Nonlinear Ill-posed Operator Equations Research Articles

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract 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.

An accelerated inexact Newton-type regularizing algorithm for ill-posed operator equations

We propose and analyze a new iterative regularization approach, called IN-SETPG, for efficiently solving nonlinear ill-posed operator equations in the Hilbert-space setting. IN-SETPG consists of an outer iteration and an inner iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method, while the inner iteration is performed by a sequential subspace optimization method based on the two-point gradient iteration. The key idea behind IN-SETPG is that, unlike the standard Landweber method, it uses multiple search directions per iteration in combination with an adaptive step size in order to reduce the total number of iterations. The regularization property of IN-SETPG has been established, i.e., the iterate converges to a solution of the nonlinear problem with exact data when the noise level tends to zero. Various numerical experiments are presented to demonstrate that, compared with the original inexact Newton iteration, IN-SETPG can achieve better reconstruction results and remarkable acceleration.

Simplified REGINN-IT method in Banach spaces for nonlinear ill-posed operator equations

Abstract In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method.

Analysis of the discrepancy principle for Tikhonov regularization with oversmoothing penalty term under low order source conditions

We study the application of Tikhonov regularization to ill-posed nonlinear operator equations. The objective of this work is to prove convergence rates for the discrepancy principle under low order source conditions of logarithmic type. We work within the framework of Hilbert scales and extend existing studies on this subject to the oversmoothing case. The latter means that the exact solution of the treated operator equation does not belong to the domain of definition of the penalty term. As a consequence, the Tikhonov functional fails to have a finite value.

Modified version of a simplified Landweber iterative method for nonlinear ill-posed operator equations

Modified version of a simplified Landweber iterative method for nonlinear ill-posed operator equations

Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance

Abstract A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs.

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

Oversmoothing Tikhonov regularization in Banach spaces * *The work of DC was financially supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The work of BH was financially supported by German Research Foundation (DFG Grant HO 1454/12-1). The work of IY was financially supported by German Research Foundation (DFG Grants YO159/2-2 and YO159/4-1).

This paper develops a Tikhonov regularization theory for nonlinear ill-posed operator equations in Banach spaces. As the main challenge, we consider the so-called oversmoothing state in the sense that the Tikhonov penalization is not able to capture the true solution regularity and leads to the infinite penalty value in the solution. We establish a vast extension of the Hilbertian convergence theory through the use of invertible sectorial operators from the holomorphic functional calculus and the prominent theory of interpolation scales in Banach spaces. Applications of the proposed theory involving ℓ 1, Bessel potential spaces, and Besov spaces are discussed.

Simplified Levenberg–Marquardt method in Banach spaces for nonlinear ill-posed operator equations

In 2011, Jin Qinian has proposed a Levenberg–Marquardt method, by making use of duality mapping and the Bregman distance, to get an approximate solution of a nonlinear ill-posed operator equation in Banach space using an a posteriori parameter choice strategy and Morozov-type stopping rule. The method considered by Jin Qinian was an extension of the method proposed by M. Hanke in 1997 for the Hilbert space case. In this paper, we suggest a modified variant of the method, namely, the simplified Levenberg–Marquardt scheme in Banach spaces. The advantage of the method considered in the paper is that, it is also applicable for the operator equation with non-smooth operator. We establish convergence of the method under a modified a posteriori parameter choice strategy which is more feasible than the one considered by Jin Qinian (2011). Numerical example to demonstrate the validity of the considered method is discussed.

Error estimates for the iteratively regularized Newton–Landweber method in Banach spaces under approximate source conditions

Abstract In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

Optimal Convergence Rates for Tikhonov Regularization in Besov Spaces

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We show order optimal rates of convergence for finitely smoothing operators and for the backwards heat equation for a range of Besov spaces using variational source conditions. We also derive order optimal rates for a white noise model with the help of variational source conditions and concentration inequalities for sharp negative Besov norms of the noise.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

For Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Holder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence for regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low-order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

Optimal convergence rates for sparsity promoting wavelet-regularization in Besov spaces

This paper deals with Tikhonov regularization for linear and nonlinear ill-posed operator equations with wavelet Besov norm penalties. We focus on penalty terms which yield estimators that are sparse with respect to a wavelet frame. Our framework includes, among others, the Radon transform and some nonlinear inverse problems in differential equations with distributed measurements. Using variational source conditions it is shown that such estimators achieve minimax-optimal rates of convergence for finitely smoothing operators in certain Besov balls both for deterministic and for statistical noise models.

