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. Convergence of approximations towards the exact solution is established for a linear equation. Keywords: Inverse source problem, damped wave equation MSC 2010: 65M32 || Lukáš Šeliga, Marián Slodička: Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Ghent, Belgium, e-mail: lukas.seliga@ugent.be, marian.slodicka@ugent.be 1 Introduction LetΩ be a bounded domain ofℝn, where n ≥ 1, with Lipschitz boundary Γ.We consider the following damped wave equation for u: utt + g(ut) − ∇ ⋅ (a∇u) + cu + K ∗ u = f + F, (1.1) where (K ∗ u)(t) = t ∫ 0 K(t − s)u(s) ds and

examples are presented to illustrate the validity and effectiveness of the proposed method. Keywords. Inverse problem, diffusion process, fractional derivative, regularization by projection. 2010 Mathematics Subject Classification. 65J20, 65M32. 1 Introduction In many industrial applications related with diffusion processes one usually wishes to determine the previous status from its present information, in particular when the initial conditions are to be found. Such a backward problem has been well ad- dressed in the classic heat equation, for instance [1, 3, 16]. This

, the source term being localized on the blood vessels. We consider the inverse problem that consists in re- covering the position of the blood vessels assuming the distribution of tumor cells. We use an adjoint method. Results relative to idealized clinical cases are discussed. Keywords. Tumor growth, vascularization, inverse problem, proliferative cells. 2010 Mathematics Subject Classification. 92C50, 65M32, 35R30. 1 Introduction Mathematical modeling of tumor growth has numerous potential applications: nu- merical simulations could be useful to study how a

regularization method. Numerical examples with noisy data are given to illustrate the eectiveness of this method. Keywords: Inverse Scattering, Born Approximation, Schrödinger Equation, Numerical Solution MSC 2010: 35P25, 35R30, 65M32 DOI: 10.1515/cmam-2015-0032 Received June 25, 2015; revised October 8, 2015; accepted November 6, 2015 1 Introduction We consider the nonlinear two-dimensional Schrödinger equation − ∆u(x) + h(x, |u(x)|)u(x) = k2u(x), x ∈ ℝ2, (1) where k > 0 is the wavenumber and h(x, s) is a rather general nonlinear function of s ∈ ℝ. This equation models the

volume method and the finite elementmethod,whose accuracies are verified against analytical solutions.Minimization of the sum of normalized least-squares differences between the calculated and measured values of the field quan- tity at the boundaries then leads to the correct parameters in the analytic model for the spatial distribution of the spatially varying material property. Keywords: Inverse problems, diffusion coefficient, parameter identification, functionally graded materials MSC 2010: 31A25, 65M32, 80A23 1 Introduction This paper deals with a general

–Levitan equation in one specic point, due to the properties of the method. That allows the Monte Carlo method to be more eective in terms of span cost, compared with regular methods of solving linear system. Results of numerical simulations are presented. Keywords: Wave equation, inverse problems, Gelfand–Levitan equations, Monte Carlo methods, regulariza- tion methods MSC 2010: 65C05, 65M32, 65R32 DOI: 10.1515/jiip-2014-0018 Received March 3, 2014; revised August 31, 2015; accepted August 31, 2015 1 Introduction This paper is focused on the numerical solution of

reduce the non-linear inverse problem to a family of linear integral equations. We consider the fast algorithm for solving the relevant linear system, based on using the block-Toeplitz structure of the matrix. The algorithm applied to the M. G. Krein equation allows to obtain the solution of the whole family of the integral equations by solving only one linear system. Results of numerical calculations are presented. Keywords: Inverse problems, Gelfand–Levitan method, M. G. Krein equation, fast Toeplitz algorithm MSC 2010: 65M32, 65R20, 65F05 DOI: 10.1515/jiip-2015

the subdomain covered by acoustic rays, which are emanated from the points of this part orthogonally to the boundary. The determination is time-optimal: the longer the observation time is, the larger the subdomain is, in which c is recovered. The numerical results are preceded with a brief exposition of the relevant variant of the BC-method. Keywords:Acoustic equation, time-domain inverse problem, determination frompart of boundary, boundary control method MSC 2010: 35R30, 65M32, 86A22 DOI: 10.1515/jiip-2015-0052 Received May 21, 2015; accepted July 3, 2015 1

variational problem is discretized by finite difference splitting methods and solved by the conjugate gradient method. Some numerical examples are presented to show the efficiency of the method. Also as a by-product of the variational method, we propose a numerical scheme for numerically estimating singular values of the solution operator in the inverse problem. Keywords: Inverse problems, ill-posed problems, initial condition, boundary observations, finite difference splitting method, variational method, singular values MSC 2010: 65M32, 65N20, 65J20 || Dedicated to

## Abstract

We concerned with the asymptotic analysis for the dynamical probe method which is a reconstruction scheme to identify an anomaly inside a heat conductor from the Neumann-to-Dirichlet map. In this paper an inclusion was considered as an anomaly and we succeeded giving an improved asymptotic behavior of the indicator function defined in terms of the Neumann-to-Dirichlet map to identify not only the location of the inclusion but also some of its physical properties simultaneously. The two major improvements made for analyzing the asymptotic behavior of the indicator function are as follows. Firstly, we can know the distance to the boundary of unknown inclusion as we probe it from its outside. This improvement can avoid overshooting the boundary points as much as possible if we probe it from outside the inclusion numerically. Secondly, we can know the value of heat conductivity of inclusion as we probe close to the inclusion even without touching it.