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Introduction

Laurent Gosse currently works at the Institute for Applied Mathematics "Mauro Picone" IAC, Italian National Research Council. Laurent does research in Analysis, Applied Mathematics and Computational Physics. Their current project is '"Caseology" for numerical analysis of kinetic models.' and 'Multi-dimensional numerical schemes based on L-splines'.

Additional affiliations

April 2013 - August 2019

March 2003 - August 2004

March 2002 - April 2002

## Publications

Publications (139)

For a four-stream approximation of the kinetic model of radiative transfer with isotropic scattering, a numerical scheme endowed with both truly-2D well-balanced and diffusive asymptotic-preserving properties is derived, in the same spirit as what was done in [14] in the 1D case. Building on former results of Birkhoff and Abu-Shumays, [4], it is po...

Dissipative kinetic models inspired by neutron transport are studied in a (1+1)-dimensional context: first, in the two-stream approximation, then in the general case of continuous velocities. Both are known to relax, in the diffusive scaling, toward a damped heat equation. Accordingly, it is shown that "uniformly accurate" L-splines discretizations...

A (2 + 2)-dimensional kinetic equation, directly inspired by the run-and-tumble modeling of chemotaxis dynamics is studied so as to derive a both "2D well-balanced" and "asymptotic-preserving" numerical approximation. To this end, exact stationary regimes are expressed by means of Laplace transforms of Fourier-Bessel solutions of associated ellipti...

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping , allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementin...

The Fokker-Planck approximation for an elementary linear, two-dimensional kinetic model endowed with a mass-preserving integral collision process is numerically studied, along with its diffusive limit. In order to set up a well-balanced discretization relying on a S-matrix, exact steady-states of the continuous equation are derived. The ability of...

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering $S$-matrix and corresponding ``truly 2D well-balanced'' numerical schemes. A first scheme is obtained by directly implemen...

A correction to this paper has been published: https://doi.org/10.1007/s42985-021-00091-x

This article proposes an efficient explicit numerical model with a relaxed stability condition for the simulation of heat, air and moisture transfer in porous material. Three innovative approaches are combined to solve the system of two differential advection-diffusion equations coupled with a purely diffusive equation. First, the DuFort-Frankel sc...

We investigate numerically a model consisting in a kinetic equation for the biased motion of bacteria following a run-and-tumble process, coupled with two reaction-diffusion equations for chemical signals. This model exhibits asymptotic propagation at a constant speed. In particular, it admits travelling wave solutions. To capture this propagation,...

The existence of travelling waves for a coupled system of hyperbolic/ parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum...

Movie displaying the shape of the slow traveling aggregate, along with its (nearly) constant numerical velocity.

Movie displaying the shape of the fast traveling aggregate, along with its (nearly) constant numerical velocity.

MathSciNet Review of the related paper, https://www.researchgate.net/publication/257312852_Effective_band-limited_extrapolation_relying_on_Slepian_series_and_regularization , avaiable on RG.

These are the slides corresponding to my Invited Talk at the NumAsp'18 Conference, see https://numasp2018.wordpress.com/

Localization phenomena (sometimes called "flea on the elephant") for the operator L ε = −ε 2 ∆u + p(x)u, p(x) being an unbounded potential, are studied both analytically and numerically, mostly in two space dimensions and within a perturbative framework. Starting from the classical harmonic potential, the effects of various perturbations is retriev...

A notion of "2D well-balanced" for drift-diffusion is proposed. Exactness at steady-state, typical in 1D, is weakened by aliasing errors when deriving "truly 2D" numerical fluxes from local Green's function. A main ingredient for proving that such a property holds is the optimality of the trapezoidal rule for periodic functions. In accordance with...

A general methodology, which consists in deriving two-dimensional finite-difference schemes which involve numerical fluxes based on Dirichlet-to-Neumann maps (or Steklov–Poincaré operators), is first recalled. Then, it is applied to several types of diffusion equations, some being weakly anisotropic, endowed with an external source. Standard finite...

Well-balanced schemes, nowadays mostly developed for both hyperbolic and kinetic equations, are extended in order to handle linear parabolic equations, too. By considering the variational solution of the resulting stationary boundary-value problem, a simple criterion of uniqueness is singled out: the $C^1$ regularity at all knots of the computation...

A genuinely two-dimensional discretization of general drift-diffusion (including in-compressible Navier-Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of "bubbles" which are deduced from available discrete data by exploiting the stationary Dirichlet-Green function of the convection-diffusion oper...

This paper is concerned with diffusive approximations of peculiar numerical schemes for several linear (or weakly nonlinear) kinetic models which are motivated by wide-range applications, including radiative transfer or neutron transport, run-and-tumble models of chemotaxis dynamics, and Vlasov-Fokker-Planck plasma modeling. The well-balanced metho...

