A family of rules for parameter choice in Tikhonov regularization of ill-posed problems with inexact noise level

We consider Tikhonov regularization of linear ill-posed problems with noisy data. The choice of the regularization parameter by classical rules, such as discrepancy principle, needs exact noise level information: these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level. We propose a general family of parameter choice rules, which includes many known rules and guarantees convergence of approximations. Quasi-optimality is proved for a sub-family of rules. Many rules from this family work well also in the case of many times under- or overestimated noise level. In the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post-estimated monotone error rule and a certain new rule from the proposed family. The advantages of the new rules are demonstrated in extensive numerical experiments.     

A two-stage method for nonlinear inverse problems

In this work we are interested in the solution of nonlinear inverse problems of the form F(x)=y$F(x)=y$. We consider a two-stage method which is third order convergent for well-posed problems. Combining the method with Levenberg–Marquardt regularization of the linearized problems at each stage and using the discrepancy principle as a stopping criterion, we obtain a regularization method for ill-posed problems. Numerical experiments on some parameter identification and inverse acoustic scattering problems are presented to illustrate the performance of the method.     

A Modified Convergence Analysis For Solving Fredholm Integral Equations of The First Kind

In this paper, we consider a modified convergence analysis for solving Fredholm integral equations of the first kind in Hilbert space setting using Tikhonov regularization. We follow a general approach which not only includes, as special case, the results of Groetsch [2] but also obtain the same with weaker assumptions.     

Inverse problem of groundwater modeling by iteratively regularized Gauss-Newton method with a nonlinear regularization term

A nonlinear minimization problem ‖F(d)−u‖?min, ‖u−u_δ‖≤δ, is a typical mathematical model of various applied inverse problems. In order to solve this problem numerically in the lack of regularity, we introduce iteratively regularized Gauss-Newton procedure with a nonlinear regularization term (IRGN-NRT). The new algorithm combines two very powerful features: iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process. The convergence analysis is carried out in the presence of noise in the data and in the modified source condition. Numerical simulations for a parameter identification ill-posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method.     

Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions

In this paper, we consider the Cauchy problem of Laplace’s equation in the neighborhood of a circle. The method of fundamental solutions (MFS) combined with the discrete Tikhonov regularization is applied to obtain a regularized solution from noisy Cauchy data. Under the suitable choices of a regularization parameter and an a priori assumption to the Cauchy data, we obtain a convergence result for the regularized solution. Numerical experiments are presented to show the effectiveness of the proposed method.     

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).     

Four parameter proximal point algorithms

Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) [7] while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) [9] and later developed by Xu (2006) [8]. The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature.     

On the regularization of projection methods for solving III-posed problems

Summary In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.     

Strong asymptotic convergence of evolution equations governed by maximal monotone operators with Tikhonov regularization

We consider the Tikhonov-like dynamics where A is a maximal monotone operator on a Hilbert space and the parameter function ε(t) tends to 0 as t→∞ with . When A is the subdifferential of a closed proper convex function f, we establish strong convergence of u(t) towards the least-norm minimizer of f. In the general case we prove strong convergence towards the least-norm point in A⁻¹(0) provided that the function ε(t) has bounded variation, and provide a counterexample when this property fails.     

Bifurcation direction and exchange of stability for variational inequalities on nonconvex sets

This paper concerns Crandall–Rabinowitz type bifurcation for abstract variational inequalities on nonconvex sets and with multidimensional bifurcation parameter. We derive formulae which determine the bifurcation direction and, in the case of potential operators, the stability of all solutions close to the bifurcation point. In particular, it follows that in some cases an exchange of stability appears similar to the case of equations, but in some other cases stable nontrivial solutions bifurcate at points where there is no loss of stability of the trivial solution. As an application we consider a system of two second order ODEs with nonconvex unilateral boundary conditions.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

A note on the implicit function theorem for quasi-linear eigenvalue problems

We consider the quasi-linear eigenvalue problem −Δ_pu=λg(u) subject to Dirichlet boundary conditions on a bounded open set Ω, where g is a locally Lipschitz continuous function. Imposing no further conditions on Ω or g, we show that for λ near zero the problem has a bounded solution which is unique in the class of all small solutions. Moreover, this curve of solutions parameterized by λ depends continuously on the parameter.     

Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.     

Multiple solutions for a class of elliptic equations with jumping nonlinearities

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.     

Drift transforms and Green function estimates for discontinuous processes

In this paper, we consider Girsanov transforms of pure jump type for discontinuous Markov processes. We show that, under some quite natural conditions, the Green functions of the Girsanov transformed process are comparable to those of the original process. As an application of the general results, the drift transform of symmetric stable processes is studied in detail. In particular, we show that the relativistic α-stable process in a bounded C^1,1-smooth open set D can be obtained from symmetric α-stable process in D through a combination of a pure jump Girsanov transform and a Feynman-Kac transform. From this, we deduce that the Green functions for these two processes in D are comparable.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

Two regularization methods for a spherically symmetric inverse heat conduction problem

In this paper, we consider a spherically symmetric inverse heat conduction problem of determining the internal surface temperature of a hollow sphere from the measured data at a fixed location inside it. This is an ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Tikhonov type’s regularization method and a Fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates.     

Convergence rate analysis of a derivative free landweber iteration for parameter identification in certain elliptic PDEs

We consider the nonlinear inverse problem of identifying a parameter from knowledge of the physical state in an elliptic partial differential equation. For a derivative free Landweber method, convergence rates are proven under a weak source condition not involving the standard Fréchet derivative of the nonlinear parameter-to-output map. This source condition is discussed both for the estimation of state- and space-dependent parameters in higher dimensions. Finally, numerical results are presented.This work was supported by the Austrian National Science Foundation FWF under grant SFB F013 project F1308.