Journal of Inverse and Ill-posed Problems 

About: Journal of Inverse and Ill-posed Problems is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Inverse problem & Inverse scattering problem. It has an ISSN identifier of 0928-0219. Over the lifetime, 1330 publications have been published receiving 13712 citations. The journal is also known as: Journal of inverse & ill-posed problems (Internet).

Recovering a potential from Cauchy data in the two-dimensional case

Abstract: In this paper we prove that the Cauchy data for the Schrödinger equation in the two-dimensional case determines a potential from Lp (for p > 2) uniquely. We also obtain a linear inversion formula for smooth potentials.

Definitions and examples of inverse and ill-posed problems

Abstract: Abstract The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.

Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems

Abstract: This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.

Reconstruction of the support function for inclusion from boundary measurements

Abstract: Abstract - First we give a formula (procedure) for the reconstruction of the support function for unknown inclusion by means of the Dirichlet to Neumann map. In the procedure we never make use of the unique continuation property or the Runge approximation property of the governing equation. Second we apply the method to a similar problem for the Helmholtz equation.

Convergence rates and source conditions for Tikhonov regularization with sparsity constraints

Abstract: This paper addresses the regularization by sparsity constraints by means of weighted $\ell^p$ penalties for $0\leq p\leq 2$. For $1\leq p\leq 2$ special attention is payed to convergence rates in norm and to source conditions. As main result it is proven that one gets a convergence rate in norm of $\sqrt{\delta}$ for $1\leq p\leq 2$ as soon as the unknown solution is sparse. The case $p=1$ needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed. For $p<1$ only preliminary results are shown. These results indicate that, different from $p\geq 1$, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for $p=0$ shows that regularization need not to happen.