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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.

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.

Discrete ill-posed problems where both the coefficient matrix and the right hand side are contaminated by noise appear in a variety of engineering applications. In this paper we consider Tikhonov regularized solutions where the regularization parameter is chosen by the generalized discrepancy principle (GDP). In contrast to Newton-based methods often used to compute such parameter, we propose a new algorithm referred to as GDP-FP, where derivatives are not required and where the regularization parameter is calculated efficiently by a fixed-point iteration procedure. The algorithm is globally and monotonically convergent. Additionally, a specialized version of GDP-FP based on a projection method, that is well-suited for large-scale Tikhonov problems, is also proposed and analyzed in detail. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms on test problems from the literature.

In this work, the problem of coil design is studied. It is assumed that the structure of the coil is known (i.e., the positions of simple circular coils are fixed) and the problem is to find current distribution to obtain the required magnetic field in a given region. The unconstrained version of the problem (arbitrary currents are allowed) can be formulated as a Least-SQuares (LSQ) problem. However, the results obtained by solving the LSQ problem are usually useless from the application point of view. Moreover, for higher dimensions the problem is ill-conditioned. To overcome these difficulties, a regularization term is sometimes added to the cost function, in order to make the solution smoother. The regularization technique, however, produces suboptimal solutions. In this work, we propose to solve the problem under study using the constrained Quadratic Programming (QP) method. The methods are compared in terms of the quality of the magnetic field obtained, and the power of the designed coil. Several 1D and 2D examples are considered. It is shown that for the same value of the maximum current the QP method provides solutions with a higher quality magnetic field than the regularization method.

In this paper, we construct a Runge-Kutta type total variation regularization for the nonlinear ill-posed problems. This method resembles Runge-Kutta type iteration with the Bregman distance as a regularization functional. In the presence of noise with noise level ? we prove this method to be convergent under appropriate stopping rule. Furthermore, some numerical experiments are presented to verify this method to be more suitable for problems with discontinuous solutions. RK type TV regularization for the nonlinear ill-posed problems is constructed.This method introduced Bregman distance as regularization functional.Under appropriate stopping rule this method with noise is convergent.Numerical tests testify the method for problems with discontinuous solutions.

In recent studies on sparse modeling, the nonconvex regularization approaches (particularly, Lq regularization with q ∈ (0,1)) have been demonstrated to possess capability of gaining much benefit in sparsity-inducing and efficiency. As compared with the convex regularization approaches (say, L1 regularization), however, the convergence issue of the corresponding algorithms are more difficult to tackle. In this paper, we deal with this difficult issue for a specific but typical nonconvex regularization scheme, the L1/2 regularization, which has been successfully used to many applications. More specifically, we study the convergence of the iterative half thresholding algorithm (the half algorithm for short), one of the most efficient and important algorithms for solution to the L1/2 regularization. As the main result, we show that under certain conditions, the half algorithm converges to a local minimizer of the L1/2 regularization, with an eventually linear convergence rate. The established result provides a theoretical guarantee for a wide range of applications of the half algorithm. We provide also a set of simulations to support the correctness of theoretical assertions and compare the time efficiency of the half algorithm with other known typical algorithms forL1/2 regularization like the iteratively reweighted least squares (IRLS) algorithm and the iteratively reweighted l1 minimization (IRL1) algorithm.

—Numerical instability is an inherent issue with power system state estimation, and much effort has been made to overcome this issue using numerical techniques to ensure stability. This article proposes a regularization-based method for solving ill-conditioned hybrid state-estimation problems in the presence of equality constraints. The iterative linear state-estimation problem is solved by using Tikhonov regularization. The trade-off between the residual norm and the regularized norm is controlled by the regularization parameter in the regularization method. The regularization method employs an L-curve criterion for optimal selection of the regularization parameter. At the post-estimation stage, the correction in voltage phasor at the zero-injection bus is applied using Kirchhoff's current law. The proposed method is analyzed and tested for IEEE 14- and 118-bus test systems, as well as for a 13-bus ill-conditioned system.

Lavrentiev regularization is a popular approach to the solution of linear discrete ill-posed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods.

