335 research outputs found
Logarithmic stability estimates for initial data in Ornstein-Uhlenbeck equation on -space
In this paper, we continue the investigation on the connection between
observability and inverse problems for a class of parabolic equations with
unbounded first order coefficients. We prove new logarithmic stability
estimates for a class of initial data in the Ornstein-Uhlenbeck equation posed
on with respect to the Lebesgue measure. The
proofs combine observability and logarithmic convexity results that include a
non-analytic semigroup case. This completes the picture of the recent results
obtained for the analytic Ornstein-Uhlenbeck semigroup on -space with
invariant measure
Numerical impulse controllability for parabolic equations by a penalized HUM approach
This work presents a comparative study to numerically compute impulse
approximate controls for parabolic equations with various boundary conditions.
Theoretical controllability results have been recently investigated using a
logarithmic convexity estimate at a single time based on a Carleman commutator
approach. We propose a numerical algorithm for computing the impulse controls
with minimal -norms by adapting a penalized Hilbert Uniqueness Method
(HUM) combined with a Conjugate Gradient (CG) method. We consider static
boundary conditions (Dirichlet and Neumann) and dynamic boundary conditions.
Some numerical experiments based on our developed algorithm are given to
validate and compare the theoretical impulse controllability results
Stable determination of coefficients in semilinear parabolic system with dynamic boundary conditions
In this work, we study the stable determination of four space-dependent
coefficients appearing in a coupled semilinear parabolic system with variable
diffusion matrices subject to dynamic boundary conditions which couple
intern-boundary phenomena. We prove a Lipschitz stability result for interior
and boundary potentials by means of only one observation component, localized
in any arbitrary open subset of the physical domain. The proof mainly relies on
some new Carleman estimates for dynamic boundary conditions of surface
diffusion type
Numerical identification of initial temperatures in heat equation with dynamic boundary conditions
We investigate the inverse problem of numerically identifying unknown initial
temperatures in a heat equation with dynamic boundary conditions whenever some
overdetermination data is provided after a final time. This is a backward
parabolic problem which is severely ill-posed. As a first step, the problem is
reformulated as an optimization problem with an associated cost functional.
Using the weak solution approach, an explicit formula for the Fr\'echet
gradient of the cost functional is derived from the corresponding sensitivity
and adjoint problems. Then the Lipschitz continuity of the gradient is proved.
Next, further spectral properties of the input-output operator are established.
Finally, the numerical results for noisy measured data are performed using the
regularization framework and the conjugate gradient method. We consider both
one- and two-dimensional numerical experiments using finite difference
discretization to illustrate the efficiency of the designed algorithm. Aside
from dealing with a time derivative on the boundary, the presence of a boundary
diffusion makes the analysis more complicated. This issue is handled in the 2-D
case by considering the polar coordinate system. The presented method implies
fast numerical results
- …