Regularization Of Ill-posed Problems Research Articles

The ℓ1−2-norm-regularized basis pursuit seismic inversion based on the exact Zoeppritz equation

Sparsity constraints have been widely adopted in the regularization of ill-posed problems to obtain subsurface properties with sparseness features. However, the target parameters are generally not sparsely distributed, and sparsity constraints lead to results that are missing information. Smooth constraints (e.g., [Formula: see text] norm) lead to the insufficient resolution of the inversion results. To overcome this issue, an effective solution is to convert the target parameters to a sparse representation, which can then be solved with sparsity constraints. For the estimation of elastic parameters, a high-resolution and reliable seismic basis pursuit inversion is developed based on the exact Zoeppritz equation. Furthermore, the [Formula: see text] norm is developed as a constraint, wherein a regularized function is minimized with the alternating direction method of the multipliers algorithm. Numerical examples and real data applications demonstrate that our method can not only improve the accuracy of the inversion results, especially the S-wave velocity and density information but also increase the resolution of the inversion results. Furthermore, the [Formula: see text]-norm constraint has better noise suppression demonstrating great potential in practical applications.

Regularization of ill-posed problems involving constant-coefficient pseudo-differential operators

This paper deals with the wavelet regularization for ill-posed problems involving linear constant-coefficient pseudo-differential operators. We concentrate on solving ill-posed equations involving these operators, which are behaving badly in theory and practice. Since a wide range of ill-posed and inverse problems in mathematical physics can be described and rewritten by the language of these operators, it has gathered significant attention in the literature. Based on a general framework, we classify ill-posed problems in terms of their degree of ill-posedness into mildly, moderately, and severely ill-posed problems in a certain Sobolev scale. Using wavelet multi-resolution approximations, it is shown that wavelet regularizers can achieve order-optimal rates of convergence for pseudo-differential operators in special Sobolev space both for the a priori and the a posteriori choice rules. Our strategy, however, turns out that both schemes yield comparable convergence rates. In this setting, ultimately, we provided some prototype examples for which our theoretical results correctly predict improved rates of convergence.

On the existence and stability of minimizers for generalized Tikhonov functionals with general similarity data

<abstract><p>We study a unified form of Tikhonov regularization for ill-posed problems with a general data similarity term. We discuss sufficient conditions on this generalized Tikhonov functional that guarantee existence and stability of solutions. Furthermore, we show that some particular cases of similarity functionals and regularization techniques can be cast into a unified theoretical framework. In particular, we consider the cases of $ p $-power norm, Bregman distance and mutual information as examples of a data similarity term, and Tikhonov regularization of order one or using a Sobolev norm, total variation penalization and powers of semi-norms associated to closed operators as examples of a regularization term. Finally, we take up the case of the image registration problem.</p></abstract>

Ab initio molecular force fields fitted in Cartesian coordinates to experimental frequencies of isotopic species using symmetry constraints: application to indole and pyrrole molecules

We present the new computational procedure for the fitting ab initio molecular force fields represented in Cartesian coordinates to the experimental frequencies with inclusion data for isotopic species and symmetry constraints. This procedure is based on the original numerical algorithms developed within theory of regularization of ill-posed problems and included in the software package SPECTRUM. New procedure allows organize the synthesis of molecular force field of the target molecule from the matrices in Cartesian coordinates of corresponding patterns and to avoid difficulties with introducing internal coordinates for a case of complicated molecules, so appears beneficial for fitting vibrational spectra of the large molecular systems, when only moderately accurate quantum chemistry methods may be applied. In this work, an earlier suggested algorithm for scaling force matrices in Cartesian coordinates has been extended to allow direct reduction by symmetry and simultaneous use of the experimental data on several isotopic species of a molecule. The molecules of pyrrole and indole (typical fragments of biological molecules) were chosen to demonstrate the efficiency of scaling procedure in Cartesian coordinates with symmetry constraints.

Oversmoothing regularization with $\ell^1$-penalty term

In Tikhonov-type regularization for ill-posed problems with noisy data, the penalty functional is typically interpreted to carry a-priori information about the unknown true solution. We consider in this paper the case that the corresponding a-priori information is too strong such that the penalty functional is oversmoothing, which means that its value is infinite for the true solution. In the case of oversmoothing penalties, convergence and convergence rate assertions for the regularized solutions are difficult to derive, only for the Hilbert scale setting convincing results have been published. We attempt to extend this setting to $\ell^1$-regularization when the solutions are only in $\ell^2$. Unfortunately, we have to restrict our studies to the case of bounded linear operators with diagonal structure, mapping between $\ell^2$ and a separable Hilbert space. But for this subcase, we are able to formulate and to prove a convergence theorem, which we support with numerical examples.

Mathematical model for axisymmetric steady flow within layered structures with finite or infinitely high bulk viscosity

A mathematical model was developed for viscous incompressible medium; this model comprises a small parameter for regularization of ill-posed problem. Starting from the analogy with strain problems of axisymmetric laminar structures which include the elastic orthotropic and elastic volume-incompressible layers with a connection to viscous laminar fluid flow, we have developed a method providing a single algorithm for computation of velocity field, stresses, and other parameters of laminar composite material. The example of calculation for the binding agent flow during pultrusion formation is presented.