Numerical algorithm for morphogen synthesis region identification with indirect image-type measurement data.

Diffusion-reaction models are used to describe development processes in the framework of morphogen theory. The images of the concentration fields for the subset of the interacting morphogens are available. In order to interpret this data in terms of the model parameters, the inverse source problem is stated. The sensitivity operator, composed of the independent adjoint problem solutions ensemble, allows transforming the inverse problem to the family of nonlinear ill-posed operator equations. The equations are solved with the Newton-Kantorovich-type algorithm. The approach is applied to the morphogen synthesis region identification problem for the model of regulation of the renewing zone size in biological tissue.

An ensemble Kalman filter approach based on operator splitting for solving nonlinear Hammerstein type ill-posed operator equations

In this paper, we consider the application of the ensemble Kalman filter for some nonlinear operator equations. According to the structural feature of the nonlinear operator, we construct an iteration process and the corresponding linear observation operator. This construction puts our problem into the frame of 3DVar. Based on the linear observation operator, the ensemble Kalman filter approach is adopted to blend the data into the dynamics to obtain an approximation of the unknown parameter. The method is applied to an inverse potential problem and a nonlinear Fredholm integral equation. The numerical results are compared with Bayesian approach, classical regularization methods including Tikhonov regularization and Landweber iteration, which shows that the proposed algorithm is effective and competitive.

A regularization method based on level sets and augmented Lagrangian for parameter identification problems with piecewise constant solutions

We propose and analyse a regularization method for parameter identification problems modeled by ill-posed nonlinear operator equations, where the parameter to be identified is a piecewise constant function taking known values.Following (De Cezaro et al 2013 Inverse Problems 29 015003), a piecewise constant level set approach is used to represent the unknown parameter, and a corresponding Tikhonov functional is defined on an appropriated space of level set functions. Additionally, a suitable constraint is enforced, resulting that minimizers of our Tikhonov functional belong to the set of piecewise constant level set functions. In other words, the original parameter identification problem is rewritten in the form of a constrained optimization problem, which is solved using an augmented Lagrangian method.We prove the existence of zero duality gaps and the existence of generalized Lagrangian multipliers. Moreover, we extend the analysis in De Cezaro et al’s work (2013 Inverse Problems 29 015003), proving convergence and stability of the proposed parameter identification method.A primal-dual algorithm is proposed to compute approximate solutions of the original inverse problem, and its convergence is proved. Numerical examples are presented: this algorithm is applied to a 2D diffuse optical tomography problem. The numerical results are compared with the ones in Agnelli et al (2017 ESAIM: COCV 23 663–83) demonstrating the effectiveness of this primal-dual algorithm.

Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

Abstract 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.

An optimal order a posteriori parameter choice strategy with modified Newton iterative scheme for solving nonlinear ill-posed operator equations

Study of inverse problems are interesting and mathematically challenging due to the fact that in most of the situation they are unstable with respect to perturbations of the data. In this paper to solve such operator equations, we propose a modified form of Gauss-Newton method combined with an a posteriori parameter choice strategy with the inexact data. Convergence and the convergence rate results are proven. We consider both a-priori and a-posteriori choice rule of parameter that guarantees the scheme converges to the exact solution. The theoretical results are illustrated through numerical examples and compared with the standard scheme to demonstrate that the scheme is stable and achieves good computational output. The salient features of our proposed scheme are: 1) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature; 2) consideration of an adaptive and numerically stable a posteriori parameter strategy that gives the same order of convergence as that of an a priori method; 3) computation of an optimal order regularisation parameter of the order O(δ2/3) using a discrepancy principle.

Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales

We study Tikhonov regularization for ill-posed non-linear operator equations in Hilbert scales. Our focus is on the interplay between the smoothness-promoting properties of the penalty and the smoothness inherent in the solution. The objective is to study the situation when the unknown solution fails to have a finite penalty value, hence when the penalty is oversmoothing. By now this case was only studied for linear operator equations in Hilbert scales. We extend those results to certain classes of non-linear problems. The main result asserts that, under appropriate assumptions, order optimal reconstruction is still possible. In an appendix we highlight that the non-linearity assumption underlying the present analysis is met for specific applications.