229 research outputs found
On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation
This paper is devoted to the analysis of some uniqueness properties of a
classical reaction-diffusion equation of Fisher-KPP type, coming from
population dynamics in heterogeneous environments. We work in a one-dimensional
interval and we assume a nonlinear term of the form where belongs to a fixed subset of . We
prove that the knowledge of at and of , at a single point
and for small times is sufficient to completely
determine the couple provided is known.
Additionally, if is also measured for ,
the triplet is also completely determined. Those
analytical results are completed with numerical simulations which show that, in
practice, measurements of and at a single point (and for ) are sufficient to obtain a good approximation of the
coefficient These numerical simulations also show that the
measurement of the derivative is essential in order to accurately
determine
Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations
In this work, we consider a one-dimensional It{\^o} diffusion process X t
with possibly nonlinear drift and diffusion coefficients. We show that, when
the diffusion coefficient is known, the drift coefficient is uniquely
determined by an observation of the expectation of the process during a small
time interval, and starting from values X 0 in a given subset of R. With the
same type of observation, and given the drift coefficient, we also show that
the diffusion coefficient is uniquely determined. When both coefficients are
unknown, we show that they are simultaneously uniquely determined by the
observation of the expectation and variance of the process, during a small time
interval, and starting again from values X 0 in a given subset of R. To derive
these results, we apply the Feynman-Kac theorem which leads to a linear
parabolic equation with unknown coefficients in front of the first and second
order terms. We then solve the corresponding inverse problem with PDE technics
which are mainly based on the strong parabolic maximum principle
Stability estimate in an inverse problem for non-autonomous Schr\"odinger equations
We consider the inverse problem of determining the time dependent magnetic
field of the Schr\"odinger equation in a bounded open subset of , with , from a finite number of Neumann data, when the boundary measurement is
taken on an appropriate open subset of the boundary. We prove the Lispchitz
stability of the magnetic potential in the Coulomb gauge class by times
changing initial value suitably
Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations
We consider the inverse problem of the simultaneous reconstruction of the
dielectric permittivity and magnetic permeability functions of the Maxwell's
system in 3D with limited boundary observations of the electric field. The
theoretical stability for the problem is provided by the Carleman estimates.
For the numerical computations the problem is formulated as an optimization
problem and hybrid finite element/difference method is used to solve the
parameter identification problem.Comment: in Inverse Problems and Imaging Volume: 9, Number: 1 February 2015.
arXiv admin note: text overlap with arXiv:1510.0752
Uniqueness and stability of time and space-dependent conductivity in a hyperbolic cylindrical domain
This paper is devoted to the reconstruction of the time and space-dependent
coefficient in an infinite cylindrical hyperbolic domain. Using a local
Carleman estimate we prove the uniqueness and a H\"older stability in the
determining of the conductivity by a single measurement on the lateral
boundary. Our numerical examples show good reconstruction of the location and
contrast of the conductivity function in three dimensions.Comment: arXiv admin note: text overlap with arXiv:1501.0138
Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary
We consider the multidimensional inverse problem of determining the
conductivity coefficient of a hyperbolic equation in an infinite cylindrical
domain, from a single boundary observation of the solution. We prove H{\"o}lder
stability with the aid of a Carleman estimate specifically designed for
hyperbolic waveguides
Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations
We consider an inverse problem of reconstructing two spatially varying
coefficients in an acoustic equation of hyperbolic type using interior data of
solutions with suitable choices of initial condition. Using a Carleman
estimate, we prove Lipschitz stability estimates which ensures unique
reconstruction of both coefficients. Our theoretical results are justified by
numerical studies on the reconstruction of two unknown coefficients using noisy
backscattered data
Inverse problem for a parabolic system with two components by measurements of one component
We consider a system of parabolic equations with first and zeroth
coupling and establish a Carleman estimate by extra data of only one component
without data of initial values. Then we apply the Carleman estimate to inverse
problems of determining some or all of the coefficients by observations in an
arbitrary subdomain over a time interval of only one component and data of two
components at a fixed positive time over the whole spatial domain. The
main results are Lipschitz stability estimates for the inverse problems. For
the Lipschitz stability, we have to assume some non-degeneracy condition at
for the two components and for it, we can approximately control the
two components of the system by inputs to only one component. Such
approximate controllability is proved also by our new Carleman estimate.
Finally we establish a Carleman estimate for a system for parabolic
equations with coupling of zeroth-order terms by one component to show the
corresponding approximate controllability with a control to one component
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