Regularization Methods For Ill-posed Problems Research Articles

Regularization Methods for Ill-Posed Problems in Quantum Optics

Regularization Methods for Ill-Posed Problems in Quantum Optics

Regularization methods for ill-posed problems of general physics

On the example of a specific physical problem of noise reduction associated with losses, dark counts, and background radiation, a summary of methods for regularizing ill-posed problems is given in the statistics of photocounts of quantum light. The mathematical formulation of the problem is presented by an operator equation of the first kind. The operator is generated by a matrix with countable elements. In the sense of Hadamard, the problem of reconstructing the number of photons of quantum light is due to the compactness of the operator of the mathematical model. A rigorous definition of a regularizing operator (regularizer) is given. The problem of stable approximation to the exact solution of the operator equation with inaccurately given initial data can be overcome by one of the most well-known regularization methods, the theoretical foundations of which were laid in the works of A.N. Tikhonov. The selection of an important class of regularizing algorithms is based on the construction of a parametric family of functions that are Borel measurable on the semiaxis and satisfy some additional conditions. The set of regularizers in this family includes most of the known regularization methods. The main ones are given in the work.

Neural network for computing GSVD and RSVD

This paper presents the neural dynamical network to compute the generalized and restricted singular value decompositions (GSVD/RSVD) in the regularization methods for ill-posed problems. The neural network model is defined by ordinary differential equations (ODE) which can be solved by many state-of-the-art techniques. The main purpose of the paper is to develop two neural network models for finding approximations of the GSVD and the RSVD. The globally asymptotic stability analyses are provided and numerical experiments illustrate our theory and methods. For small scale problems, the estimation can be as accurate as O(10-15). For ill-posed problems, the truncation regularization method implementing the GSVD/RSVD algorithms also produces accurate results.

A Mixed Regularization Method for Ill-Posed Problems

A Mixed Regularization Method for Ill-Posed Problems

Expanding the applicability of an iterative regularization method for ill-posed problems

Expanding the applicability of an iterative regularization method for ill-posed problems

Operator-theoretic and regularization approaches to ill-posed problems

A general framework of regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized in this expository-research paper: philosophy of resolution, regularization–approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined with particular reference to the problem of finding numerically minimum weighted-norm least squares solutions of first kind integral equations (and more generally of linear operator equations with non-closed range). An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The thrust of the contribution is devoted to the interdisciplinary character of operator-theoretic and regularization methods for ill-posed problems, in particular in mathematical geoscience.

A new highly anti-interference regularization method for ill-posed problems

The solution of inverse problems has many applications in mathematical physics. Regularization methods can be applied to obtain the solution of ill-conditioned inverse problems by solving a family of neighboring well-posed problems. Thus, it is significant to investigate the regularization methods to increase the accuracy and efficiency of the solution of inverse problems. In this work, a new regularization filter and the related regularization method based on the singular system theory of compact operator are proposed to solve ill-posed problems. The Cauchy problem of Laplace equation of the first kind is a kind of well-known ill-posed problem. Numerical tests show that the proposed regularization method can solve the Cauchy problems more efficiently under a proper selection of regularization parameters. Numerical results also show that the proposed method is especially effective in solving ill-posed problems with big perturbations.

A modified iterative regularization method for ill-posed problems

In this paper, we study a modified Landweber iteration method via a gradient flow equation induced by a weighted least squares functional. We investigate the proposed scheme for solving ill-posed problems under the setting of compact operator and pseudo-differential operator. The a-priori and the a-posteriori choice rules for regularization parameters are given and both rules yield the order optimal error estimates. Relative to the classical Landweber method, the new method significantly reduces the number of iterations needed to match an appropriate stopping criterion. As applications, we focus on some important ill-posed problems arising from mathematical physics. Numerical experiments are conducted for showing the validity of the scheme.

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

AbstractIn this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).MSC:65F22, 65J15, 65J22, 65M30, 47A52.

Regularization methods for ill-posed problems in multiple Hilbert scales

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, and regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases, convergence is proved and orders and optimal orders of convergence are shown. Finally, some potential applications and open problems are discussed.

