Simplified Generalized Gauss-Newton Iterative Method under Morozove Type Stopping Rule

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.

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.

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.

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.

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.

- In this paper a convergence proof is given for the continuous analog of the Gauss-Newton method for nonlinear ill-posed operator equations and convergence rates are obtained. Convergence for exact data is proved for nonmonotone operators under weaker source conditions than before. Moreover, nonlinear ill-posed problems with noisy data are considered and a priori and a posteriori stopping rules are proposed. These rules yield convergence of the regularized approximations to the exact solution as the noise level tends to zero. The convergence rates are optimal rates under the source conditions considered.

Iteratively regularized Gauss–Newton type methods for approximating quasi–solutions of irregular nonlinear operator equations in Hilbert space with an application to COVID–19 epidemic dynamics

Recurrence relations for a Newton-like method in Banach spaces

In this paper, in order to solve nonlinear ill-posed operator equations involving an m-accretive mapping on a real Banach space, that does not admit a weak sequential continuous duality mapping, we prove a strongly convergent theorem for Newton-Kantorovich iterative regularization method with a posteriori stopping rule. In our results, the Lipschitz continuity of the derivatives for the mapping is not necessary.

Modified Noor iterations for nonlinear equations in Banach spaces

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.

Convergence of the two-step combined method and uniqueness of the solution of nonlinear operator equations

Let T(w,x) = y( w ) be a nonlinear random operator equation with not necessarily unique solution. For this and similar equations, we prove results about convergence of solutions of suitable approximate problems Tn(w,x) =Yn(w) to solutions of the original equations. We do this for rather general notions of convergence for random variables. Concepts like consistency, stability, and compactness in sets of measurable functions are introduced and used. For all assumptions that are needed in the general theory, sufficient conditions are given with respect to convergence in probability and almost-sure convergence. As a specific method for constructing approximate equations we discuss "discretization schemes ", where the underlying probability space is discretized. Some results might be of interest also in different contexts; these include criteria for almost-sure convergence of measurable multifunctions and results about compactness with respect to convergence in probability and almost-sure convergence.

In this paper, we establish a novel reduced-order weak Galerkin (ROWG) finite element method for solving parabolic equation with nonlinear compression coefficient. We first present the classical weak Galerkin finite element discretization scheme and derive the optimal error estimates. Then we apply a proper orthogonal decomposition (POD) technique to develop the ROWG method, which can effectively reduce degrees of freedom and CPU time. The optimal order error estimates are also derived, and the algorithm flow is provided. Finally, some numerical experiments illustrate the performance of the ROWG method. The numerical results show that the proposed ROWG method is efficient for solving nonlinear parabolic equations.

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Duran (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.