A general methodology, which consists in deriving two-dimensional finite-difference schemes which involve numerical fluxes based on Dirichlet-to-Neumann maps (or Steklov-Poincaré operators), is first recalled. Then, it is applied to several types of diffusion equations, some being weakly anisotropic, endowed with an external source. Standard finite...

Inverse 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 pos...

Well-balanced schemes, nowadays well-known for 1D hyperbolic equations with source terms and systems of balance laws, are extended to strictly parabolic equations, first in 1D, then in 2D on Cartesian computational grids. The construction heavily relies on a particular type of piecewise-smooth interpolation of discrete data known as ℒ-splines. In 1...

The existence of travelling waves for a coupled system of hyperbolic/ parabolic equations is established in the case of a finite number of velocities in the kinetic equation. This finds application in collective motion of chemotactic bacteria. The analysis builds on the previous work by the first author (arXiv:1607.00429) in the case of a continuum...

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax-Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretizat...

Classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models: a model of chemotaxis involving a biased velocity-redistribution integral term, and a Vlasov-Fokker-Planck (VFP) system. Both are coupled to an attractive elliptic equation pro...

A model consisting in a kinetic equation for " run-and-tumble " biased bacteria motion , coupled with two reaction-diffusion equations for chemical signals is studied. It displays time-asymptotic propagation at constant velocity, i.e. aggregated traveling (exponential) layers. To capture them for various parameters, a well-balanced setup is based o...

This volume gathers contributions reflecting topics presented during an INDAM workshop held in Rome in May 2016. The event brought together many prominent researchers in both Mathematical Analysis and Numerical Computing, the goal being to promote interdisciplinary collaborations. Accordingly, the following thematic areas were developed:
1. Lagrang...

In this note we review and recast some recent results on the existence of nonnstandard solutions to the compressible Euler equations as to make possible a preliminary numerical investigation. In particular, we are interested in studying numerically the forward in time evolution of some Lipschitz initial data which allow for nonnstandard solutions (...

La reformulation du système Euler 2d incompressible sous la forme d’une inclusion différentielle par De Lellis et Székelyhidi (cf. e.g. Bull. Amer. Math. Soc. vol. 49, 347-375) a permis d’appliquer le h-principe à plusieurs familles d’équations de la mécanique des fluides en 2d. De telles techniques fournissent alors une myriade de solutions faible...

A new numerical scheme is set up for the approximation of the weak solution of a two-dimensional damped heat equation with forcing. Its peculiarity is that it derives from a collection of local L-splines, centered at each node of the computational grid and interpolating discrete grid values: by applying Steklov-Poincaré operator, an approximate rad...

http://congressi.iac.cnr.it/wiaa2016 The workshop, organized under the sponsorship of the Istituto Nazionale di Alta Matematica, in collaboration with the Istituto delle Applicazioni del Calcolo of the Italian National Research Council, aims at bringing altogether skilled European researchers in both Mathematical Analysis and Numerical Computing fi...

Numerical resolution of two-stream kinetic models in strong aggregative setting is considered. To illustrate our approach, we consider an 1D kinetic model for chemotaxis in hyperbolic scaling and the high field limit of the Vlasov-Poisson-Fokker-Planck system. A difficulty is that, in this aggregative setting, weak solutions of the limiting model b...

Slides of a talk given during november 2015 at Malaga Univ. in the realm of the "CoCoNuT meetings". It presents most of the results obtained in the SIAP paper "Locally inertial [.,.]" (in particular what type of approximations are needed to recover Mann's toy model of gravity) along with a few animations which illustrate the WB character of the 1+1...

This animation displays the 3D perspective of the 2D Riemann problem (proposed by Jiang-Tadmor) as computed by the 2D scheme and displayed in Sect. 4.3 (Figs 4.7 & 4.8) of the paper.

This animation displays the contour lines of the 2D Riemann problem (proposed by Jiang-Tadmor) as computed by the 2D scheme and displayed in Sect. 4.3 (Figs 4.7 & 4.8) of the paper.

In this chapter we address a semilinear system of two equations, in one space dimension, related to the wave equation with space-dependent damping. An approximation scheme is defined, of Well-Balanced type; for this scheme an error estimate is devised by means of the stability analysis for hyperbolic systems.

In this chapter we analyze the scheme, which was introduced in the previous chapter, by means of a classical Kuznetsov approach. An alternative qualitative estimate, in terms of time and mesh size, is therefore devised. The two estimates are compared, revealing complementary aspects.

In this final chapter we address some problems to which the analysis could be extended. In view of the possible extension of the well-balanced approach to a two-dimensional situation, numerical simulations of two-dimensional Riemann problems for the linear wave equation are shown, together with possible difficulties arising in the application of a...