Nonlinear regularization methods for ill-posed problems with piecewise constant or strongly varying solutions

In this paper we consider nonlinear ill-posed problems with piecewise constant or strongly varying solutions. A class of nonlinear regularization methods is proposed, in which smooth approximations to the Heavyside function are used to reparameterize functions in the solution space by an auxiliary function of levelset type. The analysis of the resulting regularization methods is carried out in two steps: first, we interpret the algorithms as nonlinear regularization methods for recovering the auxiliary function. This allows us to apply standard results from regularization theory, and we prove convergence of regularized approximations for the auxiliary function; additionally, we obtain the convergence of the regularized solutions, which are obtained from the auxiliary function by the nonlinear transformation. Second, we analyze the proposed methods as approximations to the levelset regularization method analyzed in [Frühauf F, Scherzer O and Leitão A 2005 Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators SIAM J. Numer. Anal. 43 767–86], which follows as a limit case when the smooth functions used for the nonlinear transformations converge to the Heavyside function. For illustration, we consider the application of the proposed algorithms to elliptic Cauchy problems, which are known to be severely ill-posed, and typically allow only for limited reconstructions. Our numerical examples demonstrate that the proposed methods provide accurate reconstructions of piecewise constant solutions also for these severely ill-posed benchmark problems.

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.

Application of regularization methods to damage detection in large scale plane frame structures using incomplete noisy modal data

This paper presents an approach for detecting local damage in large scale frame structures by utilizing regularization methods for ill-posed problems. A direct relationship between the change in stiffness caused by local damage and the measured modal data for the damaged structure is developed, based on the perturbation method for structural dynamic systems. Thus, the measured incomplete modal data can be directly adopted in damage identification without requiring model reduction techniques, and common regularization methods could be effectively employed to solve the developed equations. Damage indicators are appropriately chosen to reflect both the location and severity of local damage in individual components of frame structures such as in brace members and at beam–column joints. The Truncated Singular Value Decomposition solution incorporating the Generalized Cross Validation method is introduced to evaluate the damage indicators for the cases when realistic errors exist in modal data measurements. Results for a 16-story building model structure show that structural damage can be correctly identified at detailed level using only limited information on the measured noisy modal data for the damaged structure.

Optimal control as a regularization method for ill-posed problems

We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. We include convergence proofs of the regularization method and error estimates. We illustrate our method through problems in signal processing and parameter identification using an efficient Riccati solver. Our numerical results are compared to the same examples solved using Tikhonov regularization.

Optimal control as a regularization method for ill-posed problems

We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. We include convergence proofs of the regularization method and error estimates. We illustrate our method through problems in signal processing and parameter identification using an efficient Riccati solver. Our numerical results are compared to the same examples solved using Tikhonov regularization.

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.

A Regularization Homotopy Method for the Inverse Problem of 2-D Wave Equation and Well Log Constraint Inversion

Combining the Tikhonov regularization method for ill-posed problems with the widely convergent homotopy method applied to the inversion process of operator identification, a new strategy−regularization-homotopy method is proposed for the inversion of 2-D wave equation, which is suitable for the nonlinear, ill-posed and multi-extremum seismic inverse problem. In order to restrain noises and improve the quality of inversion, a well log constraint regularization-homotopy method is further constructed. Lots of numerical simulations indicate effectiveness of our methods.

Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems

In the convergence theory of regularization methods for ill-posed problems, so far deterministic error concepts have dominated, which leads to worst-case error estimates. Since this is sometimes not desirable, we aim at providing a framework for proving convergence rates in the Prokhorov metric for the regularization of ill-posed problems with stochastic noise. This allows us to assess uncertainty in the sense of a confidence region for the probability that the deviation between the exact and regularized solutions stays below a given bound with given probability. We exemplify this method for the special case of Tikhonov regularization for linear ill-posed problems and apply the result to the problem of deblurring an image contaminated by random blurring.

Hysteresis Parameter Identification With Limited Experimental Data

The Preisach operator and its variants have been successfully used in the modeling of hysteresis observed in ferromagnetic, magnetostrictive, and piezoelectric materials. However, in designing with these smart materials, one has to determine a density function for the Preisach operator by using the input-output behavior of the material at hand. In this paper, we describe a method for numerically determining an approximation of the density function when there is not enough experimental data to uniquely solve for the actual density function by Mayergoyz's method. We present theoretical justification for our method by establishing links to regularization methods for ill-posed problems. We also present numerical results where we estimate an approximate density function from data published in the literature for a magnetostrictive actuator and two electroactive polymers.

A Sobolev space analysis of linear regularization methods for ill-posed problems

Abstract - The paper provides a general framework for studying regularization methods in Sobolev scales. We show that linear regularization methods can be interpreted as smoothing of the generalized inverse or equivalently as the generalized inverse applied to smoothed data. In contrast to [16] we use Sobolev spaces instead of spaces generated by the operator under consideration. Further we give conditions on the degree of the smoothing based on estimates in Sobolev norms. We also show that the regularization methods form a semi group. Conditions for the order optimality of the methods are provided. Some very useful conditions are presented to guarantee that a mollifier of convolution type builds up a regularization method. Finally we verify the results for the Radon transform as the model in computerized tomography.