. 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: firstname.lastname@example.org, email@example.com
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)
(K ∗ u)(t) =
K(t − s)u(s) ds
examples are presented to illustrate the validity and effectiveness of the
Keywords. Inverse problem, diffusion process, fractional derivative, regularization by
2010 Mathematics Subject Classification. 65J20, 65M32.
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.
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
Received June 25, 2015; revised October 8, 2015; accepted November 6, 2015
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
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
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-
MSC 2010: 65C05, 65M32, 65R32
Received March 3, 2014; revised August 31, 2015; accepted August 31, 2015
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
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
MSC 2010: 35R30, 65M32, 86A22
Received May 21, 2015; accepted July 3, 2015
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
Keywords: Inverse problems, ill-posed problems, initial condition, boundary observations, finite difference
splitting method, variational method, singular values
MSC 2010: 65M32, 65N20, 65J20
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.