Ill-posed Operator 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.

Tractability of linear ill-posed problems in Hilbert space

We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.

Regularization with two differential operators and its application to inverse problems

We introduce a regularization method with two differential operators for solving a linear ill-posed operator equation system. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence and convergence rates of the regularized solution are also obtained. As an application, we apply the regularization to a simultaneous inversion of the source term and the initial value problem for a heat conduction equation, numerical experiments are presented to demonstrate the effectiveness of the proposed method.

On inertial iterated Tikhonov methods for solving ill-posed problems

In this manuscript we propose and analyze an implicit two-point type method (or inertial method) for obtaining stable approximate solutions to linear ill-posed operator equations. The method is based on the iterated Tikhonov (iT) scheme. We establish convergence for exact data, and stability and semi-convergence for noisy data. Regarding numerical experiments we consider: (i) a 2D inverse potential problem, (ii) an image deblurring problem; the computational efficiency of the method is compared with standard implementations of the iT method.

Optimal regularized hypothesis testing in statistical inverse problems

Testing of hypotheses is a well studied topic in mathematical statistics. Recently, this issue has also been addressed in the context of inverse problems, where the quantity of interest is not directly accessible but only after the inversion of a (potentially) ill-posed operator. In this study, we propose a regularized approach to hypothesis testing in inverse problems in the sense that the underlying estimators (or test statistics) are allowed to be biased. Under mild source-condition type assumptions, we derive a family of tests with prescribed level α and subsequently analyze how to choose the test with maximal power out of this family. As one major result we prove that regularized testing is always at least as good as (classical) unregularized testing. Furthermore, using tools from convex optimization, we provide an adaptive test by maximizing the power functional, which then outperforms previous unregularized tests in numerical simulations by several orders of magnitude.

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.

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.

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.

An Implicit Iteration Method for Solving Linear Ill-Posed Operator Equations

An Implicit Iteration Method for Solving Linear Ill-Posed Operator Equations

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh, Appl. Math. Comput. 392, 125701, 2021] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

A modified iterative Lavrentiev method for nonlinear monotone ill-posed operators

A modified iterative Lavrentiev method for nonlinear monotone ill-posed operators

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

Error estimates for Golub–Kahan bidiagonalization with Tikhonov regularization for ill–posed operator equations

Linear ill-posed operator equations arise in various areas of science and engineering. The presence of errors in the operator and the data often makes the computation of an accurate approximate solution difficult. In this paper, we compute an approximate solution of an ill-posed operator equation by first determining an approximation of the operators of generally fairly small dimension by carrying out a few steps of a continuous version of the Golub–Kahan bidiagonalization process to the noisy operator. Then Tikhonov regularization is applied to the low-dimensional problem so obtained and the regularization parameter is determined by solving a low-dimensional nonlinear equation. The effect of the errors incurred in each step of the solution process is analyzed. Computed examples illustrate the theory presented.

Simplified Levenberg–Marquardt Method in Hilbert Spaces

Abstract In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ ( x ) = y {F(x)=y} , where F : D ⁢ ( F ) ⊂ X → Y {F:D(F)\subset X\rightarrow Y} is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } {\{x_{n}^{\delta}\}} is defined as x n + 1 δ = x n δ - ( α n ⁢ I + F ′ ⁢ ( x 0 ) * ⁢ F ′ ⁢ ( x 0 ) ) - 1 ⁢ F ′ ⁢ ( x 0 ) * ⁢ ( F ⁢ ( x n δ ) - y δ ) . x^{\delta}_{n+1}=x^{\delta}_{n}-(\alpha_{n}I+F^{\prime}(x_{0})^{*}F^{\prime}(x% _{0}))^{-1}F^{\prime}(x_{0})^{*}(F(x^{\delta}_{n})-y^{\delta}). Here { α n } {\{\alpha_{n}\}} is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ ( x 0 ) {F^{\prime}(x_{0})} denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ ( F ) {x_{0}\in D(F)} for the exact solution x † {x^{\dagger}} and ( F ′ ⁢ ( x 0 ) ) * {(F^{\prime}(x_{0}))^{*}} denote the adjoint of F ′ ⁢ ( x 0 ) {F^{\prime}(x_{0})} . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 {x_{0}} . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method.

Alternating step size method for solving ill-posed linear operator equations in energetic space

An alternating step size method for solving ill-posed linear operator equations is introduced in energetic space, a subspace of the Hilbert space. Focus is on the case of a positive bounded self-conjugated operator under the assumption that the error for the right hand side of the equation is available. The advantage of the energetic norm is that does not require smoothness of the exact solution x. Convergence of the new method is discussed and shown to be faster for similar error estimates. Furthermore, the conditions for convergence in the energetic norm are derived which also imply convergence in the original Hilbert space norm. The new method is demonstrated on several numerical examples of the integral equation in L20,1 space, taken from the class of inverse problems in potential theory, and convergence speed compared with that of the fixed step size method. Finally, a novel real-word application of the integral equation is proposed: deblurring averages of trial-to-trial jittered EEG responses. When applying the Kosambi–Hilbert Torsion (KTH) to align these responses, the jitter is more effectively reduced and alignment improves with the deblurred average as a reference than with the original average.

Finite dimensional realization of fractional Tikhonov regularization method in Hilbert scales

One of the intuitive restrictions of infinite dimensional Fractional Tikhonov Regularization Method (FTRM) for ill-posed operator equations is its numerical realization. This paper addresses the issue to a considerable extent by using its finite dimensional realization in the setting of Hilbert scales. Using adaptive parameter choice strategy, we choose the regularization parameter and obtain an optimal order error estimate. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales.

Solving equations with semimartingale noise

Abstract In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.