We discuss statistical tests in inverse problems when the original equation is replaced by a discretized one, i.e. a linear system of equations. Previous studies revealed that using the discretization level as regularizing procedure is possible, but its application is limited unless discretization is restricted to the singular value decomposition, see C. Marteau and P. Mathe, General regularization schemes for signal detection in inverse problems, 2013. General linear regularization may circumvent this, and we propose a regularization of the discretized equations. The discretization level may be chosen adaptively, which may save computational budget. This results in tests which are known to yield the optimal separation rate up to some constant in many cases.

SUMMARY This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi–Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this quantity is recovered during the Arnoldi algorithm. Some theoretical estimates are derived in order to motivate our approach. Many numerical experiments performed on classical test problems as well as image deblurring problems are presented. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a numerical algorithm is proposed based on the method of fundamental solutions (MFS) for the Cauchy problem connected with the Laplace equation in ℝ2. The major advantage of the modified MFS is to keep a very basic natural property, i.e. the invariance under trivial coordinate changes in the problem description. The method combines Newton's method and classic Tikhonov regularization to solve an inverse problem. The numerical convergence, accuracy, and stability of the method with respect to increasing the number of source points, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples, even for high noise levels.

The regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.

We investigate the simultaneous regularization and discretization of an optimal control problem with pointwise control constraints. Typically such problems exhibit bang---bang solutions: the optimal control almost everywhere takes values at the control bounds. We derive discretization error estimates that are robust with respect to the regularization parameter. These estimates can be used to make an optimal choice of the regularization parameter with respect to discretization error estimates.

Acoustic imaging aims at localization and characterization of sound sources using microphone arrays. In this paper a new regularization method for acoustic imaging by inverse approach is proposed. The method first relies on the singular value decomposition of the plant matrix and on the projection of the measured data on the corresponding singular vectors. In place of regularization using classical methods such as truncated singular value decomposition and Tikhonov regularization, the proposed method involves the direct definition of the filter factors on the basis of a thresholding operation, defined from the estimated measurement noise. The thresholding operation is achieved using modified filter functions. The originality of the approach is to propose the definition of a filter factor which provides more damping to the singular components dominated by noise than that given by the Tikhonov filter. This has the advantage of potentially simplifying the selection of the best regularization amount in inverse problems. Theoretical results show that this method is comparatively more accurate than Tikhonov regularization and truncated singular value decomposition.

The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the context of Tikhonov regularization. Also, other regularization methods, including truncated GSVD, are considered. We compare the computational efforts required by the simplified GSVD method and the A -weighted generalized inverse introduced by Elden.

This paper introduces two new algorithms, belonging to the class of Arnoldi--Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the $p$-norm, $p\geq 1$. The regularization matrix can be updated both at each step and after some iterations have been performed, leading to two different approaches: the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible Krylov subspaces; the second one is based on restarting the Arnoldi algorithm. Numerical examples are given in order to show the effectiveness of these new methods, and comparisons with some other already existing algorithms are made.

The widespread applicability of the multipenalty regularization is limited by the fact that theoretically optimal rate of reconstruction for a given problem can be realized by a one-parameter counterpart, provided that relevant information on the problem is available and taken into account in the regularization. In this paper, we explore the situation where no such information is given, but still accuracy of optimal order can be guaranteed by employing multipenalty regularization. Our focus is on the analysis and the justification of an a posteriori parameter choice rule for such a regularization scheme. First we present a modified version of the discrepancy principle within the multipenalty regularization framework. As a consequence we provide a theoretical justification to the multipenalty regularization scheme equipped with the a posteriori parameter choice rule. We then establish a fast numerical realization of the proposed discrepancy principle based on a model function approximation. Finally, we pro...

A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.

The electrical impedance equation is considered an ill-posed problem where the solution to the forward problem is more easy to achieve than the inverse problem. This work tries to improve convergence in the forward problem method, where the Pseudoanalytic Function Theory by means of the Taylor series in formal powers is used, incorporating a regularization method to make a solution more stable and to obtain better convergence. In addition, we include a comparison between the designed algorithms that perform proposed method with and without a regularization process and the autoadjustment parameter for this regularization process.

In this paper, we consider the problem for determining an unknown source in the heat equation. The Tikhonov regularization method in Hilbert scales is presented to deal with ill-posedness of the problem and error estimates are obtained with a posteriori choice rule to find the regularization parameter. The smoothness parameter and the a priori bound of exact solution are not needed for the choice rule. Numerical tests show that the proposed method is effective and stable.

