Solution Of Nonlinear Inverse Problem Research Articles

An Alternating Direction Multiplier Method for the Inversion of FDEM Data

In this paper, we focus on the numerical solution of nonlinear inverse problems in applied geophysics. Our aim is to reconstruct the structure of the soil, i.e., either its electrical conductivity or the magnetic permeability distribution, by inverting frequency domain electromagnetic data. This is a very challenging task since the problem is nonlinear and severely ill-conditioned. To solve the nonlinear inverse problem, we propose an alternating direction multiplier method (ADMM), we prove its convergence, and propose an automated strategy to determine the parameters involved. Moreover, we present two heuristic variations of the ADMM that either improve the accuracy of the computed solutions or lower the computational cost. The effectiveness of the different proposed methods is illustrated through few numerical examples.

An Efficient Implementation of the Gauss–Newton Method Via Generalized Krylov Subspaces

The solution of nonlinear inverse problems is a challenging task in numerical analysis. In most cases, this kind of problems is solved by iterative procedures that, at each iteration, linearize the problem in a neighborhood of the currently available approximation of the solution. The linearized problem is then solved by a direct or iterative method. Among this class of solution methods, the Gauss–Newton method is one of the most popular ones. We propose an efficient implementation of this method for large-scale problems. Our implementation is based on projecting the nonlinear problem into a sequence of nested subspaces, referred to as Generalized Krylov Subspaces, whose dimension increases with the number of iterations, except for when restarts are carried out. When the computation of the Jacobian matrix is expensive, we combine our iterative method with secant (Broyden) updates to further reduce the computational cost. We show convergence of the proposed solution methods and provide a few numerical examples that illustrate their performance.

Simulation study for cross-sectional absorption distribution in turbid medium using spatially resolved backscattered light with lateral scanning and one-dimensional solution of nonlinear inverse problem

For a practical technique of cross-sectional imaging of animal bodies, we developed a new method using spatially resolved backscattered light. This method is based on the solution of the one-dimensional nonlinear inverse problem, and on the lateral scan of the source–detector pair along the object surface. Using this method, unknown variables in inverse problems can be reduced more greatly than when using conventional methods. A stable solution for the inverse problem becomes possible. The possibility of using the proposed method was assessed using simulation analysis. The results verified that cross-sectional imaging from several to 10 millimeter depths is possible for animal tissue. This analysis clarified the specific spatial resolution and accuracy in the estimated absorption coefficient. Distortionless imaging was confirmed. Results suggest that the proposed method represents new options as a stable and practical method for biological cross-sectional imaging.

Superlens Enhanced 2-D Microwave Tomography With Contrast Source Inversion Method

The contrast source inversion (CSI) algorithm is one of the primary techniques used for solution of non-linear inverse problems in microwave tomography. In this paper, we describe a modification of the CSI method adapted to imaging of the 2-D objects in the presence of the focusing media under TM-polarization. The focusing media is presented in the form of the Veselago lens. The data domain and imaging domain are properly positioned with respect to the location of the lens. Specifically, the sensors are located at the focal points of the lens with respect to the location of the individual pixels discretizing the contrast source. Such positioning of the source and observation locations in the presence of the lens, eliminates rank deficiency in the formulation of the inverse problem and results in significant improvements to both convergence speed of underlying conjugate gradient iterations and the accuracy of the image reconstruction in the CSI method.

Tikhonov regularization in Hilbert scales under conditional stability assumptions

Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a priori and a posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.

A proximal iteratively regularized Gauss–Newton method for nonlinear inverse problems

Abstract In this paper we discuss the construction, convergence analysis, and implementation of a proximal iteratively regularized Gauss–Newton method for the solution of nonlinear inverse problems with a specific regularization that linearly combines the L 2 {L^{2}} -norm and L 1 {L^{1}} -norm penalties. This regularization combines two very powerful features: the advantages of L 1 {L^{1}} -norm based penalty which impose less smoothing on the reconstruction parameter, and the general L 2 {L^{2}} -norm stabilizing term which can lead to smaller errors in some cases. However, non-linearity and non-smoothness of the problem make it challenging to find an efficient numerical solution. By using the proximal mapping, we derive a generalization of the iteratively regularized Gauss–Newton algorithm to handle such nonsmooth objective functions. Analysis on local convergence is carried out in the presence of observation noise. Parameter identification in numerical simulations of partial differential equations demonstrates the efficiency of the proposed method.

Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems

Abstract 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 [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

An implicit Landweber method for nonlinear ill-posed operator equations

In this paper, we are interested in the solution of nonlinear inverse problems of the form F ( x ) = y . We propose an implicit Landweber method, which is similar to the third-order midpoint Newton method in form, and consider the convergence behavior of the implicit Landweber method. Using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. We conclude with numerical examples confirming the theoretical results, including comparisons with the classical Landweber iteration and presented modified Landweber methods.

Approximate solution of nonlinear inverse problems by fixed-point iteration

In this paper we propose a derivative-free iterative method for the approximate solution of a nonlinear inverse problem Fx = y. In this method the iterations are defined as Gxk+1 = Gxk + (Sy − SFxk), where G is an easily invertible operator and S is an operator from a data space to a solution space. We give general suggestions for the choice of operators G and S and show a practically relevant example of an inverse problem where such a method is succesfully applied. We carry out analysis of the proposed method for linear inverse problems. Using the recently introduced balancing principle we construct a stopping rule. Under reasonable assumptions, we show that this stopping rule leads to the regularization algorithm. Numerical results for a test example show its satisfactory behavior.

