization, Arnoldi-Tikhonov methods can deliver more accurate approximations by easily incorporating some information about the behavior of the solution within the reconstruction process. In connection with Arnoldi-Tikhonov methods, many open questions still remain, the most significant ones being the choice of the regularization parameters and the choice of the regularization matrices. The first issues are addressed in Chapter 4, where two new efficient and original parameter selection strategies to be employed with the Arnoldi-Tikhonov methods are derived and extensively tested; still in Chapter 4, a novel extension of the Arnoldi-Tikhonov method to the multi-parameter Tikhonov regularization case is described. Finally, in Chapter 5, two efficient and innovative schemes to approximate the solution of nonlinear regularized problems are presented: more precisely, the regularization terms originally defined by the 1-norm or by the Total Variation functional are approximated by adaptively updating suitable regularization matrices within the Arnoldi-Tikhonov iterations. Along this thesis, the results of many numerical experiments are presented in order to show the performance of the newly proposed methods, and to compare them with the already existing strategies

Mathematical Problems Arising When Connecting Kinetic to Fluid Regimes

In this dissertation we study two problems that are related to the question of how to obtain appropriate macroscopic descriptions of a gas from its microscopic formulation. Mathematically, fluid equations formulate the macroscopic dynamics of a gas while kinetic equations are used to study the microscopic world. One can derive fluid equations from kinetic equations through formal asymptotic expansions like those of Hilbert or Chapman-Enskog. The first problem we study relates to the justification of the steps in those formal expansions, while the second relates to the well-posedness of a resulting fluid system. The first problem we study is that of establishing a Fredholm alternative for the linearized Boltzmann collision operator. The Fredholm alternative is used in both the formal asymptotic derivations and the rigorous justifications of fluid approximations to the Boltzmann equation. Results of this type have been obtained for collision kernels satisfying the Grad angular cutoff assumption. However, because DiPerna-Lions' renormalized solutions for the Boltzmann equation are established for more general collision kernels, it is interesting to extend the Fredholm property of the linearized Boltzmann operator to these collision kernels. We show that under a weak cutoff assumption, the linearized Boltzamnn operator does satisfy the Fredholm alternative. The second problem we study is the well-posedness of a dispersive fluid system that is formally obtained via an asymptotic expansion of the Boltzmann equation as a first correction to the compressible Navier-Stokes system. This system is degenerate in both dissipation and dispersion. Therefore the theory for strictly dispersive systems does not apply directly. To prove the well-posedness of this degenerate system, we need to study different smoothing effects for different components of the solution. We show that using the regularization effects including dispersion and dissipation, this system has a unique smooth solution for a finite time

4D imaging in tomography and optical nanoscopy

Diese Dissertation gehört zu den Gebieten mathematische Bildverarbeitung und inverse Probleme. Ein inverses Problem ist die Aufgabe, Modellparameter anhand von gemessenen Daten zu berechnen. Solche Probleme treten in zahlreichen Anwendungen in Wissenschaft und Technik auf, z.B. in medizinischer Bildgebung, Biophysik oder Astronomie. Wir betrachten Rekonstruktionsprobleme mit Poisson Rauschen in der Tomographie und optischen Nanoskopie. Bei letzterer gilt es Bilder ausgehend von verzerrten und verrauschten Messungen zu rekonstruieren, wohingegen in der Positronen-Emissions-Tomographie die Aufgabe in der Visualisierung physiologischer Prozesse eines Patienten besteht. Standardmethoden zur 3D Bildrekonstruktion berücksichtigen keine zeitabhängigen Informationen oder Dynamik, z.B. Herzschlag oder Atmung in der Tomographie oder Zellmigration in der Mikroskopie. Diese Dissertation behandelt Modelle, Analyse und effiziente Algorithmen für 3D und 4D zeitabhängige inverse Probleme. This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics or astronomy. We mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In the latter case, the task is to reconstruct images from blurred and noisy measurements, whereas in positron emission tomography the task is to visualize physiological processes of a patient. In 3D static image reconstruction standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in tomography or cell motion in microscopy. This thesis is a treatise on models, analysis and efficient algorithms to solve 3D and 4D time-dependent inverse problems