In this paper, we consider the problem of image restoration with Tikhonov regularization as a convex constrained minimization problem. Using a Kronecker decomposition of the blurring matrix and the Tikhonov regularization matrix, we reduce the size of the image restoration problem. Therefore, we apply the conditional gradient method combined with the Tikhonov regularization technique and derive a new method. We demonstrate the convergence of this method and perform some numerical examples to illustrate the effectiveness of the proposed method as compared to other existing methods.

This contribution is an extension of the work initiated in [1], presenting a strategy for the calibration of the local volatility. Due to Morozov's discrepancy principle [6], the Tikhonov regularization problem introduced in [7] is understood as an inequality-constrained minimization problem. An Uzawa procedure is proposed to replace this latter by a sequence of unconstrained problems dealt with in the modified Thikonov regularization procedure in [1]. Numerical tests confirm the consistency of the approach and the significant speed-up of the process of local volatility determination., Cette contribution dans ce papier est une extension des travaux initiés dans [1], qui pré-sente une stratégie pour l'estimation de la volatilité locale. En raison du principe de la différence de Morozov [6], le problème de la régularisation de Tikhonov introduite dans [7] est reformulé comme un problème de minimisation de l'inégalité des contraintes. Une procédure Uzawa est proposé de remplacer ce dernier par une séquence de problèmes non contraints traités dans la procédure de régularisation Thikonov modifié dans [1]. Des tests numériques confirment la cohérence de l'approche et l'importante accélérer le processus de détermination de la volatilité locale.

In this paper, we consider the inverse problem of determining the heat source, which depends only on spatial variable in one-dimensional heat equation in a bounded domain where data is given at some fixed time. A conditional stability result is given, and a quasi-boundary value regularization method is also provided. For this regularization solution, the Holder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable. Copyright (c) 2013 John Wiley & Sons, Ltd.

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices $$A^\mathrm{T}\!A$$ (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered.

In this paper, a new improved regularization method is proposed to identify dynamic loads in practical engineering problems. Dynamic loads are expressed as the functions of time and the forward model for dynamic load identification is established through the discretized convolution integral of loads and the corresponding unit-pulse response functions of system. With measured responses containing noises, a regularization operator is proposed to construct a novel regularization method, and its regular property is proved. The improved regularization operator and L-curve method are combined to overcome the ill-condition of load reconstruction and to obtain the stable and approximate solutions of inverse problems. The theory is successfully applied to a mathematical problem and a vehicle hood problem in numerical simulations, which demonstrates the efficiency and robustness of the presented method.

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.

The method of Tikhonov regularization is commonly used to obtain regularized solutions of ill-posed linear inverse problems. We use its natural connection to optimal Bayes estimators to determine optimal experimental designs that can be used with Tikhonov regularization; they are designed to control a measure of total relative efficiency. We present an iterative/semidefinite programming hybrid method to explore the configuration space efficiently. Two examples from geophysics are used to illustrate the type of applications to which the methodology can be applied.

We calibrate the local volatility surface for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second-order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both volatility models with known analytical solutions of European option prices and real market option data.

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a sequential variant of the discrepancy principle is analysed. In many cases, such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here, we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

[1] Improved estimation of hydrometeorological states from down-sampled observations and background model forecasts in a noisy environment has been a subject of growing research in the past decades. Here we introduce a unified variational framework that ties together the problems of downscaling, data fusion, and data assimilation as ill-posed inverse problems. This framework seeks solutions beyond the classic least squares estimation paradigms by imposing a proper regularization, expressed as a constraint consistent with the degree of smoothness and/or probabilistic structure of the underlying state. We review relevant smoothing norm regularization methods in derivative space and extend classic formulations of the aforementioned problems with particular emphasis on land surface hydrometeorological applications. Our results demonstrate that proper regularization of downscaling, data fusion, and data assimilation problems can lead to more accurate and stable recovery of the underlying non-Gaussian state of interest with improved performance in capturing isolated and jump singularities. In particular, we show that the Huber regularization in the derivative space offers advantages, compared to the classic solution and the Tikhonov regularization, for spatial downscaling and fusion of non-Gaussian multisensor precipitation data. Furthermore, we explore the use of Huber regularization in a variational data assimilation experiment while the initial state of interest exhibits jump discontinuities and non-Gaussian probabilistic structure. To this end, we focus on the heat equation motivated by its fundamental application in the study of land surface heat and mass fluxes.

