Newton-Kantorovich iterative regularization and generalized discrepancy principle for nonlinear ill-posed equations involving accretive mappings

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.

Modified Noor iterations for nonlinear equations in Banach spaces

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.

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.

We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with $\|y-y^\d \|\leq \d $ resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

Recurrence relations for a Newton-like method in Banach spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = exp n +1τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .

Let X X be a real Banach space with X ∗ {X^\ast } uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E) \[ u ′ ( t ) + A ( t ) u ( t ) = G ( t , u t ) , u 0 = ϕ , t ∈ [ 0 , T ] . u\prime (t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T]. \] The operators A ( t ) : D ⊂ X → X A(t):D \subset X \to X are m m -accretive and G ( t , ϕ ) G(t,\phi ) is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators A ^ ( t ) \hat A(t) and the corresponding use of nonlinear evolution operator theory.

Multiple incorrect classifications of vertical small ISO80 QC seed items located under the center of the MM2×2 sensor have occurred during cued AGC operations for FUDS MMRP remedial actions. For these failures the sensor QC tests and background data meet quality requirements and did not cause the misclassifications. This study examined the MM2×2 raw data and inversion results to determine why the QC seed items were not correctly classified as TOI. These classification failures are rare, but have several consistent characteristics, 1) the type of item, 2) the orientation of the item, and 3) the location of the item under the center of the MM2×2 sensor. This presentation describes experimental and modeling efforts that were performed to determine the cause of the misclassification and the extent of the problem. Namely, the primary and secondary EM signals are modeled and illustrated for the MM2×2 for this problematic configuration (vertical smallISO80 located 40cm below the center of the sensor). The modeling confirmed the expected symmetry of horizontal components of the primary magnetic field directly below the center of the sensors array and the relatively weak vertical component at that location. For this configuration, the target produces: a) the same vertical component in all four z-receivers, 2) equal and opposite sign horizontal components in x and y receivers, resulting the three independent equations and underdetermined (nine unknowns and three equations) ill-posed non-linear equations, see Figure 1. Three variables, item type; orientation; and item location were modified to evaluate how deviations from the perfectly symmetric configuration impact the misclassification problem. The results of this modeling and its relevance to classification with the MM2×2 on MMRP projects will be discussed in this presentation.

Let X be a real Banach space with a uniformly convex dual, X*, and let C be a nonempty closed convex and bounded subset of X. Let T: C → C be a strongly accretive and a continuous mapping. For any f ∈ C, let S: C → C be defined by Sx = f + x – Tx for each x ∈ C. Then, the iteration process xo ∈ C,under suitable conditions on the real sequence converges strongly to a solution of the equation Tx = f in C. Furthermore, if T is strongly accretive and Lipschitz with Lipschitz constant L ≥ 1 then the iteration process x0 ∈ C,under suitable conditions on the real sequences and converges strongly to a solution of the equation Tx = f in C. Explicit error estimates are obtained.

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.

Analytic Approximations of Uniformly Continuous Functions in Real Banach Spaces

This chapter provides an extended analysis of the Lavrentiev regularization for nonlinear ill-posed problems. It presents convergence results and error bounds for the nonlinear ill-posed problems. The chapter discusses the implementation of adaptive choice rule by providing a starting point for the iteration approximating an unique solution and a parameter algorithm fulfilling a balancing principle for the problems. It also presents numerical examples for the nonlinear ill-posed problems.

This paper introduces a hybrid spectral-conjugate gradient (SCG) method to solve nonlinear monotone operator equations efficiently. The proposed method incorporates a hybrid parameter that encom- passes the Polak–Ribi`ere–Polyak (PRP), Liu-Storey (LS), Fletcher-Reeves (FR), and conjugate descent (CD) meth- ods as particular instances. Additionally, we derive the spectral parameter to ensure that the search direction adheres to the sufficient descent condition. The search direction is also designed to be bounded, and under spe- cific conditions, we demonstrate that the sequence produced by our hybrid SCG algorithm converges toward a solution. Furthermore, to underscore the effectiveness of our proposed method, we conducted extensive numer- ical experiments comparing its performance against that of existing algorithms. These experiments were based on a selection of benchmark nonlinear monotone operator equations, highlighting our proposed algorithm’s superior efficiency and potential in practical applications.

A new class of generalized set-valued implicit variational inclusions in banach spaces with an application