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In this paper we present an algorithm to bound parameters estimated from very small data sets and use this method to suggest optimal times for data collection. The method is deterministic in that we assume that one has error bars on the data. The basic idea is based on bounded error parameter identification methods and we use this framework to assign a measure of quality to the estimated parameter. Exploiting the dynamics of the underlying mathematical model allows us to describe the quality of the estimated parameter in a way that, due to the small number of data points, is not appropriate for traditional statistical techniques. The algorithm in this paper differs from traditional bounded error parameter identification techniques in that one computes the membership set by solving an inverse problem rather than a forward interval problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified inhomogeneous Helmholtz equation Δ u( x, y) u( x, y) f( x, y), x , 0 y 1, with inhomogeneous Cauchy data being given at y 0. The problem is known to be ill-posed, as the solution (if exists) does not depend continuously on the given data. We propose a regularization method to obtain a stable approximate solution of the problem and get some error estimates. Finally, a numerical example shows the effectiveness of the proposed method. [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]

This paper addresses the regularization of linear inverse problems with so-called sparsity constraints in terms of ℓ P-penalty terms. Error estimates and convergence rates are derived. The application of the generalized gradient projection method and the semismooth Newton method to the according minimization problem is shown. Numerical experiments show the efficiency of the semismooth Newton method. [ABSTRACT FROM AUTHOR]

Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well. [ABSTRACT FROM AUTHOR]

In this paper we report on a new iterative method for regularizing a nonlinear Hammerstein type operator equation in Hilbert spaces. The proposed Newton–Lavrentiev method is a combination of Lavrentiev regularization and a Newton's iteration. Under the assumptions that the operator F is continuous Fréchet differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulfills a smoothness condition, we will give a convergence rate result. [ABSTRACT FROM AUTHOR]

A problem of an approximate solving of Symm's integral equation for an infinitely smooth closed boundary is considered. For a fully discrete projection method, error estimates are found in the metric of Sobolev spaces. A discretization parameter is chosen by balancing principle. [ABSTRACT FROM AUTHOR]

In this paper, we consider the possibility of a posteriori error estimates for linear and nonlinear ill-posed operator equations. Given an auxiliary finite-dimensional problem , that approximates the original infinite model , with a certain level of accuracy, we try to estimate the distance between z, an approximate solution to , and , the exact solution to . The problem is assumed to accumulate different sources of error (discretization, measurements, etc.), and the computed solution z is assumed to satisfy the equation within a nonzero tolerance . We conduct both a theoretical and numerical study of a posteriori error analysis. [ABSTRACT FROM AUTHOR]

For solving linear ill-posed problems with noisy data, regularization methods are required. In this paper we study regularization under general noise assumptions containing large noise and small noise as special cases. We derive order optimal error bounds for an extended Tikhonov regularization by using some pre-smoothing. This accompanies recent results by the same authors, Regularization under general noise assumptions, Inverse Problems 27:3, 035016, 2011. [ABSTRACT FROM AUTHOR]

A Cauchy problem for elliptic equations with nonhomogeneous Neumann data in a cylindrical domain is investigated in this paper. For the theoretical aspect the a-priori and a-posteriori parameter choice rules are suggested and the corresponding error estimates are obtained. About the numerical aspect, for a simple case results given by two methods based on the discrete Sine transform and the finite difference method are presented; an idea of left-preconditioned GMRES (Generalized Minimum Residual) method is proposed to deal with the high dimensional case to save the time; a view of dealing with a general domain is suggested. Some ill-posed problems regularized by the quasi-boundary-value method are listed and some rules of this method are suggested. [ABSTRACT FROM AUTHOR]

In this paper solutions of ill-posed problems with some a priori information about the exact solution are considered. For the first group of such problems it is supposed that the exact solution is a bounded piecewise convex function on some bounded segment [ a, b]. It is shown that the set of these functions is a compact set in LP[ a, b] and an approximate solution tends to the exact one uniformly on some subset of [ a, b]. Sourcewise represented functions form the second group of the problems. For this case it is possible to find a so-called a posteriori error estimation of an approximate solution. The method of extending compacts may help to estimate this a posteriori error. [ABSTRACT FROM AUTHOR]

For solving linear ill-posed problems regularization methods are required when the right hand side and the operator is with some noise. In the present paper regularized approximations are obtained by Tikhonov regularization, by regularized total least squares (RTLS) and by dual regularized total least squares (DRTLS). We discuss computational aspects and provide order optimal error bounds under two assumptions characterizing the smoothness of the unknown solution and the smoothing properties of the forward operator. The derived error bounds extend results from [S. Lu, S.V. Pereverzev, and U. Tautenhahn, Regularized total least squares: computational aspects and error bounds, Johann Radon Institute for Computational and Applied Mathematics, Report no. 2007-30, 2007.] where the above regularization methods are studied for finitely smoothing forward operators. We illustrate our theory by a special inverse heat conduction problem. [ABSTRACT FROM AUTHOR]

