Lavrentiev Regularization Research Articles

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

Abstract In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

On ill-posedness concepts, stable solvability and saturation

Abstract We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

Lavrentiev's regularization method for nonlinear ill-posed equations in Banach spaces

In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the implementation of Lavrentiev regularization method. Using general Hölder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

An analytical solution of the ill–posed Cauchy problem for Lame equation in a rectangle

In this paper we present an analytical method for the inverse Cauchy problem of Lame equation of the elasticity theory. A rectangular domain is frequently used in engineering structures and we only consider the analytical solution in a two-dimensional rectangle, wherein a missing boundary condition is recovered from a full measurement of the stresses and displacements on an accessible boundary. The essence of the method consists in solving three independent Cauchy problems for the Laplace and Poisson equations. For each of them the Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we use a Lavrentiev regularization method and the termwise separable property of kernel function allows us to obtain a closed-form regularized solution. As a result, for the displacement components, we obtain solutions in the form of a sum of series with three regularization parameters. The uniform convergence and error estimation of the regularized solutions are proved.

Nyström type subsampling analyzed as a regularized projection

In the statistical learning theory the Nyström type subsampling methods are considered as tools for dealing with big data. In this paper we consider Nyström subsampling as a special form of the projected Lavrentiev regularization, and study it using the approaches developed in the regularization theory. As a result, we prove that the same capacity independent learning rates that are guaranteed for standard algorithms running with quadratic computational complexity can be obtained with subquadratic complexity by the Nyström subsampling approach, provided that the subsampling size is chosen properly. We propose a priori rule for choosing the subsampling size and a posteriori strategy for dealing with uncertainty in the choice of it. The theoretical results are illustrated by numerical experiments.

Optimal rates for Lavrentiev regularization with adjoint source conditions

Optimal rates for Lavrentiev regularization with adjoint source conditions

Multiscale collocation methods for ill-posed integral equations with a modified posteriori parameter selection

Multiscale collocation methods are developed for solving ill-posed Fredholm integral equations of the first kind in Banach spaces, if the associated resolvent integral operator fulfils a condition with respect to a interval. We apply a multiscale collocation method with a matrix compression strategy to discretize the integral equation of the second kind obtained by using the Lavrentiev regularization from the original ill-posed integral equation and then use the multilevel augmentation method to solve the resulting discrete equation. A modified a posteriori parameter choice strategy is presented, which leads to optimal convergence rates. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations

Abstract Mahale and Nair [12] considered an iterated form of Lavrentiev regularization for obtaining stable approximate solutions for ill-posed nonlinear equations of the form F ⁢ ( x ) = y ${F(x)=y}$ , where F : D ⁢ ( F ) ⊆ X → X ${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate under a general source condition. However, the iterations defined in [12] require calculation of Fréchet derivatives at each iteration. In this paper, we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fréchet derivative only at the point x 0 ${x_{0}}$ , i.e., at the initial approximation of the exact solution x † ${x^{\dagger}}$ . Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fréchet derivative at the point x 0 ${x_{0}}$ , instead at the unknown exact solution x † ${x^{\dagger}}$ as in [12].

Converse Results, Saturation and Quasi-optimality for Lavrentiev Regularization of Accretive Problems

This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.

Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space

In the literature on singular perturbation (Lavrentiev regularization) for the stable approximate solution of operator equations with monotone operators in the Hilbert space the phenomena of conditional stability and local well-posedness or ill-posedness are rarely investigated. Our goal is to present some studies which try to bridge this gap. So we present new results on the impact of conditional stability on error estimates and convergence rates for the Lavrentiev regularization and distinguish for linear problems well-posedness and ill-posedness in a specific manner motivated by a saturation result. Taking into account that the behavior of the bias (regularization error in the noise-free case) is crucial, general convergence rates, including logarithmic rates, are derived for linear operator equations by means of the method of approximate source conditions. This allows us to extend well-known convergence rate results for the Lavrentiev regularization that were based on general source conditions to the case of non-selfadjoint linear monotone forward operators for which general source conditions fail. Examples presenting the self-adjoint multiplication operator as well as the non-selfadjoint fractional integral operator and Cesàro operator illustrate the theoretical results. Extensions to the nonlinear case under specific conditions on the nonlinearity structure complete the paper.

Lavrentiev's regularization method in Hilbert spaces revisited

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations

George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution xαδ of the method of Lavrentiev regularization, i.e., F(x)+α(x-x0)=yδ, approximating the solution xˆ of the ill-posed problem F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator defined on a real Hilbert space X. The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem.

On the method of Lavrentiev regularization and majorizing sequence for solving ill-posed problems with monotone operators

In this paper we consider the Lavrentiev regularization method for obtaining stable approximate solution to nonlinear ill-posed operator equations F (x) = y where F : D(F ) X ! X is a nonlinear monotone operator dened on a real Hilbert space X. We assume that only a noisy data y 2 X withky y k are available. Under the assumption that F is Lipschitz continuous, the iteration x n; converges to the unique solution x of the equation F (x) + (x x0) = y (x0 := x 0; ). It is known that (Tautanhahn (2002)) x converges to the solution ^ of F (x) = y: The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. Under a general source condition on x0 ^ x we proved that the errorkx n; ^

Lavrentiev-type regularization methods for Hermitian problems

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.

A derivative free iterative method for the implementation of Lavrentiev regularization method for ill-posed equations

In this work, we develop a derivative free iterative method for the implementation of Lavrentiev regularization for approximately solving the nonlinear ill-posed operator equation F(x) = y. Convergence analysis shows that the method converges quadratically. Apart from being totally free of derivatives, the method, under a general source condition provide an optimal order error estimate. We use the adaptive method introduced in Pereverzyev and Schock (SIAM J. Numer. Anal. 43, 2060---2076, 2005) for choosing the regularization parameter. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem.

An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems

In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator or F′(x0) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data yδ∈X with ‖y-yδ‖⩽δ are available. Further we assume that Fréchet derivative F′ of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x0-xˆ, the error ‖xˆ-xn,αδ‖ between the regularized approximation xn,αδ(x0,αδ≔x0) and the solution xˆ is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem.

POD a-posteriori error analysis for optimal control problems with mixed control-state constraints

In this work linear-quadratic optimal control problems for parabolic equations with mixed control-state constraints are considered. These problems arise when a Lavrentiev regularization is utilized for state constrained linear-quadratic optimal control problems. For the numerical solution a Galerkin discretization is applied utilizing proper orthogonal decomposition (POD). Based on a perturbation method it is determined how far the suboptimal control, computed on the basis of the POD method, is from the (unknown) exact one. Numerical examples illustrate the theoretical results. In particular, the POD Galerkin scheme is applied to a problem with state constraints.

Fractional regularization matrices for linear discrete ill-posed problems

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.

Newton Lavrentiev regularization for ill-posed operator equations in Hilbert scales

In this paper we consider the two step method for approximately solving the ill-posed operator equation F(x)=f, where F:D(F)⊆X→X, is a nonlinear monotone operator defined on a real Hilbert space X, in the setting of Hilbert scales. We derive the error estimates by selecting the regularization parameter α according to the adaptive method considered by Pereverzev and Schock in (2005), when the available data is fδ with ‖f-fδ‖⩽δ. The error estimate obtained in the setting of Hilbert scales {Xr}r∈R generated by a densely defined, linear, unbounded, strictly positive self adjoint operator L:D(L)⊂X→X is of optimal order.

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.