In this chapter we illustrate our approach to the error estimate analysis, for a scalar, space-dependent, non-resonant balance law. A wave-front tracking scheme is analyzed, leading to a generic linear dependence in time of the error. Numerical illustrations are given for accretive case and for the case of periodic forcing.

In this chapter we analyze some simple examples, which suggest that the error quantification should take into account of the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error (so-called Local Truncation Error).

Illustration of the animation of kinetic variables as computed in Sect. 4.2 (see Figure 2 of the paper).

Illustration of the animation of kinetic variables as computed in Sect. 4.1 (see Figure 1 of the paper).

Time evolution of the kinetic density f(t,x,v) for a 1+1 dimensional vlasov-fokker-planck kinetic model in a box with specular boundary conditions in the x-variable.

Time evolution of the macroscopic current/flux for a 1+1 dimensional vlasov-fokker-planck kinetic model in a box with specular boundary conditions in the x-variable.

Time evolution of the macroscopic density for a 1+1 dimensional vlasov-fokker-planck kinetic model in a box with specular boundary conditions in the x-variable.

Three main topics were raised in this discussion session which took place on the 19th of June at 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...

Well-balanced schemes for two types of (1 + 1)-dimensional kinetic models are set up relying on scattering matrices derived from the explicit so–called " elementary solutions " of the corresponding stationary equations. A " matrix balancing " (or " matrix equilibration ") technique is evoked in order to ensure the overall mass (and positivity) pres...

Sharp and local L 1 a-posteriori error estimates are established for so–called "well-balanced" BV (hence possibly discontinuous) numerical approximations of 2 × 2 space-dependent Jin-Xin relaxation systems under sub-characteristic condition. Ac-cording to the strength of the relaxation process, one can distinguish between two complementary regimes:...

An elementary model of $1+1$-dimensional general relativity, known as ``$\R=\T$'' and mainly developed by Mann {\it et al.} \cite{Mbh1,Mbh2,MMSS,MST,MR,MM,SM}, is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of self-gravitating gas coupled to a Liouville equation for the m...

MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws) of the Godunov numerical scheme for scalar conservation laws are shown to admit a rather simple reformulation when recast in the formalism of the Haar multi-resolution analysis of L 2 (R). By pursuing this wavelet reformulation, a seemingly new MUSCL-WB scheme is derived fo...

Well-balanced schemes were introduced to numerically enforce consistency with long-time behavior of the underlying continuous PDE. When applied to linear kinetic models, like the Goldstein–Taylor system, this construction generates discretizations which are inconsistent with the hydrodynamic stiff limit (despite it captures diffusive limits quite w...

When numerically simulating a kinetic model of n+nn+ semiconductor device, obtaining a constant macroscopic current at steady-state is still a challenging task. Part of the difficulty comes from the multiscale, discontinuous nature of both p|n junctions which create spikes of electric field and enclose a channel where corresponding depletion layers...

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the L2 nor...

A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux \cite{blr} in the context of a staggered Lax-Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by me...

A posteriori L^1 error estimates (in the sense of [11, 20]) are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary 2 × 2 system w...

The ability of Well-Balanced (WB) schemes to capture very accurately steady-state regimes of non-resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by Greenberg and LeRoux (1996) [15] (see also the anterior WB Glimm scheme in E, 1992 [8]). This paper aims at showing, by means of rigorous Ct0(Lx1) esti...

Numerical approximation of one-dimensional kinetic models for directed motion of bacterial populations in response to a chemical gradient, usually called chemotaxis, is considered in the framework of well-balanced (WB) schemes. The validity of one-dimensional models have been shown to be relevant for the simulation of more general situations with s...

Substantial effort has been drawn for years onto the development of (possibly high-order) numerical techniques for the scalar homogeneous conservation law, an equation which is strongly dissipative in L1 thanks to shock wave formation. Such a dissipation property is generally lost when considering hyperbolic systems of conservation laws, or simply...

This shorter chapter is devoted to the numerical study of a particular Fokker-Planck equation (also called Vlasov-Lorentz model in [7]),$$ {\partial}_tf+v{\partial}_xf=\sigma \partial v\left(\left(1-{v}^2\right){\partial}_vf\right),f\left(t=0,x,v\right)={f}_0\left(x,v\right).

Our aim is now to summarize several rigorous results concerning BV solutions of general hyperbolic systems of balance laws with source terms about which no dissipation hypothesis is made. Results of this kind owe to the pioneering 1979 paper by Tai-Ping Liu [38], already quoted in Chapter 4, inside which an astute extension of the Glimm scheme [21]...