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

Many questions in science and engineering often give rise to linear ill-posed inverse problems. To enable meaningful approximate solutions of the inverse problems, various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations. One of the most popular regularization techniques is iterated Tikhonov regularization, which has attracted considerable attention due to its interesting applications in practical. However, this regularization often suffers from inaccurate results and low computational efficiency in some situations. In this paper, we proposed an accelerated predictor-corrector iterated Tikhonov regularization. This method combined the classical iterated Tikhonov regularization with modified Euler method, which could improve computational efficiency without sacrificing numerical accuracy. The convergence rate of the accelerated regularization was investigated theoretically when the right-hand side was corrupted by noise. In particular, we derived an error estimate of optimal order for the convergence of the accelerated regularization. Both one-and two-dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated Tikhonov regularization.

nverse problems are important interdisciplinary subject, which receive more and more attention in recent years in the areas of mathematics, computer science, information science and other applied natural sciences. There is close relationship between inverse problems and ill-posedness. Regularization is an important strategy when computing the ill-posed problems to maintain the stability of the computation.This paper compares a new regularization method,which is called Adaptive regularization, with the traditional Tikhonov regularization method. The conclusion that Adaptive regularization method is a stronger regularization method than the traditional Tikhonov regularization method can be made by computing some numerical examples.

We consider an ill-posed equation in a Hilbert space with a noisy operator and a noisy right-hand side. The noise level information is given in a general form, as a norm of a certain operator applied to the noise. We derive the monotone error rule (ME-rule) for the choice of the regularization parameter in many methods, giving parameter such that the error is monotonically increasing for larger parameters in the Tikhonov method and for smaller stopping indices in iteration methods. Regularization methods considered include -scale regularization in (iterated) Tikhonov method and in iteration methods (Landweber method, CG type methods, semi-iterative methods). We also consider modifications of the ME-rule and show in numerical experiments (test problems from Hansen’s Regularization Toolbox, including the sideways heat equation) their advantages over the discrepancy principle.

s of MMA2013 & AMOE2013, May 27–30, 2013, Tartu, Estonia c © 2013 University of Tartu TWO-PARAMETER DISCREPANCY PRINCIPLE FOR COMBINED PROJECTION AND TIKHONOV REGULARIZATION OF ILL-POSED PROBLEMS TERESA REGIŃSKA Institute of Mathematics, Polish Academy of Sciences Śniedeckich 8, 00-950 Warsaw, Poland E-mail: reginska@impan.pl Let A ∈ L(X) be a compact operator in a Hilbert space X. We consider the operator equation Au = f (1) with noisy right hand side f, ‖f − fδ‖ ≤ δ. If the dimension of the range of A is not finite, then (1) is the ill-posed problem (also in the least squares sense). In practice we deal with a finitedimensional approximation of (1), i.e. with the family of equations parameterized by n, Anun = f δ n, where An ∈ L(Xn) are linear operators acting on finite dimensional spaces Xn. Following the notation used in literature, this finite-dimensional problem is called ”discrete ill-posed problem”, which underlines the ill-posedness of the main problem. In this presentation we consider a combination of finite dimensional projection and Tikhonow regularization (AnAn + α)u δ n,α = A ∗ nf . (2) So, the method is generated by two parameters: the dimension n of the projection and Tikhonov regularization parameter α. The novelty of the present approach is that both the parameters are treated as independent regularization ones. We focus our attention on an a-posteriori parameter choice rule. Following the approach presented in [1], we introduce the following discrepancy set DS(δ) := {n, α : n ∈ N, α ∈ R, ‖Aun,α − fδ‖ = Cδ}. (3) For the case of discretization by truncated SVD, the set DS(δ) is described in details. The main results concern the order of convergence, when δ → 0 and they are obtained under the standard source condition u = (A∗A)μv and ‖v‖ ≤ ρ. It is proved that if μ ≤ 12 , then for a combination of Tikhonov regularization and truncated SVD as well as LSQ projection method, the convergence is of optimal order for regularization parameters belonging to DS(δ). However, for μ > 1 2 , the convergence rate is sub-optimal and it is shown that generally this result cannot be improved. Therefore, the twoparameter method has the same order of convergence as Tikhonov method (without discretization) with α chosen by the standard discrepancy principle.

