Lavrentiev Regularization Research Articles

New Trends in Applying LRM to Nonlinear Ill-Posed Equations

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ(u)=v, where κ:D(κ)⊆X⟶X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn’s paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems.

Homeier-like methods for regularization of nonlinear ill-posed equations in Hilbert space

Homeier-like methods have been extensively studied in the literature for solving non-linear operator equation of the form F(x)=y in the Banach or Hilbert space setting. Motivated by the work of George et al. (2023), we extend some class of Homeier-like methods to approximate ill-posed operator equation in Hilbert space. Two Homeier-like iterative schemes are proposed for approximating the nonlinear ill-posed operator equation F(x)=y in Hilbert space, where F is monotone. Lavrentiev regularization method and recurrence relation are employed for convergence analysis. The regularization parameter is chosen by apriori parameter choice strategy with new source condition as studied by George et al. (2023). We also supply a numerical example for comparison study of the proposed method with some of the Lavrentiev regularization methods in the literature.

Directional differentiability for shape optimization with variational inequalities as constraints

For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.

Iterative method for solving linear operator equation of the first kind

In this work, we study the iterative method for solving linear operator equation of the first kind. We present a new version of method based on the applied the iterative performance on the modified Lavrentiev method. This method is used to resolve a linear operator problem of the first kind. The suggested iterative can used to compute approximate solutions with high quality than the (standard) modified Lavrentiev regularization method. We also compared the new iterative method (modified Lavrentiev) with Landweber iterative method. The numerical testing shows the efficiency of the new iterative method in its application to resolve the inverse heat equation when trying to find the boundary value function.•Studying of new iteration algorithm and mathematical experimentations show the efficiency of the new iteration method.•Iteration method is depended on decomposed the main linear operator by using polar decomposition in order to obtain unitary operator.•The new unitary operator increases the convergence of iteration.

Approximate Solution of Boundary Value Problem for Heat Equation after Represented by Volterra Integral Equation of the First Kind

In this work, we study the regularization method for solving the Boundary Value Problem (BVP) for heat equation. The discretization method applied with two variables on Volterra integral equation in order to covert the problem into a linear operator equation after applied the separation of variables method to solve the partial differential equation. The regularization way used to obtain the estimate solution by using the Lavrentiev regularization method.

Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Choosing a Regularization Parameter in the Problem of Analytical Continuation of Gravitational Fields (Separation of Anomalies Generated by Shallow and Deep Sources)

Abstract—We present a filtration tomography technique for isolating components of the gravity field anomalies generated by inhomogeneities of the horizontally layered density model. The filtration algorithm of field separation relies on the solution of the forward and inverse problems of analytical continuation of harmonic functions through the horizontal boundary plane. We applied the regularizing algorithms to analytical continuation of the gravity field “down” to its generating sources. The fields successively recalculated upward and downward relative to preset depths allowed us to partition the initial (total) field as the sum of the fields generated in the layers based on the properly selected adaptive regularization parameter α. For the sake of stability of the inverse problem solution in the analytical continuation of the observed gravity field to a certain depth, we used the Lavrentiev regularization scheme involving the L-curve method (for selecting the adaptive regularization parameter). The smoothing regularization parameter values obtained from the preset successive depth intervals and grid step for the observed field are shown to be optimal for dividing the observed field into components corresponding to different depths. The developed algorithms for massively parallel computing systems and their application to a group of different heights were numerically implemented on the Uran supercomputer.

A New Parameter Choice Strategy for Lavrentiev Regularization Method for Nonlinear Ill-Posed Equations

In this paper, we introduced a new source condition and a new parameter-choice strategy which also gives the known best error estimate. To obtain the results we used the assumptions used in earlier studies. Further, we studied the proposed new parameter-choice strategy and applied it to the method (in the finite-dimensional setting) considered in George and Nair (2017).

Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem

A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities (VI) is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture describes an elastic body with a Barenblatt cohesive crack under the inequality condition of non-penetration at the crack faces. Based on the Lagrange approach and using smooth penalization with the Lavrentiev regularization, a formula for the shape derivative is derived. The explicit formula contains both primal and adjoint states and is useful for finding descent directions for a gradient algorithm to identify an optimal crack shape from a boundary measurement. Numerical examples of destructive testing are presented in 2D.

A Convergence Analysis of the Iterated Lavrentiev Regularization Method under a Lipschitz Condition

Mahale and Nair (2009) considered a iterated Lavrentive regularization method for obtaining stable approximate solution of nonlinear ill-posed operator equation F(x) = y, where is a nonlinear monotone operator on Hilbert space X. In this paper, we do convergence analysis of the method under the weaker non-linearity condition than the one considered by Mahale and Nair. We obtain convergence and rate of convergence results for the method merely under Lipschitz’s condition. At the end of the paper, we give a numerical example to demonstrate our algorithm.

Приближенное решение многомерных интегральных уравнений методами Монте Карло и квази-Монте Карло

The article considers an approach based on the random cubature method for solving both single and multidimensional singular integral equations, Volterra and Fredholm equations of the 1st kind, for ill-posed problems in the theory of integral equations, etc. A variant of the quasi-Monte Carlo method is studied. The integral in an integral equation is approximated using the traditional Monte Carlo method for calculating integrals. Multidimensional interpolation is applied on an arbitrary set of points. Examples of applying the method to a one-dimensional integral equation with a smooth kernel using both random and low-dispersed pseudo-random nodes are considered. A multidimensional linear integral equation with a polynomial kernel and a multidimensional nonlinear problem – the Hammerstein integral equation – are solved using the Newton method. The existence of several solutions is shown. Multidimensional integral equations of the first kind and their solution using regularization are considered. The Monte Carlo and quasi-Monte Carlo methods have not been used to solve such problems in the studied literature. The Lavrentiev regularization method was used, as well as random and pseudo-random nodes obtained using the Halton sequence. The problem of eigenvalues is solved. It is established that one of the best methods considered is the Leverrier-Faddeev method. The results of solving the problem for a different number of quadrature nodes are presented in the table. An approach based on parametric regularization of the core, an interpolation-projection method, and averaged adaptive densities are studied. The considered methods can be successfully applied in solving spatial boundary value problems for areas of complex shape. These approaches allow us to expand the range of problems in the theory of integral equations solved by Monte Carlo and quasi-Monte Carlo methods, since there are no restrictions on the value of the norm of the integral operator. A series of examples demonstrating the effectiveness of the method under study is considered.

Solving of an Inverse Boundary Value Problem for the Heat Conduction Equation by Using Lavrentiev Regularization Method

In this paper, the inverse problem for the boundary value of the heat equation is posed and solved. It is well known that this problem classified as an ill-posed problem. The boundary value problem can be represented as an integral equation of the first kind by using the separation of variables method. The discretization of the integral equation allowed us to reduce the integral equation to a system of linear algebraic equations or a linear operator equation of the first kind on Hilbert spaces. In order to find an approximation solution, we need to apply a regularization algorithm. In this type of equation and through the regularization step we faced a non-injective operator problem. The Lavrentev regularization method was used to obtain the solution instead of the Tikhonov regularization method.

