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In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., L ∞ {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of quasireversibility whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. This is in fact the direction we will take but will do so with a nonlocal operator; an equation of fractional order in time for which the backwards problem is known to be "almost well posed."We shall look at various possible options and strategies but our conclusion for the best of these will exploit the linearity of the problem to break the inversion into distinct frequency bands and to use a different fractional order for each. The fractional exponents will be chosen using the discrepancy principle under the assumption we have an estimate of the noise level in the data. An analysis of the method is provided as are some illustrative numerical examples. [ABSTRACT FROM AUTHOR]

The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to overly restrictive conditions. Here the model and the observation are considered simultaneously as one large system, with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing not only the reduced and all-at-once formulations but also the well-known and highly versatile variational approach (also called the Kohn{Vogelius functional approach) as special cases, is to formulate the inverse problem as a minimization problem|instead of an equation|for the state and parameter. Regularization can be incorporated via imposing constraints and/or adding regularization terms to the objective. In this paper, after giving a motivation by formulating the electrical impedance tomography (EIT) problem by means of the classical Kohn{Vogelius functional, we explore the regularization aspects for such variational formulations in an abstract setting. Indeed, combination of regularization by constraints and by penalization leads to new methods that are applicable without solving forward problems. In particular, for the EIT problem we will consider a method employing box constraints in a very natural manner to incorporate the discrepancy principle for regularization parameter choice as well as a priori information on the searched for conductivity. [ABSTRACT FROM AUTHOR]

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus, the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz []. We here present a convergence analysis of the overall method as well as numerical experiments. [ABSTRACT FROM AUTHOR]

Parameter identification problems typically consist of a model equation, e.g., a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularizat ion method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this paper we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type, and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-once and reduced versions and provide some numerical comparison. [ABSTRACT FROM AUTHOR]

In this paper we consider the thermoacoustic inverse problem of identifying the oscillatory heat release from pressure measurements. We consider the spatially onedimensional and time harmonic case. Three different regularization methods for the stable solution of this ill-posed inverse problem are proposed: Lavrent'ev's method, regularization by discretization, and a method based on an explicit formula combined with regularized numerical differentiation. For these methods, the results of numerical experiments are documented. [ABSTRACT FROM AUTHOR]

Abstract: Due to the principle of regularization by restricting the number of degrees of freedom, truncating the Cholesky factorization of a symmetric positive definite matrix can be expected to have a stabilizing effect. Based on this idea, we consider four different approaches for regularizing ill-posed linear operator equations. Convergence in the noise free case as well as—with an appropriate a priori truncation rule—in the situation of noisy data is analyzed. Moreover, we propose an a posteriori truncation rule and characterize its convergence. Numerical tests illustrate the theoretical results. Both analysis and computations suggest one of the four variants to be favorable to the others. [Copyright &y& Elsevier]

In this paper, we propose a multilevel method for solving nonlinear ill-posed operator equations By minimizing the distance to some initial guess under the constraint of a discretized version of the operator equation for different levels of discretization, we define a sequence of regularized approximations to the exact solution, which is shown to be stable and convergent for arbitrary initial guess and can be computed via a multilevel procedure that altogether yields a globally convergent method. Moreover, this approach enables one to relax restrictions on the nonlinearity of the forward operator, as were used in previous work on regularization methods for nonlinear ill-posed problems. [ABSTRACT FROM AUTHOR]