The solution of nonlinear inverse problems and the Levenberg-Marquardt method

Although the Levenberg-Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems, its theoretical basis has not been described adequately in the literature. This is unfortunate, because Levenberg and Marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method. The idea of damping the solution was introduced by Levenberg, who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively. This result is not obvious, although it is taken for granted. Moreover, Levenberg derived a solution more general than the one currently used. Marquardt started with the current equation and showed that it interpolates between the ordinary least-squares-method and the steepest-descent method. In this tutorial, the two papers are combined into a unified presentation, which will help the reader gain a better understanding of what happens when solving nonlinear problems. Because the damped least-squares and steepest-descent methods are intimately related, the latter is also discussed, in particular in its relation to the gradient. When the inversion parameters have the same dimensions (and units), the direction of steepest descent is equal to the direction of minus the gradient. In other cases, it is necessary to introduce a metric (i.e., a definition of distance) in the parameter space to establish a relation between the two directions. Although neither Levenberg nor Marquardt discussed these matters, their results imply the introduction of a metric. Some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth. Finally, the work done by early researchers that rediscovered the damped least-squares method is put into a historical context.

A two-stage method for nonlinear inverse problems

In this work we are interested in the solution of nonlinear inverse problems of the form F ( x ) = y . We consider a two-stage method which is third order convergent for well-posed problems. Combining the method with Levenberg–Marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. Numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method.

On optimization techniques for solving nonlinear inverseproblems

This paper considers optimization techniques for the solution ofnonlinear inverse problems where the forward problems, likethose encountered in electromagnetics, are modelled bydifferential equations. Such problems are often solved byutilizing a Gauss-Newton method in which the forward modelconstraints are implicitly incorporated. Variants of Newton'smethod which use second-derivative information are rarelyemployed because their perceived disadvantage in computationalcost per step offsets their potential benefits of fasterconvergence. In this paper we show that, by formulating theinversion as a constrained or unconstrained optimizationproblem, and by employing sparse matrix techniques, we cancarry out variants of sequential quadratic programming and thefull Newton iteration with only a modest additional cost. Byworking with the differential equation explicitly we are able torelate the constrained and the unconstrained formulations anddiscuss the advantages of each. To make the comparisons meaningfulwe adopt the same global optimization strategy for allinversions. As an illustration, we focus upon a 1Delectromagnetic (EM) example simulating a magnetotelluric survey.This problem is sufficiently rich that it illuminates most ofthe computational complexities that are prevalent inmulti-source inverse problems and we therefore describe itssolution process in detail. The numerical results illustratethat variants of Newton's method which utilize second-derivativeinformation can produce a solution in fewer iterations and, insome cases where the data contain significant noise, requiringfewer floating point operations than Gauss-Newton techniques.Although further research is required, we believe that thevariants proposed here will have a significant impact ondeveloping practical solutions to large-scale 3D EMinverse problems.

Empirical modeling of the solar corona using genetic algorithms

Many remote sensing applications encountered in astronomy and space science involve the solution of nonlinear inverse problems. These are often difficult to solve because of nonlinearities, ill‐behaved integration kernels, and amplification of data noise associated with the inversion of the integral operator. In some cases these difficulties are severe enough to warrant repeated evaluations of the forward problem as an alternate approach to formal inversion. Because a forward approach is intrinsically repetitive and time consuming, an efficient and flexible forward technique is required for this avenue to be practical. We show how a forward technique based on a genetic algorithm allows us to fit magnetostatic models of the solar minimum corona to observations in white light to a degree that would otherwise have been computationally prohibitive. In addition, and perhaps equally important, the method also allows the determination of global error estimates on the model parameters defining the best fit solution.

Bayesian statistics of non-linear inverse problems: example of the magnetotelluric 1-D inverse problem

SUMMARY We present a practical algorithm for determining the Bayesian solution of non-linear inverse problems with a limited number of parameters. This approach allows the use of very general conditional probability density functions (pdfs of the data given the model parameters) and a priori pdfs (prior beliefs upon the parameters). The results consist in the a posteriori marginal pdfs of the model parameters. The marginal pdfs describe the additional information (if any) brought by the data to the prior knowledge about the model parameters. We have paid a special attention to the numerical calculation of the a posteriori pdfs and we propose a solution which enables the simultaneous determination of the numerical estimates of the a posteriori pdfs and their uncertainties. With the help of a practical example (the 1-D magnetotelluric inverse problem) with synthetic and real data, we show that in most cases, the a posteriori marginal pdfs are complicated functions and cannot be predicted from the data pdf or the a priori pdfs. Consequently, the standard analyses applied to the same data sets such as the maximum likelihood techniques or the asymptotic estimation technique would lead to severely biased estimates of the parameters. This remark is likely to be true for any non-linear inverse problem.

Regularized numerical solution of nonlinear inverse heat-conduction problem

The construction of an algorithm for a numerical solution of the nonlinear inverse problem is discussed for the case of a generalized one-dimensional heat-conduction equation in a region with moving boundaries. The algorithm is regularized in the Tikhonov manner.