On a final value problem for a nonlinear fractional pseudo-parabolic equation

<p style='text-indent:20px;'>In this paper, we investigate a final boundary value problem for a class of fractional with parameter <inline-formula><tex-math id="M1">$ \beta $</tex-math></inline-formula> pseudo-parabolic partial differential equations with nonlinear reaction term. For <inline-formula><tex-math id="M2">$ 0<\beta < 1, $</tex-math></inline-formula> the solution is regularity-loss, we establish the well-posedness of solutions. In the case that <inline-formula><tex-math id="M3">$ \beta >1 $</tex-math></inline-formula>, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.</p>

Regularization of ill-posed operator equations: an overview

We discuss and illustrate the stability issues associated with ill-posed linear operator equations and the need to have regularization methods for obtaining stable approximate solutions. As illustration, we describe two simple regularization methods, namely, the Lavrentiev regularization in the setting of a Banach spaces and Tikhonov regularization in the setting of Hilbert spaces, and discuss on the corresponding error estimates. We also indicate procedures to obtain error estimates under milder general source conditions and also to obtain better error estimates under modified methods. The discussed procedures include some of the work of the author as well.

An Efficient Nonmonotone Method for State-Constrained Elliptic Optimal Control Problems

This paper presents a novel numerical strategy based on combination of an adaptive semismooth Newton (ASN) method and the Lavrentiev regularization technique for the solution of elliptic optimal control problems with state constraints. Using the global convergence proof for a nonmonotone semismooth Newton method, we will exploit an adaptive nonmonotone line search method such that the nonmonotonicity degree of this method can be increased when the results are far from the optimum solution and it can be reduced when they are close to the optimizer. In this strategy, the role of the Lavrentiev regularization technique is converting the original optimal control problem to a regularized optimal control problem. Using the finite difference discretization scheme and a Newton–Cotes rule, the regularized optimal control problem is converted to a bound constrained optimization problem (BCOP). Then the ASN method is implemented to solve the resulting BCOP. Numerical results show the efficiency of the proposed procedure.

A Regularization Approach for an Inverse Source Problem in Elliptic Systems from Single Cauchy Data

In this article, we investigate the problem of identifying the source term f in the elliptic system from a single noisy measurement couple of the Neumann and Dirichlet data (j, g) with noise level In this context, the diffusion matrix Q is given. A variational method of Tikhonov-type regularization with specific misfit term of Kohn–Vogelius-type and quadratic stabilizing penalty term is suggested to tackle this linear inverse problem. The method also appears as a variant of the Lavrentiev regularization. For the occurring linear inverse problem in infinite dimensional Hilbert spaces, convergence and rate results can be found from the general theory of classical Tikhonov and Lavrentiev regularization. Using the variational discretization concept, where the PDE is discretized with piecewise linear and continuous finite elements, we show the convergence of finite element approximations to a sought source function. Moreover, we derive an error bound and corresponding convergence rates provided a suitable range-type source condition is satisfied. For the numerical solution, we propose a conjugate gradient method. To illustrate the theoretical results, a numerical case study is presented which supports our analytical findings.

Newton–Kantorovich regularization method for nonlinear ill-posed equations involving $$m-$$accretive operators in Banach spaces

In this paper, we study nonlinear ill-posed problems involving $$m-$$accretive mappings in Banach spaces. We consider Newton–Kantorovich regularization method for the implementation of Lavrentiev regularization method. Using general Holder type source condition we obtain an error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) for choosing the regularization parameter.

A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem

In this paper, Elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the Lavrentiev regularization is employed to deal with the state constraints. To numerically discretize the resulted problem, full piecewise linear finite element discretization is employed. Estimation of the error produced by regularization and discretization is done. The error order of full discretization is not inferior to that of variational discretization because of the Lavrentiev-regularization. Taking the discretization error into account, algorithms of high precision do not make much sense. Utilizing efficient first-order algorithms to solve discretized problems to moderate accuracy is sufficient. Then a heterogeneous alternating direction method of multipliers (hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two different weighted norms in two subproblems respectively. Additionally, to get more accurate solution, a two-phase strategy is presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the hADMM. Numerical results not only verify error estimates but also show the efficiency of the hADMM and the two-phase strategy.

Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems

For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejer property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed.

An analytical method for the inverse Cauchy problem of Lame equation in a rectangle

In this paper, we present an analytical computational method for the inverse Cauchy problem of Lame equation in 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 the full measurement of 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.