144 research outputs found
Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems
We consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numerically and prove that the algorithm computes the right solution.Inverse problems, finite Markov moment problem, exponential transform.
Resolution of the finite Markov moment problem
We expose in full detail a constructive procedure to invert the so--called
"finite Markov moment problem". The proofs rely on the general theory of
Toeplitz matrices together with the classical Newton's relations
Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems
We consider a class of finite Markov moment problems with arbitrary number of
positive and negative branches. We show criteria for the existence and
uniqueness of solutions, and we characterize in detail the non-unique solution
families. Moreover, we present a constructive algorithm to solve the moment
problems numerically and prove that the algorithm computes the right solution
Chirplet approximation of band-limited, real signals made easy
In this paper we present algorithms for approximating real band-limited
signals by multiple Gaussian Chirps. These algorithms do not rely on matching
pursuit ideas. They are hierarchial and, at each stage, the number of terms in
a given approximation depends only on the number of positive-valued maxima and
negative-valued minima of a signed amplitude function characterizing part of
the signal. Like the algorithms used in \cite{gre2} and unlike previous
methods, our chirplet approximations require neither a complete dictionary of
chirps nor complicated multi-dimensional searches to obtain suitable choices of
chirp parameters
Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation
The final publication is available at Springer via https://doi.org/10.1007/978-3-319-49262-9_7Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a Nwave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presentedAcknowledgements This work was partially supported by the Advanced Grant 694126-DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain
A two-dimensional flea on the elephant phenomenon and its numerical visualization
First Published in Multiscale Modeling and Simulation in 17.1 (2019): 137-166, published by the Society for Industrial and Applied Mathematics (SIAM)Localization phenomena (sometimes called flea on the elephant) for the operator Lvarepsilon = varepsilon 2Δ u + p(x)u, p(x) being an asymmetric double well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for Lvarepsilon, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layersEnrique Zuazua’s research was supported by the Advanced
Grant DyCon (Dynamical Control) of the European Research Council Executive
Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO
(Spain) and the ICON project of the French ANR-16-ACHN-0014. L.G. thanks Profs.
François Bouchut and Roberto Natalini for some technical discussion
Stringent error estimates for one-dimensional, space-dependent 2×2 relaxation systems
International audienceSharp and local {\it a-posteriori} error estimates are established for so--called "well-balanced" (hence possibly discontinuous) numerical approximations of space-dependent Jin-Xin relaxation systems under sub-characteristic condition. According to the strength of the relaxation process, one can distinguish between two complementary regimes: 1/ a weak relaxation, where local errors are shown to be of first order in \DX and uniform in time, 2/ a strong one, where numerical solutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov's theory indicates a behavior of the error in t\cdot \sqrt{\DX}. The uniformly first-order accuracy in weak relaxation regime is obtained by carefully studying interaction patterns and building up a seemingly original variant of Bressan-Liu-Yang's functional, able to handle solutions of arbitrary size for these particular inhomogeneous systems. The complementary estimate in strong relaxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problem
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