Method For Nonlinear Ill-posed Problems Research Articles

Parameter Choice Strategy that Computes Regularization Parameter before Computing the Regularized Solution

The modeling of many problems of practical interest leads to nonlinear ill-posed equations (for example, the parameter identification problem (see the Numerical section)). In this article, we introduce a new source condition (SC) and a new parameter choice strategy (PCS) for the Tikhonov regularization (TR) method for nonlinear ill-posed problems. The new PCS is introduced using a new SC to compute the regularization parameter (RP) before computing the regularized solution. The theoretical results are verified using a numerical example.

A simplified iteratively regularized projection method for nonlinear ill-posed problems

In this paper, we propose a projection method for the numerical solution of the simplified iteratively regularized Gauss-Newton method of nonlinear integral equations for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive the order optimality. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

A fast two-point gradient algorithm based on sequential subspace optimization method for nonlinear ill-posed problems

In this paper, we propose a fast two-point gradient algorithm for solving nonlinear ill-posed problems, which is based on the sequential subspace optimization method. The key idea, in contrast to the standard two-point gradient method, is to use multiple search directions in each iteration without extra computation, and to get the step size by metric projection. Moreover, a modified discrete backtracking search algorithm is proposed to select the combination parameters in the accelerated two-point gradient method. Under the basic assumptions for iterative regularization methods, we establish the convergence results of the method in the noise-free case. Furthermore, stability and regularity are presented when the algorithm terminated by the discrepancy principle for the case of noisy data. Finally, some numerical simulations are presented, which exhibit that the proposed method leads to a significant reduction of the iteration numbers and the overall computational time.

Convergence Rate of Runge-Kutta-Type Regularization for Nonlinear Ill-Posed Problems under Logarithmic Source Condition

We prove the logarithmic convergence rate of the families of usual and modified iterative Runge-Kutta methods for nonlinear ill-posed problems between Hilbert spaces under the logarithmic source condition, and numerically verify the obtained results. The iterative regularization is terminated by the a posteriori discrepancy principle.

Inexact Newton regularization combined with two-point gradient methods for nonlinear ill-posed problems * *This research is partially supported by NSFC grant 11971408 and NSFC/ANR joint program 51661135011/ ANR-16-CE40-0026-01.

In this paper, we propose an inexact Newton regularization combined with two-point gradient methods for nonlinear ill-posed problems. The basic idea of the proposed method is to linearize the equation around each outer iteration and subsequently apply a so-called two-point gradient method in the inner loop to accelerate the iterative process. Under suitable assumptions, we show that the iteration sequence generated by the proposed algorithm converges to a solution of the related problem in the noiseless situation. Furthermore, the stability and regularization properties of the proposed algorithm are analyzed in the noise-data case. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method.

On a Family of Gradient-Type Projection Methods for Nonlinear Ill-Posed Problems

We propose and analyze a family of successive projection methods whose step direction is the same as the Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family encompasses the Landweber method, the minimal error method, and the steepest descent method; thus, providing an unified framework for the analysis of these methods. Moreover, we define new methods in this family, which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to successively project onto these sets. Numerical experiments are presented for a nonlinear two-dimensional elliptic parameter identification problem, validating the efficiency of our method.

Nesterov’s accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional

In this paper, we consider Nesterov’s accelerated gradient method for solving nonlinear inverse and ill-posed problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide convergence analysis of this method for ill-posed problems based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.

A direct method for nonlinear ill-posed problems

We propose a direct method for solving nonlinear ill-posed problems in Banach-spaces. The method is based on a stable inversion formula we explicitly compute by applying techniques for analytic functions. Furthermore, we investigate the convergence and stability of the method and prove that the derived noniterative algorithm is a regularization. The inversion formula provides a systematic sensitivity analysis. The approach is applicable to a wide range of nonlinear ill-posed problems. We test the algorithm on a nonlinear problem of travel-time inversion in seismic tomography. Numerical results illustrate the robustness and efficiency of the algorithm.

