By making use of tools from convex analysis, we formulate an inexact NewtonLandweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme. The inner scheme provides increments by applying Landweber iteration with non-smooth uniformly convex penalty terms to local linearized equations. The o...

The aim of this section is to define some generalization of the notion of formal group. More precisely, we consider the analog of formal groups with coefficients belonging to a Hopf algebra. We also study some example of a formal group over a Hopf algebra, which generalizes the formal group of geometric cobordisms. Recently some important connections between the Landweber-Novikov algebra and th...

This article presents a numerical method to find the heat flux in an inhomogeneous inverse heat conduction problem with linear boundary conditions and an extra specification at the terminal. The method is based upon applying the satisfier function along with the Ritz-Galerkin technique to reduce the approximate solution of the inverse problem to the solution of a system of algebraic equations. ...

Many iterative methods that are used to solve Ax = b can be derived as quasi-Newton methods for minimizing the quadratic function 1 2 xAAx−xAb. In this paper, several such methods are considered, including conjugate gradient least squares (CGLS), Barzilai-Borwein (BB), residual norm steepest descent (RNSD) and Landweber (LW). Regularization properties of these methods are studied by analyzing t...

In this paper, we investigate an inverse problem to determine an unknown source term that has a separable-variable form in the time-fractional diffusion equation, whereby the data is obtained at a certain time. This problem is ill-posed, and we use the Landweber iterative regularization method to solve this inverse source problem. Two kinds of convergence rates are obtained by using an a priori...

We describe a method to speed up microelectromechanical system (MEMS) mirror scanning by > 20x, while also improving scan accuracy. We use Landweber deconvolution to determine an input voltage which would produce a desired output, based on the measured MEMS impulse response. Since the MEMS is weakly nonlinear, the observed behavior deviates from expectations, and we iteratively improve our inpu...

During the past the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators...

The set of all words in the alphabet {l, r) forms the full binary tree T. If x E T then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of Tand over arbitrary subsets of T. We prove that there is no monadic second-order form...