Lavrentiev Regularization Research Articles

On the semilocal convergence of a two-step Newton-like projection method for ill-posed equations

We present new semilocal convergence conditions for a two-step Newton-like projection method of Lavrentiev regularization for solving ill-posed equations in a Hilbert space setting. The new convergence conditions are weaker than in earlier studies. Exampl

An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

On the Laplacian operation with applications in magnetic resonance electrical impedance imaging

Consider the numerical computation of Laplacian operation from the noisy data of a given function f(x) defined in Ω ⊂ ℝ2. By expressing this ill-posed problem as a Fredholm integral equation of the first kind, we construct the approximate solution from a PDE boundary value problem directly, which can be considered as the Lavrentiev regularization in essence. The optimal error estimate of this regularization for an appropriate a posteriori choice strategy of the regularization parameter is given under some source conditions of the exact Laplacian operation. An iterative algorithm for computing this a posteriori regularization parameter under the framework of model function method is also given with its convergence analysis. The advantage of the proposed scheme is that we treat the Laplacian operation as a whole, rather than the partial derivatives of each variable, and therefore the Laplacian operation can be computed in a bounded domain with arbitrary boundary shape for noisy data specified at randomly scattered points. At last, we apply this scheme to the iterative algorithm of reconstructing the conductivity in magnetic resonance electrical impedance tomography, where the input data is the magnetic flux data for which the Laplacian operation must be carried out. The validity of the proposed scheme is shown numerically for this new medical imaging technology.

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multi- grid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical ex- periments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

Two-Step Modified Newton Method for Nonlinear Lavrentiev Regularization

A two step modified Newton method is considered for obtaining an approximate solution for the nonlinear ill-posed equation F(x)=f when the available data are fδ with ‖f−fδ‖≤δ and the operator F is monotone. The derived error estimate under a general source condition on x0−x^ is of optimal order; here x0 is the initial guess and x^ is the actual solution. The regularization parameter is chosen according to the adaptive method considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

Multigrid Solution of a Lavrentiev-Regularized State-Constrained Parabolic Control Problem

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multi- grid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical ex- periments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

An Analytical Method for the Inverse Cauchy Problem of Laplace Equation in a Rectangular Plate

ABSTRACTThe present paper reveals an analytically computational method for the inverse Cauchy problem of Laplace equation. For the sake of analyticity, and also for the frequent use of rectangular plate in engineering structure, we only consider the analytical solution in a two-dimensional rectangular domain, wherein a missing boundary condition is recovered from a partial measurement of the Neumann data on an accessible boundary. The Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we consider a Lavrentiev regularization amended to a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us to obtain a closed-form solution of the regularization type. The uniform convergence and error estimation of the regularization solution are proven. The numerical examples show the effectiveness and robustness of the new method.

Multilevel augmentation methods for nonlinear ill-posed problems

We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.

Error estimates for joint Tikhonov and Lavrentiev regularization of constrained control problems

We consider joint Tikhonov and Lavrentiev regularization of control problems with pointwise control- and state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of these results we show that, if the solution of the unconstrained problem has no active constraints, the same holds for the Tikhonov regularized solution if the regularization parameter is small enough and a certain source condition is fulfilled.

Numerical analysis of Leray–Tikhonov deconvolution models of fluid motion

This report develops and studies a new family of Navier–Stokes equation regularizations: Leray–Tikhonov regularizations with time relaxation models. This new family of turbulence models is based on a modification (consistent with the large scales) of Tikhonov–Lavrentiev regularization. With this approach, we obtain an approximation of the unfiltered solution by one filtering step. We introduce the modified Tikhonov deconvolution operator and study its mathematical properties. We also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it.

Iterated Lavrentiev regularization for nonlinear ill-posed problems

We consider an iterated form of Lavrentiev regularization, using a null sequence $(\alpha_k)$ of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form $F(x)=y$, where $F:D(F)\subseteq X\to X$ is a nonlinear operator and $X$ is a Hilbert space. Recently, Bakushinsky and Smirnova [``Iterative regularization and generalized discrepancy prinicple for monotone operator equations", Numer. Funct. Anal. Optim. 28 (2007) 13-25] considered an a posteriori strategy to find a stopping index $k_{\delta}$ corresponding to inexact data $y^{\delta}$ with $\|y-y^{\delta} \| \leq \delta$ resulting in the convergence of the method as $\delta \to 0$. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on $(\alpha_k)$ is weaker than that considered by Bakushinsky and Smirnova. doi:10.1017/S1446181109000418

Electrical Impedance Tomography Image Reconstruction Using Iterative Lavrentiev and L-Curve-Based Regularization Algorithm

Electrical impedance tomography (EIT) is a technique for determining the electrical conductivity and permittivity distribution inside a medium from measurements made on its surface. The impedance distribution reconstruction in EIT is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods are often used in solving inverse problems. However, for EIT image reconstruction, the generalized Tikhonov regularization methods may lose the boundary information due to its smoothing operation. In this paper, we propose an iterative Lavrentiev regularization and L-curve-based algorithm to reconstruct EIT images. The regularization parameter should be carefully chosen, but it is often heuristically selected in the conventional regularization-based reconstruction algorithms. So, an L-curve-based optimization algorithm is used for selecting the Lavrentiev regularization parameter. Numerical analysis and simulation results are performed to illustrate EIT image reconstruction. It is shown that choosing the appropriate regularization parameter plays an important role in reconstructing EIT images.

