On the local convergence of inexact Newton-type methods under residual control-type conditions

AbstractA local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study

Unified Ball Convergence of Inexact Methods For Finding Zeros with Multiplicity

We present an extended ball convergence of inexact methods for approximating a zero of a nonlinear equation with multiplicity m; where m is a natural number. Many popular methods are special cases of the inexact method

Second order semi-smooth Proximal Newton methods in Hilbert spaces

We develop a globalized Proximal Newton method for composite and possibly non-convex minimization problems in Hilbert spaces. Additionally, we impose less restrictive assumptions on the composite objective functional considering differentiability and convexity than in existing theory. As far as differentiability of the smooth part of the objective function is concerned, we introduce the notion of second order semi-smoothness and discuss why it constitutes an adequate framework for our Proximal Newton method. However, both global convergence as well as local acceleration still pertain to hold in our scenario. Eventually, the convergence properties of our algorithm are displayed by solving a toy model problem in function space.Comment: 31 pages, 4 figure

Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems

We present globally convergent nonsmooth Schur–Newton methods for the solution of discrete multicomponent Cahn–Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments

A novel two-point gradient method for Regularization of inverse problems in Banach spaces

In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, $L^1$ functions etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practical applications involving the reconstruction of several important features of solutions such as piecewise constancy and sparsity. We carefully discuss the choices for important parameters, such as combination parameters and step sizes involved in the design of the method. Additionally, we discuss an example to validate our assumptions.Comment: Submitted in Applicable Analysi

Convergence rates for variational regularization of inverse problems in exponential families

We consider statistical inverse problems with statistical noise. By using regularization methods one can approximate the true solution of the inverse problem by a regularized solution. The previous investigation of convergence rates for variational regularization with Poisson and empirical process data is shown to be suboptimal. In this thesis we obtain improved convergence rates for variational regularization methods of nonlinear ill-posed inverse problems with certain stochastic noise models described by exponential families and derive better reconstruction error bounds by applying deviation inequalities for stochastic process in some function spaces. Furthermore, we also consider iteratively regularized Newton-method as an alternative while the operator is non-linear. Due to the difficulty of deriving suitable deviation inequalities for stochastic processes in some function spaces, we are currently not able to obtain optimal convergence rates for variational regularization such that we state our desired result as a conjecture. If our conjecture holds true, then we can immediately obtain our desired results

On the iterative regularization of non-linear illposed problems in L∞L^{\infty }

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in L∞. This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular Lp-spaces with 1<p<∞. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smoothed Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded in L∞ and the regularizer, delivered by a discrepancy principle, converges weakly-⋆ to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis

Novel Inverse-Scattering Methods in Banach Spaces

The scientific community is presently strongly interested in the research of new microwave imaging methods, in order to develop reliable, safe, portable, and cost-effective tools for the non-invasive/non-destructive diagnostic in many fields (such as medicine, civil and industrial engineering, \u2026). In this framework, microwave imaging techniques addressing the full three-dimensional nature of the inspected bodies are still very challenging, since they need to cope with significant computational complexity. Moreover, non-linearity and ill-posedness issues, which usually affects the related inverse scattering problems, need to be faced, too. Another promising topic is the development of phaseless methods, in which only the amplitude of the electric field is assumed to be measurable. This leads to a significant complexity reduction and lower cost for the experimental apparatuses, but the missing information on the phase of the electric field samples exacerbates the ill-posedness problems. In the present Thesis, a novel inexact-Newton inversion algorithm is proposed, in which the iteratively linearized problems are solved in a regularized sense by using a truncated Landweber or a conjugate gradient method developed in the framework of the l^p Banach spaces. This is an improvement that allows to generalize the classic framework of the l^2 Hilbert spaces in which the inexact-Newton approaches are usually defined. The applicability of the proposed imaging method in both the 3D full-vector and 2D phaseless scenarios at microwave frequencies is assessed in this Thesis, and an extensive validation of the proposed imaging method against both synthetic and experimental data is presented, highlighting the advantages over the inexact-Newton scheme developed in the classic framework of the l^2 Hilbert spaces

Optimal sensor placement: A robust approach

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests