Volume 23, Issue 3

A problem of solving a (non)linear operator equation, F ( u ) = y , F : 𝒳 → 𝒴, on a pair of Hilbert spaces is addressed. A generalized Gauss–Newton scheme u n +1 = ξ n - θ( F '* ( u n ) F '( u n ),τ n ) F '* ( u n ){ F ( u n ) - y δ - F '( u n )( u n - ξ n )}, u 0 ,ξ n ∈ 𝒳, is investigated. Three basic groups of generating functions are considered. A nonstandard approximation of pseudoinverse through a “gentle” iterative truncation is presented. Its optimality on the class of generating functions with the same correctness coefficient is proven. A novel a posteriori stopping rule, designed to accommodate noise in both the data and the source condition, is justified. Practical aspects are illustrated with numerical simulations for a large-scale linear image de-blurring system as well as a nonlinear inverse scattering model.

We use a positive parameter to develop a differentiable perturbed reconstruction model to solve the L1–L2 magnetic resonance image (MRI) reconstruction problem. We use Bregman iterative formulation to solve the differentiable perturbed L1–L2 model, and lagged diffusivity fixed-point iteration to solve the minimization problem in the Bregman iteration. Two Fourier transforms and an inverse Fourier transform are used to accelerate L1–L2 MRI reconstruction. Real MR images are used to test the method in numerical experiments. The results demonstrate that the proposed method is very efficient for L1–L2 MRI reconstruction.

In this paper, we study the propagation of elastic waves in porous media governed by the Biot equations in the low frequency range. We prove the existence and uniqueness result both for the direct problem and the inverse one, which consists in identifying the unknown scalar function f ( t ) in the body density force f ( t )𝐠(𝐱, t ) acting on a poroelastic body when some additional measurement is available.

Solving an autoconvolution equation is a nonlinear ill-posed inverse problem. Besides standard methods for general nonlinear problems several customized methods for deautoconvolution are available. Recently, a new decomposition approach for solving ill-posed quadratic equations, e.g. autoconvolutions, has been proposed. In this article we compare the new approach to the TIGRA method of Ramlau and to the local regularization method of Dai and Lamm. Numerical tests show that the new method yields better approximations to the unknown true solution than existing methods in comparable computation time.

The aim of this paper is to discuss and illustrate the fact that conditionally well-posed problems stand out among all ill-posed problems as being regularizable via an operator independent of the level of errors in input data. We give examples of corresponding purely data driven regularizing algorithms for various classes of conditionally well-posed inverse problems and optimization problems in the context of deterministic and stochastic error models.

We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wulff problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semidefinite relaxation method which is based on a solution to a convex semidefinite problem obtained by a semidefinite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of finite-dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wulff problem. Several computational examples, including those corresponding to boundaries of real snowflakes, and discussion on the rate of convergence of numerical method are also presented in this paper.

The present paper is devoted to the problem of recovering smooth functions from their Fourier coefficients. We consider the functions of two variables. Herewith this problem is ill-posed. The problem is studied in the case of given smoothness of the functions as well as when a parameter of smoothness is supposed to be unknown.