In this paper, an inverse problem for space-fractional backward diffusion equation, which is highly ill-posed, is considered. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α ∈ (0,2]. We show that such a problem is severely ill-posed, and further present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

This paper introduces a novel method to combine total variation and @?"2 regularizations to reconstruct piecewise smooth signals. The main idea is to consider the signal as a sum of two components: a piecewise constant component and a smooth component. For the solution of ill-posed problems, the Tikhonov method with a special stabilizer in the form of a sum of two different stabilizers is used: the total variation for the first component and the Sobolev norm for the second one. An iteratively re-weighted least squares technique is used as a fast and an efficient algorithm for minimization of the Tikhonov functional. A method is also presented for determining the regularization parameters. Numerical experiments, among the many performed, in denoising, deblurring, and compressed sensing demonstrate high performance of the new regularization for reconstruction of piecewise-smooth solutions with sharp discontinuities.

Tikhonov regularization for the inverse heat conduction problem (IHCP) is considered a “whole domain” or “batch” method, meaning that observations are needed over the entire time domain of interest, and that calculations must be performed all-at-once in a batch. This paper examines the structure of the Tikhonov regularization problem and concludes that the method can be interpreted as a sequential filter formulation for continuous processing of data. Several general observations regarding features of the filter formulation are noted. Two error norms are discussed: one regarding temperature and one regarding heat flux. It is shown that these metrics can be split into two parts: one dependent on the heat flux history (bias error) and one dependent on the measurement noise (random error). Two examples demonstrate that the optimal selection of the regularization parameter to minimize the heat flux error yields results similar to the classical Morozov principle defined through temperature error, and that the results are relatively insensitive to the precise selection of the parameter.

In this paper, the inverse problem of identifying a space-dependent source for the time-fractional diffusion equation is investigated. Such a problem is obtained from the classical diffusion equation in which the time derivative is replaced with a Caputo derivative of order @[email protected]?(0,1]. We show that such a problem is ill-posed and apply the Tikhonov regularization method and a simplified Tikhonov regularization method to solve it based on the solution given by the separation of variables. Convergence estimates are presented under an a priori parameter choice rule and an a posteriori parameter choice rule, respectively. Finally, numerical examples are given to show that the regularization methods are effective and stable.

In this paper we consider a general equilibrium problem in a Hilbert space defined on a closed and convex set. We show a Tikhonov-type regularization method that can be extended for the equilibrium problem. Under mild assumptions we establish the equivalence between the existence of solution of the original problem and the boundedness of the sequence generated by regularized problems.

This paper has two objectives. One is to review the Backus-Gilbert (BG) method. Backus and Gilbert investigate continuous inverse problems in a radially symmetric earth using an approach completely different from that of standard inversion methods. If [Formula: see text] is an earth property of interest, the BG method provides average values of [Formula: see text] over some radial intervals. If an interval [Formula: see text] is relatively short and centered near a particular radius [Formula: see text], the BG average will be a local average of [Formula: see text] near [Formula: see text] with a resolving length of [Formula: see text]. Backus and Gilbert also introduce the concept of trade-off between resolution and error, which is essential in inverse theory regardless of the inversion method. In spite of its importance, inverse theory texts neither describe the BG method in full nor show how to apply it in all its generality. Because the three papers that introduce the BG method are mathematically difficult, this paper provides background material and a detailed derivation of the method, as well as two completely worked-out synthetic 1D examples. In addition, the application of the method to disciplines other than solid-earth geophysics is discussed. The second objective is to derive the minimum-norm solution for the continuous case, also introduced by Backus and Gilbert. Their derivation is little known, and it is completely different from a better known, simpler derivation that has some logical difficulties. The minimum-norm solution lacks a constraint that the BG averages have, which may lead to results affected by significant error. To address this problem, a constrained minimum-norm solution is derived here. The two examples for the BG method were used to compute minimum-norm solutions. Interestingly, these examples show that the BG averages can be closer to the true values than the corresponding minimum-norm solutions, although these results should not be considered general.

