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A fast subspace optimization method for nonlinear inverse problems in Banach spaces with an application in parameter identification
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On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework
No full textParameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography
On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework
No full textParameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems (in divergence form) with Dirichlet, Neumann, Robin or mixed boundary conditions. In particular, we show that the corresponding parameter-to-state operators fulfil, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography
Sequentielle Unterraum-Optimierung für nichtlineare inverse Probleme mit einer Anwendung in der Terahertz-Tomographie
No full textWe introduce a sequential subspace optimization (SESOP) method for the iterative solution of nonlinear inverse problems in Hilbert spaces, based on the well-known methods for linear problems. The key idea is to use multiple search directions per iteration. Their length is determined by the nonlinearity and the local character of the forward operator. This choice admits a geometric interpretation after which the method is originally named: The current iterate is projected sequentially onto (intersections of) stripes, which emerge from affine hyperplanes whose respective normal vectors are given by the search directions and contain the solution set of the unperturbed inverse problem. We prove convergence and regularization properties and present a fast method using two search directions, which is evaluated by solving a simple nonlinear problem. Furthermore, we extend our methods for complex Hilbert spaces and apply it to solve the inverse problem of terahertz tomography, a nonlinear parameter identification problem based on the Helmholtz equation, which consists in the nondestructive testing of dielectric media. The tested object is illuminated by an electromagnetic Gaussian beam and the goal is the reconstruction of the complex refractive index from measurements of the electric field. We conclude with some numerical reconstructions from synthetic data.In der vorliegenden Arbeit stellen wir eine Erweiterung der sequentiellen Unterraum-Optimierung (SESOP) zur Lösung nichtlinearer inverser Probleme in Hilberträumen vor, welche auf den bereits bekannten Verfahren für lineare Probleme basiert. Dabei handelt es sich um eine iterative Methode, bei der in jedem Schritt mehrere Suchrichtungen verwendet werden. Die Berechnung der Schrittweite berücksichtigt die Nichtlinearität des Vorwärtsoperators und lässt eine anschauliche geometrische Interpretation zu, welche dem Verfahren ursprünglich ihren Namen gab: Die aktuelle Iterierte wird sequentiell auf (den Schnitt von) Streifen projiziert. Diese Streifen gehen aus affinen Hyperebenen hervor und enthalten die Lösungsmenge des inversen Problems bei exakten Daten. Wir zeigen Konvergenz- und Regularisierungseigenschaften des Verfahrens. Insbesondere geben wir ein schnelles Verfahren mit zwei Suchrichtungen an und evaluieren die Methode anhand eines einfachen Beispiels. Anschließend weiten wir die Methode auf komplexe Hilberträume aus und verwenden diese zur Lösung des inversen Problems der Terahertz-Tomographie. Dabei wird ein nichtleitendes, nichtmagnetisches Objekt mithilfe eines elektromagnetischen Gaußstrahls abgetastet. Das Ziel ist die Rekonstruktion des komplexen Brechungsindex aus Messungen des elektrischen Feldes. Dieses inverse Problem modellieren wir als Parameteridentifikationsproblem mithilfe der Helmholtzgleichung. Schließlich erzeugen wir für verschiedene Objekte synthetische Daten und rekonstruieren daraus den komplexen Brechungsindex
Parameter identification for elliptic boundary value problems: an abstract framework and applications
No full textAbstractParameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. In this work, we present a general functional analytic framework suited for the study of a huge class of parameter identification problems including a variety of elliptic boundary value problems with Dirichlet, Neumann, Robin or mixed boundary conditions in Hilbert and Banach spaces and possibly complex-valued parameters. In particular, we show that the corresponding parameter-to-state operators fulfill, under suitable conditions, the tangential cone condition, which is often postulated for numerical solution techniques. This framework particularly covers the inverse medium problem and an inverse problem that arises in terahertz tomography
Sequential subspace optimization for recovering stored energy functions in hyperelastic materials from time-dependent data
No full textMonitoring structures of elastic materials for defect detection by means of ultrasound waves (Structural Health Monitoring, SHM) demands for an efficient computation of parameters which characterize their mechanical behavior. Hyperelasticity describes a nonlinear elastic behavior where the second Piola-Kirchhoff stress tensor is given as a derivative of a scalar function representing the stored (strain) energy. Since the stored energy encodes all mechanical properties of the underlying material, the inverse problem of computing this energy from measurements of the displacement field is very important regarding SHM. The mathematical model is represented by a high-dimensional parameter identification problem for a nonlinear, hyperbolic system with given initial and boundary values. Iterative methods for solving this problem, such as the Landweber iteration, are very time-consuming. The reason is the fact that such methods demand for several numerical solutions of the hyperbolic system in each iteration step. In this contribution we present an iterative method based on sequential subspace optimization (SESOP) which in general uses more than only one search direction per iteration and explicitly determines the step size. This leads to a significant acceleration compared to the Landweber method, even with only one search direction and an optimized step size. This is demonstrated by means of several numerical tests
Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces
No full textWe consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness
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