Regularization of Ill-Posed Problems with Both Data and Operator are Perturbed

In this note, ill-posed problems are studied in the case when mapping A (not necessarily linear) is given only approximately. We prove the existence of three sequences: a sequence of operators $$\left( A_{n}\right) _{n}$$ , a sequence of arguments $$\left( x_{n}\right) _{n}$$ , and a sequence of second members $$\left( u_{n}\right) _{n}$$ converging on the exact values of the operator equation $$Ax=u$$ .

Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization

Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely ill-posed problems developed in the setting of convection-dominated convection–diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson’s equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

Restoration of a potential from noisy spectral data

Interest in the inverse Sturm–Liouville problem is motivated by its numerous applications in mathematics and computational physics. To solve a complete inverse problem, one needs two exact spectra, which are usually not known in experimental spectroscopy. Accordingly, a problem of interest is to restore the potential from a finite set of noisy spectral data. A new variational method for solving inverse spectral problems is proposed, which is based on the regularization of ill-posed problems. The method takes into account the measurement error of the spectrum and restores the potential without using simplifying assumptions that it belongs to a certain functional class. The method has been tested on potentials involving smooth segments and jump discontinuities.

A study of the accuracy and computing stability of a regularized method identification of nonlinear system

A study the accuracy and noise immunity of a method of deterministic identification of nonlinear dynamic systems in the form of Volterra kernels based on the differentiation of the feedback parameter is the amplitude of the test signals. The calculation of derivatives is reduced to the solution of the corresponding linear integral Volterra equation of the first kind. The computational stability of the method of identification is provided by using the method of regularization of ill-posed problems A.N. Tikhonov. To smooth the estimates of the Volterra kernels used wavelet filtering.Tabl: 1. Refs.: 13 titles.Keywords: deterministic identification; nonlinear dynamical systems; Volterra kernel; regularization; wavelet filtering.

Sparsity-enforcing regularisation and ISTA revisited

About two decades ago, the concept of sparsity emerged in different disciplines such as statistics, imaging, signal processing and inverse problems, and proved to be useful for several applications. Sparsity-enforcing constraints or penalties were then shown to provide a viable alternative to the usual quadratic ones for the regularisation of ill-posed problems. To compute the corresponding regularised solutions, a simple, iterative and provably convergent algorithm was proposed and later on referred to as the iterative soft-thresholding algorithm. This paper provides a brief review of these early results as well as that of the subsequent literature, albeit from the authors’ limited perspective. It also presents the previously unpublished proof of an extension of the original framework.

Discretized Lavrent’ev regularization for the autoconvolution equation

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

Regularization of ill-posed problems by using stabilizers in the form of the total variation of a function and its derivatives

Abstract Under the assumption that the solution of a linear operator equation is presented in the form of a sum of several components with various smoothness properties, a modified Tikhonov regularization method is studied. The stabilizer of this method is the sum of three functionals, where each one corresponds to only one component. Each such functional is either the total variation of a function or the total variation of its derivative. For every component, the convergence of approximate solutions in a corresponding normed space is proved and a general discrete approximation scheme for the regularizing algorithm is justified.

Numerical estimates for the regularization of nonautonomous ill-posed problems

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data. Because direct computation of the solution becomes difficult in this situation, many authors have alternatively approximated the solution by the solution of a closely defined well posed problem. In this paper, we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient. In the process, we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.

Стохастическая регуляризация некорректных задач теплопереноса

Стохастическая регуляризация некорректных задач теплопереноса

Numerical estimates for the regularization of nonautonomous ill-posed problems

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data. Because direct computation of the solution becomes difficult in this situation, many authors have alternatively approximated the solution by the solution of a closely-defined well-posed problem. In this paper, we demonstrate this process of regularization for the backward heat equation with a time-dependent diffusion coefficient, among other nonautonomous ill-posed problems. In the process, we provide two different approximate well-posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.

The index function and Tikhonov regularization for ill-posed problems

In this paper, we study the regularizing properties of the conditional stability estimates in ill-posed problems. First, we analyze how conditional stability estimates occur, and which properties the corresponding index functions must obey. In addition, we adapt the convergence analysis for the Tikhonov regularization in Banach spaces where the difference between the approximated solution and the exact one in metric measure is taken into account. We conclude this study with a comparison of stability estimates and variational inequalities, another emerging tool in Banach space regularization.

Two-parameter discrepancy principle for combined projection and Tikhonov regularization of ill-posed problems

Abstract. For solving a linear ill-posed problem, a combination of Tikhonov regularization and a finite dimensional projection method is considered. In the present paper we treat the dimension of the projection as the second parameter of regularization. The method using the discrepancy set of all regularization parameter pairs which satisfy a certain discrepancy principle is investigated. For the case of truncated SVD and LSQ projection methods, under the standard source conditions, an order of convergence is derived. This order of convergence is optimal under the same conditions as those for Tikhonov regularization with the discrepancy principle and without additional discretization.

Method for the regularization of ill-posed problems of recognition in CCTV

Discusses a method for the regularization of the functional quality. Functionality can provide a correct formulation and solution of the problem of pattern recognition in a CCTV

Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces

Nonstationary iterated Tikhonov regularization is an efficient method for solving ill-posed problems in Hilbert spaces. However, this method may not produce good results in some situations since it tends to oversmooth solutions and hence destroy special features such as sparsity and discontinuity. By making use of duality mappings and Bregman distance, we propose an extension of this method to the Banach space setting and establish its convergence. We also present numerical simulations which indicate that the method in Banach space setting can produce better results.