We study error minimizing relaxation (EMR) strategies for use in Landweber and Kaczmarz type iterations applied to linear systems with or without convex constraints. Convergence results based on operator theory are given, assuming exact data. The advantages and disadvantages of these relaxation strategies on a noisy and ill-posed problem are illustrated using examples taken from the field of image reconstruction from projections. We also consider combining EMR with penalization. [ABSTRACT FROM AUTHOR]

The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize 'approximation level' to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis. [ABSTRACT FROM AUTHOR]

We are interested on an inverse problem of distributed parameter estimation in a partial differential equation (PDE) from measures of the PDE's solution. The considered parameter is supposed to be a piecewise constant function. Identifying the parameterization consists on identifying both values of the parameter and shapes of zones where the parameter is constant. We develop a posteriori error estimators for the considered inverse problem and we present a new algorithm, result of the combination between adaptive parameterization technique, leading to overcome the underdetermination problem and, an adaptive mesh technique, guided by a posteriori error estimators, providing more precise results. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data. [ABSTRACT FROM AUTHOR]

We consider ill-posed inverse problems for linear operator equations in Banach lattices with a priori information that the exact solution belongs to a compact set. We provide an error estimate for an approximate solution to the ill-posed problem. We also show the existence of a supremum and infimum of the set of approximate solutions and their convergence to the exact solution. After finite-dimensional approximation the problem of computation of the bounds is reduced to a linear programming problem. [ABSTRACT FROM AUTHOR]

Suppose that a uniqueness theorem is valid for an ill-posed problem. It is shown then that the distance between the exact solution and terms of a minimizing sequence of the Tikhonov functional is less than the distance between the exact solution and the first guess. Unlike the classical case when the regularization parameter tends to zero, only a single value of this parameter is used. Indeed, the latter is always the case in computations. Next, this result is applied to a specific coefficient inverse problem. A uniqueness theorem for this problem is based on the method of Carleman estimates. In particular, the importance of obtaining an accurate first approximation for the correct solution follows from Theorems 7 and 8. The latter points towards the importance of the development of globally convergent numerical methods as opposed to conventional locally convergent ones. A numerical example is presented. [ABSTRACT FROM AUTHOR]

We consider a class of finite difference schemes for approximating solutions to ill-posed Cauchy problems for first order linear operator differential equations in a Hilbert space. Both the operator and the initial state in the problems are supposed to be noisy. Using an appropriate coordination between the mesh width and error levels, we improve previous error estimates for approximations generated by the schemes. [ABSTRACT FROM AUTHOR]

We consider an ill-posed Cauchy problem for an abstract linear differential equation of the first order in a Hilbert space. Both the operator and the initial state are supposed to be noisy. For solving the problem, a class of finite difference methods is used. Coordinating the mesh width with error levels we obtain logarithmic error estimates for the methods of this class. [ABSTRACT FROM AUTHOR]

The authors study regularization of non-linear inverse problems by projection methods. Non-linearity is controlled by some range invariance assumption. Emphasis is on approximation theoretic properties of the discretization which determine the convergence rates. Instead of using source conditions for the true solution to represent the error, the authors show how distance functions with respect to some benchmark smoothness are able to replace this. Some examples indicate how the results can be applied. [ABSTRACT FROM AUTHOR]

We consider regularization of linear ill-posed problems Au = f in Hilbert spaces. Approximations ur to the solution u* can be constructed by the Tikhonov method or by the Lavrentiev method, by iterative or by other methods. We assume that instead of f ∈ R(A) noisy data f are available with the approximately given noise level δ: in process δ → 0 it holds ∥f - f∥/δ ≤ c with unknown constant c. We propose a new a-posteriori rule for the choice of the regularization parameter r = r(δ) guaranteeing ur(δ) → u* for δ → 0. Note that such convergence is not guaranteed for the parameter choice given by the L-curve rule, by the GCV-rule, by the quasioptimality criterion and also for discrepancy principle ∥Aur - f∥ = bδ with b < c. The error estimates are given, which in case ∥f - f∥ ≤ δ are quasioptimal and order-optimal. [ABSTRACT FROM AUTHOR]

A generalized method of Lavrent'ev regularization involving an unbounded operator is studied. Assuming some condition in terms of the inverse of the unbounded operator, error estimates of scale-type for the regularized solution are derived. The method is applied to ill-posted problems containing selfadjoint operators and Volterra equations of the first kind. [ABSTRACT FROM AUTHOR]