ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

Recently, Mahale and Nair considered a simplified generalized Gauss-Newton iterative method for getting an approximate solution for the nonlinear ill-posed operator equation under the modified general source condition. The advantage of this method and the source condition over the classical Gauss-Newton iterative method is that the iterations and source condition involve calculation of the Fréchet derivative only at the point x 0, i.e., at the initial approximation for the exact solution x † of the nonlinear ill-posed operator equation F(x) = y. Motivated by the work of Qinian Jin and Tautenhan, error analysis of the simplified Gauss-Newton iterative method is done in this article under a Morozove-type stopping rule, which is much simpler than the stopping rule considered in the article of Mahale and Nair. An order optimal error estimate is obtained under a modified general source condition which also involves calculation of the Fréchet derivative at the point x 0.

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

A finite-element discretization, independent of the location of the interface, is proposed and analysed for linear elliptic and parabolic interface problems. We establish error estimates of optimal order in the H 1 -norm and almost optimal order in the L 2 -norm for elliptic interface problems. An extension to parabolic interface problems is also discussed and an optimal error estimate in the L 2 (0, T; H1 (Ω))-norm and an almost optimal order estimate in the L 2 (0, T; L 2 (Ω))-norrn are derived for the spatially discrete scheme. A fully discrete scheme based on the backward Euler method is analysed and an optimal order error estimate in the L 2 (0, T; H I (Ω))-norm is derived. The interfaces are assumed to be of arbitrary shape and smooth for our purpose.

Recently in the work of George, 2010, we considered a modified Gauss-Newton method for approximate solution of a nonlinear ill-posed operator equationF(x)=y, whereF:D(F)⊆X→Yis a nonlinear operator between the Hilbert spacesXandY. The analysis in George, 2010 was carried out using a majorizing sequence. In this paper, we consider also the modified Gauss-Newton method, but the convergence analysis and the error estimate are obtained by analyzing the odd and even terms of the sequence separately. We use the adaptive method in the work of Pereverzev and Schock, 2005 for choosing the regularization parameter. The optimality of this method is proved under a general source condition. A numerical example of nonlinear integral equation shows the performance of this procedure.

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a 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. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

We develop and analyze a new discontinuous finite volume method for second order elliptic problems with tensor coefficients. An optimal order error estimate is obtained in a mesh dependent norm. An L2 -error estimate is also obtained.

We prove optimal order error estimates in the H1 norm for the $\Cq_1$ isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses, improving previously known results. Choose one diagonal and divide the element into two triangles. We show that, if the chosen diagonal is the longest one, then the constant in the error estimate depends only on the maximum angle of the two triangles. Otherwise, the constant depends on that maximum angle and on the ratio between the two diagonals. In particular, we obtainthe optimal order error estimate under the maximum angle condition as in the case of triangular elements. Consequently, the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals.

We consider linear ill-posed problems in Hilbert spaces with a noisy right hand side and a given noise level. To solve non-self-adjoint problems by the (it-erated) Tikhonov method, one effective rule for choosing the regularization parameter is the monotone error rule (Tautenhahn and Hamarik, Inverse Problems, 1999, 15, 1487– 1505). In this paper we consider the solution of self-adjoint problems by the (iterated) Lavrentiev method and propose for parameter choice an analog of the monotone error rule. We prove under certain mild assumptions the quasi-optimality of the proposed rule guaranteeing convergence and order optimal error estimates. Numerical examples show for the proposed rule and its modifications much better performance than for the modified discrepancy principle.

In this paper, we establish a novel reduced-order weak Galerkin (ROWG) finite element method for solving parabolic equation with nonlinear compression coefficient. We first present the classical weak Galerkin finite element discretization scheme and derive the optimal error estimates. Then we apply a proper orthogonal decomposition (POD) technique to develop the ROWG method, which can effectively reduce degrees of freedom and CPU time. The optimal order error estimates are also derived, and the algorithm flow is provided. Finally, some numerical experiments illustrate the performance of the ROWG method. The numerical results show that the proposed ROWG method is efficient for solving nonlinear parabolic equations.