Convergence analysis of a two-point gradient method for nonlinear ill-posed problems

We perform a convergence analysis of a two-point gradient method which is based on Landweber iteration and on Nesterov’s acceleration scheme. Additionally, we show the usefulness of this method via two numerical example problems based on a nonlinear Hammerstein operator and on the nonlinear inverse problem of single photon emission computed tomography.

Modified Minimal Error Method for Nonlinear Ill-Posed Problems

Abstract An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate.

Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems

ABSTRACTThis work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Hölder-type sourcewise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.

Some approximate Gauss–Newton-type methods for nonlinear ill-posed problems

This paper treats numerical methods for solving the nonlinear ill-posed equation F(x) = 0, where the operator F is a Fr´ echet differentiable operator from one Hilbert space into another Hilbert space. Two parametric approximate Gauss-Newton- type methods are developed, a local convergence theorem is proved under certain conditions on a test function and the required solution, and some computational aspects are discussed. The validity of the theoretical convergence rate estimates is illustrated by the numerical results of solving two sample problems, one in a finite-dimensional and the other in an infinite-dimensional Hilbert space.

A convergence analysis of nonlinear implicit iterative method for nonlinear ill-posed problems

Implicit iterative method acquires good effect in solving linear ill-posed problems. We have ever applied the idea of implicit iterative method to solve nonlinear ill-posed problems, under the restriction that α is appropriate large, we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence, numerical results show that the implicit iterative method for nonlinear ill-posed problems is efficient. In this paper, we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when α k are determined by Hanke criterion.

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.

On convergence of regularized modified Newton's method for nonlinear ill-posed problems

Abstract In this paper we consider regularized modified Newton's method for approximately solving the nonlinear ill-posed problem F(x) = y, where the right hand side is replaced by noisy data yδ ∈ Y with ‖y – yδ ‖ ≤ δ and F : D(F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F′ of F is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition on , the error between the regularized approximation and the solution of optimal order.

Convergence results of a modified regularized gradient method for nonlinear ill-posed problems

A modified iteratively regularized gradient method and its continuous version are proposed for nonlinear ill-posed problems, in which the Tikhonov regularization term is generated by a linear operator. The linear operator may have some physical meaning. And by employing the linear operator, scaling the problem can be avoided in the case that the nonlinear operator in the problem has larger gradient. Adopting a posteriori and a priori stopping rule respectively, we establish the convergence results by using a modified approximate source condition. The numerical results show that the linear operator effects the performance greatly.

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.

Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems

In this paper, we study convergence of two different iterative regularization methods for nonlinear ill-posed problems in Banach spaces. One of them is a Landweber type iteration, the other one the iteratively regularized Gauss–Newton method with an a posteriori chosen regularization parameter in each step. We show that a discrepancy principle as a stopping rule renders these iteration schemes regularization methods, i.e., we prove their convergence as the noise level tends to zero. The theoretical findings are illustrated by two parameter identification problems for elliptic PDEs.

A simplified generalized Gauss-Newton method for nonlinear ill-posed problems

Iterative regularization methods for nonlinear ill-posed equations of the form $F(x)= y$, where $F: D(F) \subset X \to Y$ is an operator between Hilbert spaces $X$ and $Y$, usually involve calculation of the Fréchet derivatives of $F$ at each iterate and at the unknown solution $x^\dagger$. In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Fréchet derivative of $F$ only at an initial approximation $x_0$ of the solution $x^\dagger$. The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at $x_0$. The conditions under which the results of this paper hold are weaker than those considered by Kaltenbacher (1998) for an analogous situation for a special case of the source condition.

Iterative Runge–Kutta-type methods for nonlinear ill-posed problems

We present a regularization method for solving nonlinear ill-posed problems by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical regularization method. We prove that the developed iterative regularization method converges to a solution under certain conditions and with a general stopping rule. Some particular iterative regularization methods are numerically implemented. Numerical results of the examples show that the developed Runge–Kutta-type regularization methods yield stable solutions and that particular implicit methods are very efficient in saving iteration steps.