The purpose of this paper is to study Tikhonov regularization methods for inverse variational inequalities. A rather weak coercivity condition is given which guarantees that the solution set of regularized inverse variational inequality is nonempty and bounded. Moreover, the perturbation analysis for the solution set of regularized inverse variational inequality is established. As an application, we show that solutions of regularized inverse variational inequalities form a minimizing sequence of the D-gap function under a mild condition.

An iterative method is investigated for a nonlinear ill-posed Hammerstein type operator equation KF(x)=f. We use a center-type Lipschitz condition in our convergence analysis instead of the usual Lipschitz condition. The adaptive method of Pereverzev and Schock (SIAM J. Numer. Anal. 43(5):2060–2076, 2005) is used for choosing the regularization parameter. The optimality of this method is proved under a general source condition involving the Frechet derivative of F at some initial guess x 0. A numerical example of nonlinear integral equation shows the efficiency of this procedure.

Second discrepancy principle was found as an effective rule for choosing the regularization parameter in Tikhonov’s method applied for solving linear operator equations of the first kind. The rule is formulated as an additional non-linear equation involving second discrepancy to be satisfied by the regularized solution and regularization parameter together. Two constant parameters appearing in the definition of this criterion may be chosen freely within some limits, preserving asymptotic convergence properties of the method when errors in data tend to zero. This work presents a heuristic approach to determining both constants: they are found on the basis of numerical simulations exploiting realistic model representations with inexact data of problems which are to be solved and accounting for appropriate error levels. The approach is expressed as an approximate optimization problem defined on a class of inverse problems studied. The formulation of the method is given together with a numerical illustration....

In this paper, we study the statistical properties of the method of regularization with radial basis functions in the context of linear inverse problems. Radial basis function regularization is widely used in machine learning because of its demonstrated effectiveness in numerous applications and computational advantages. From a statistical viewpoint, one of the main advantages of radial basis function regularization in general and Gaussian radial basis function regularization in particular is their ability to adapt to varying degrees of smoothness in a direct problem. We show here that similar approaches for inverse problems not only share such adaptivity to the smoothness of the signal but also can accommodate different degrees of ill-posedness. These results render further theoretical support to the superior performance observed empirically for radial basis function regularization.

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example.

SUMMARY In a recent paper an algorithm for large-scale Tikhonov regularization in standard form called GKB-FP was proposed and numerically illustrated. In this paper, further insight into the convergence properties of this method is provided, and extensions to general-form Tikhonov regularization are introduced. In addition, as alternative to Tikhonov regularization, a preconditioned LSQR method coupled with an automatic stopping rule is proposed. Preconditioning seeks to incorporate smoothing properties of the regularization matrix into the computed solution. Numerical results are reported to illustrate the methods on large-scale problems. Copyright © 2013 John Wiley & Sons, Ltd.

The problem of low-rank factorization of matrices with missing entries appears in many application fields for data analysis. Due to the high missing of entries in the matrices, this problem is highly ill-posed. It is a challenging task to give an efficient algorithm for obtaining a physically meaning full solution to the ill-posed problem. This paper surveys our recent develops on this topic, including theoretical analysis on solutions to the ill-posed problem and the regularization. Two novel regularization methods, entry-constraint regularization and inducible regularization, are given for the problem with moderate dada missing and high-level data missing, respectively. Corresponding algorithms for solving the regularized problems are also discussed and illustrated on some simulation data sets and real-world examples in large scale from applications.

The solution of linear discrete ill-posed problems is very sensitive to perturbations in the data. Confidence intervals for solution coordinates provide insight into the sensitivity. This paper presents an efficient method for computing confidence intervals for large-scale linear discrete ill-posed problems. The method is based on approximating the matrix in these problems by a partial singular value decomposition of low rank. We investigate how to choose the rank. Our analysis also yields novel approaches to the solution of linear discrete ill-posed problems with solution norm or residual norm constraints.

It is well known that the problem of fractional differentiation n is an ill -posed problem. So far there exists many approximation methods for solving this problem. In this paper we prove a stability estimate for a problem of frac tional differentiation. Based on the obtained stability estimate, we present a Tikhonov regularization method and obtain the error estimate. According to the optimality theory of regularization, the error estimates are order optimal. Numerical experiment s hows that the regularization works well.