A dynamic frictionless contact problem between an elasto-visco-plastic body and a deformable obstacle is numerically studied in this paper. The contact is modeled using the normal compliance contact condition involving adhesion effect of contact surfaces and the thermo-elasto-visco-plastic constitutive law is employed to simulate the thermal effects. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We provide variational formulation which can be put into a general model of system defined by a second order evolution equality, a first order evolution equation and a first order differential equation. The existence and the uniqueness of the weak solution were established. The proofs are based on evolution equations with monotone operators, differential equations and fixed point theorem. The applied approach leads to a system involving variational equalities and we focuse on the study of fully discrete approximation scheme where both the spatial and temporal variables are discretized. The finite difference method and finite element method are popular numerical methods to discretize problems involving ordinary and partial differential equations. In this work, we will use the first-order finite differences to approximate time derivatives. Moreover, our discussion will be limited to the case of uniform partitions of the time interval and we will use the finite elements for spatial discretizations of variational equalities. For a numerical scheme, we prove the existence and uniqueness of the solutions, and derive optimal order error estimates under certain regularity assumption on the solution of the continuous problem. The additional assumptions on the solution regularity are needed for optimal order error estimates and may not always be valid. Nevertheless, weaker solution regularity assumptions can be used to derive error estimates with lower convergence orders.

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Duran (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P 1–P 1 pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H 1-norm for velocity and an estimate in the L 2-norm for pressure are obtained. An optimal error estimate in the L 2-norm for the velocity is derived under an additional assumption on the body force.

BSTRACT. In this paper, we determine the pα-duals of certain newly introduced difference se- quence spaces, namely ℓ∞(u, △m, q), c0(u, △m, q), and c(u, △m, q). Furthermore, we investigate v v v their topological properties and establish the conditions under which these spaces are perfect. 1. Introduction and preliminaries Assume that ω denotes the space of all sequences. Throughout this chapter, we adopt the notation q = (qk) to denote a sequence of positive real numbers. Some generalized sequence spaces, such as ℓ∞(q), c(q), and c0(q), have been studied by various authors [9–11]. Ahmad and Mursaleen [1] introduced the concept of some new generalized difference sequence spaces, defined as follows: △ℓ∞(q) = {x ∈ ω : △x ∈ ℓ∞(q)}. △c(q) = {x ∈ ω : △x ∈ c(q)}. △c0(q) = {x ∈ ω : △x ∈ c0(q)}. Where △xk = xk − xk+1 . After that, Et and Bas¸arer [8] defined these difference sequence spaces in a more general form by introducing the operator △mon these spaces.

In this article, our aim is to introduce a new generalized class of vector valued sequence space $F(E_k, \Delta^m_\nu, M,p,q)$ using Orlicz function $M$, where $(E_k)_{k=1}^{\infty}$ is the class of all seminormed space ~$(E_k,q_k)$ ~with $E_{k+1}\subseteq E_k$. It is assumed that $F$ is a normal, $AK$-sequence space with absolutely monotone paranorm $g_F$ and $p=(p_k)$ is a bounded sequence of positive real numbers. Here it is also proved that the space is a complete paranormed space under the paranorm $g$ along with cetrain inclusion relations.

We introduce and analyze a post-processing for continuous variational space-time discretizations of wave problems. The post-processing lifts the fully discrete approximation in time from a continuous to a continuously differentiable one. Further, it increases the order of convergence. The discretization in time is based on the Gauss–Lobatto quadrature formula which is essential for ensuring the improved convergence behavior. Error estimates of optimal order in various norms are proved. A bound of superconvergence at the discrete time nodes is included. To show the error estimates, a special approach is developed. First, error estimates for the time derivative of the post-processed solution are proved. In a second step these results are used to show the error estimates for the post-processed solution itself. The need for this approach comes through the structure of the wave equation. Stability properties of its solution preclude us from using absorption arguments for the control of certain error quantities. A further key ingredient of this work is the construction of a time-interpolate of the exact solution that is needed in an essential way. Finally, a conservation of energy property is shown for the post-processed solution which is an important feature for approximation schemes to wave equations. The error estimates are confirmed by numerical experiments.