20 research outputs found
On the singularities of fractional differential systems, using a mathematical limiting process based on physical grounds
Fractional systems are associated with irrational transfer functions for which nonunique analytic continuations are available (from some right-half Laplace plane to a maximal domain). They involve continuous sets of singularities, namely cuts, which link fixed branching points with an arbitrary path. In this paper, an academic example of the 1D heat equation and a realistic model of an acoustic pipe on bounded domains are considered. Both involve a transfer function with a unique analytic continuation and singularities of pole type. The set of singularities degenerates into uniquely defined cuts when the length of the physical domain becomes infinite. From a mathematical point of view, both the convergence in Hardy spaces of some right-half complex plane and the pointwise convergence are studied and proved
Lipschitz stability in an inverse problem for the wave equation
We are interested in the inverse problem of the determination of the
potential from the measurement of the
normal derivative on a suitable part of the
boundary of , where is the solution of the wave equation
set in and
given Dirichlet boundary data. More precisely, we will prove local uniqueness
and stability for this inverse problem and the main tool will be a global
Carleman estimate, result also interesting by itself
Modelling Spatial Compositional Data: Reconstructions of past land cover and uncertainties
In this paper, we construct a hierarchical model for spatial compositional
data, which is used to reconstruct past land-cover compositions (in terms of
coniferous forest, broadleaved forest, and unforested/open land) for five time
periods during the past years over Europe. The model consists of a
Gaussian Markov Random Field (GMRF) with Dirichlet observations. A block
updated Markov chain Monte Carlo (MCMC), including an adaptive Metropolis
adjusted Langevin step, is used to estimate model parameters. The sparse
precision matrix in the GMRF provides computational advantages leading to a
fast MCMC algorithm. Reconstructions are obtained by combining pollen-based
estimates of vegetation cover at a limited number of locations with scenarios
of past deforestation and output from a dynamic vegetation model. To evaluate
uncertainties in the predictions a novel way of constructing joint confidence
regions for the entire composition at each prediction location is proposed. The
hierarchical model's ability to reconstruct past land cover is evaluated
through cross validation for all time periods, and by comparing reconstructions
for the recent past to a present day European forest map. The evaluation
results are promising and the model is able to capture known structures in past
land-cover compositions
A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data
We propose a globally convergent computational technique for the nonlinear
inverse problem of reconstructing the zero-order coefficient in a parabolic
equation using partial boundary data. This technique is called the "reduced
dimensional method". Initially, we use the polynomial-exponential basis to
approximate the inverse problem as a system of 1D nonlinear equations. We then
employ a Picard iteration based on the quasi-reversibility method and a
Carleman weight function. We will rigorously prove that the sequence derived
from this iteration converges to the accurate solution for that 1D system
without requesting a good initial guess of the true solution. The key tool for
the proof is a Carleman estimate. We will also show some numerical examples
Decay of Hankel singular values of analytic control systems
We show that control systems with an analytic semigroup and control and observation operators that are not too unbounded have a Hankel operator that belongs to the Schatten class S-p for all positive p. This implies that the Hankel singular values converge to zero faster than any polynomial rate. This in turn implies fast convergence of balanced truncations. As a corollary, decay rates for the eigenvalues of the controllability and observability Gramians are also provided. Applications to the heat equation and a plate equation are given