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.Inverse problems, finite Markov moment problem, exponential transform.

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

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

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

Numerical aspects of nonlinear Schrodinger equations in the presence of caustics

The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schrodinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proven rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed.Comment: 20 pages. To appear in Math. Mod. Meth. Appl. Sc

Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System

Several singular limits are investigated in the context of a $2 \times 2$ system arising for instance in the modeling of chromatographic processes. In particular, we focus on the case where the relaxation term and a $L^2$ projection operator are concentrated on a discrete lattice by means of Dirac measures. This formulation allows to study more easily some time-splitting numerical schemes

Main topics of the NumHyp-2015' discussion session

Three main topics were raised in this discussion session, which took place on the 19th of June at the NumHyp-2015 meeting: nonlinear resonance for 1D systems of balance laws, dispersive extensions of standard hyperbolic conservation laws, and the validation of weakly dispersive shallow water wave models. An introductory overview with many bibliographic references is provided for all these topics. Based on kinetic formulation, a numerical strategy that can overcome resonance issues is presented, and a well-balanced (WB) technique for Vlasov-Fokker-Planck equations is outlined. This WB scheme relies on the spectral representation of stationary solutions.Comment: 25 pages, 7 figures, 1 table, 83 reference