Lavrentiev Regularization and Balancing Principle for Solving Ill-Posed Problems with Monotone Operators

AbstractThe paper considers a method for solving nonlinear ill-posed problems with monotone operators. The approach combines the Lavrentiev method, the fixedpoint method, and the balancing principle for selection of the regularization parameter. The method’s optimality has been proved for some set of smooth solutions. A test example proves the efficiency of the proposed method.

ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

AbstractWe consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with $\|y-y^\d \|\leq \d $ resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

Optimal distributed control of the wave equation subject to state constraints

AbstractThe Lavrentiev regularization method is a tool to improve the regularity of the Lagrange multipliers in pde constrained optimal control problems with state constraints. It has already been used for problems with parabolic and elliptic systems. In this paper we consider Lavrentiev regularization for problems with a hyperbolic system, namely the scalar wave equation. We show that also in this case the regularization yields multipliers in the Hilbert space L2. We present numerical examples, where we compare the Lavrentiev regularization, Lavrentiev Prox regularization, a fixed point iteration to improve feasibility, and a penalty method.

AN ERROR ANALYSIS OF LAVRENTIEV REGULARIZATION IN LEARNING THEORY

In this paper we study how Lavrentiev regularization can be used in the context of learning theory, especially in regularization networks that are closely related to support vector machines. We briefly discuss formulations of learning from examples in the context of ill-posed inverse problem and regularization. We then study the interplay between the Lavrentiev regularization of the concerned continuous and discretized ill-posed inverse problems. As the main result of this paper, we give an improved probabilistic bound for the regularization networks or least square algorithms, where we can afford to choose the regularization parameter in a larger interval.

Solving the Inverse Problems of Laplace Equation to Determine the Robin Coefficient/Cracks' Position Inside a Disk

We consider an inverse problem of Laplace equation by recoverning boundary value on the inner circle of a two-dimensional annulus from the overde- termined data on the outer circle. The numerical results can be used to determine the Robin coefficient or crack's position inside a disk from the measurements of Cauchy data on the outer boundary. The Fourier series is used to formulate the first kind Fredholm integral equation for the unknown data f(q) on the inner circle. Then we consider a Lavrentiev regularization, by adding an extra term a f(q) to obtain the second kind Fredholm integral equation. The termwise separable prop- erty of kernel function allows us to obtain a closed-form regularized solution, of which the uniform convergence and error estimation are proved. Then we apply this method to the inverse Cauchy problem, the unknown shape of zero-potential problem, the problem of detecting crack position, as well as the problem of un- known Robin coefficient. These numerical examples show the effectiveness of the new method in providing excellent estimates of the unknown data.

On convergence of Lavrentiev regularization for semilinear parabolic optimal control problems with pointwise control and state constraints

AbstractWe consider Lavrentiev regularization for a class of semilinear parabolic optimal control problems with control constraints and pointwise state constraints and review convergence results for local solutions under Slater type assumptions as well as quadratic growth conditions. Moreover, we state a local uniqueness result for local optima under the assumptions of strict separability of the active sets as well as a second order sufficient condition for the regularized solution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Parameter choice in the Lavrentiev regularization of the data completion problem

We use the Lavrentiev method for the regularization of the severely ill-posed data completion problem, put under a variational Steklov-Poincare form. In the choice of the regularizing parameter, we check a 'super-optimal' a priori convergence criterion. Consequently, the solutions obtained by the regularization provide a minimizing sequence of the Kohn-Vogelius function, with a 'quadratic' decaying of the minimum value of it toward zero, instead of the linear rate predicted by the general theory. We call such a result the 'super-convergence' of the approximated 'incompatibility measure' of the variational problem. Finally, we apply the a posteriori Morozov discrepancy principle to the Kohn-Vogelius functional and show how it yields a regularizing strategy. We achieve by some numerical illustrations to support our analysis.

Extrapolation of Tikhonov and Lavrentiev regularization methods

We consider solution of linear ill-posed problem Au = f by Tikhonov method and by Lavrentiev method. For increasing the qualification and accuracy of these methods we use extrapolation, taking for the approximate solution linear combination of n ≥ 2 approximations of Tikhonov or Lavrentiev methods with different parameters and with proper coefficients. If the solution u* belongs to ℛ((A*A)n) and instead of f noisy data fδ with ||fδ − f|| ≥ δ are available, maximal guaranteed accuracy of Tikhonov and Lavrentiev approximations is O(δ2/3) and O(δ1/2), respectively, versus accuracy O(δ2n/(2n + 1)) and O(δn/(n + 1)) of corresponding extrapolated approximations. We propose several new rules for a posteriori choice of the regularization parameter, including modifications of the monotone error rule. Extensive numerical experiments show that in case u* ε ℛ(A*) the extrapolated Tikhonov approximation with a posteriori parameter choice (not using any smoothness information) is typically more accurate than Tikhonov approximation with optimal parameter.