In the preceding chapter, it has been shown that the inclusion of a Vlasov-type acceleration term inside the framework of well-balanced schemes for linear relaxation kinetic models leads to complications. There is an alternative: namely, when considering a Fokker-Planck approximation of the relaxation term, the steady-state equation can be reduced...

Chemotaxis describes the directed movement of a large population of micro-organisms (cells, bacteria, …) in response to a gradient of a chemical substance, usually referred to as the chemo-attractant. Indeed the swimming displacement of certain flagellated bacteria can be described by straight-line “runs” suddenly interrupted by “tumbles” of very s...

The main goal of the present Chapter is to emphasize the qualitative difference between Time-Splitting (TS, also called Fractional Step, FS) and Well-Balanced (WB) numerical schemes when it comes to computing the entropy solution [26] of a simple scalar, yet non-resonant, balance law:$$ {\partial}_tu+{\partial}_xf(u)=k(x)g(u),\kern1.68em 0\le k\in...

This introductory chapter aims at positioning the book’s primary topics according to both a scientific and an historic context; loosely speaking, the objective here is more trying to unify seemingly different sectors in numerical analysis rather than being very specific (this will come later on). In particular, one can figure out the main ideas exp...

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear...

The equations studied in Chapters 11 and 12, Vlasov-BGK and Vlasov-Fokker-Planck, are genuinely bi-dimensional problems. Thanks to their special structure, one can succeed in solving them by means of essentially one-dimensional algorithms because the formalism of elementary solutions can be extended up to some numerically tolerable approximations....

Even more than the formers ones, the present chapter will emphasize that well-balanced methods consist essentially in recycling astutely homogeneous techniques in order to take advantage of their strong stability properties in the more delicate context of non-homogeneous systems: for instance, the well-known fact that the Godunov scheme has zero nu...

Electronic transport can be studied within the framework of kinetic theory, being itself closely related to homogenization limits of quantum models, cf. e.g. [7, 38], because it deals with a statistical description which makes sense in view of the many electrons in a typical semiconductor. Accordingly, one deals with f (t, x, k) ∈ [0,1], a distribu...

The so-called method of artificial viscosity has been introduced in the seminal 1950 paper by Richtmyer and Von Neumann [24], where a Lagrangian hyperbolic system of gas dynamics is approximated by finite differences on staggered grids (the specific volume and the velocity aren’t known at the same points). In order to stabilize the Fourier modes of...

Within certain applications, one faces a differential model which is to be solved for instance on the totality of the real line ℝ, with convenient decay properties at infinity. This can happen for quantum models, where the characteristic scale is small so that a component of macroscopic length becomes infinite. It also arises on the other side of t...

The mathematical theory of scalar conservation laws has reached a state of completion: existence, uniqueness, regularity and stability with respect to initial data have been established in various settings (BV theory, compensated compactness, kinetic formulation, relaxation approximation, etc…). Here, we aim at presenting the special features holdi...

This chapter deals mainly with the numerical analysis of the following one-dimensional system of semilinear equations,$$ \begin{array}{ccc} {\partial}_t{f}^{\pm}\pm {\partial}_x{f}^{\pm }=G\left({f}^{+},{f}^{-}\right), & x\in \mathbb{R}, & t>0, \end{array}

This chapter is entirely devoted to the exposition of the method of elementary solutions, which has been introduced and developed during the 50’s–60’s mainly by Chandrasekhar, Case, Cercignani, Siewert and Zweifel. In particular, Chandrasekhar’s discrete ordinates approximation has been refined by Siewert and his collaborators into a so-called anal...

In [37], Perthame and Simeoni proposed a well-balanced scheme for shallow water equations which main core consists in solving a modified Vlasov equation,$$ {\partial}_tf+v{\partial}_xf-{\partial}_xV(x)\;{\partial}_vf={\displaystyle \sum_{n\in \mathbb{N}}\left(M\left(\rho, v-u\right)-f\right)\delta \left(t- n\varDelta t\right),}

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear...

Efficient recovery of smooth functions which are s-sparse with respect to the basis of so-called prolate spheroidal wave functions from a small number of random sampling points is considered. The main ingredient in the design of both the algorithms we propose here consists in establishing a uniform L
∞ bound on the measurement ensembles which const...

The original well-balanced (WB) framework \cite{grl,mcom} relying on nonconservative (NC) products \cite{lft} is set up in order to efficiently treat the so--called Cattaneo model of chemotaxis in 1D \cite{hs}. It proceeds by concentrating the source terms onto Dirac masses: this allows to handle them by NC jump relations based on steady-state equa...

We focus on the numerical simulation of a one-dimensional so--called Cattaneo model of chemotaxis dynamics in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme \cite{siam}. We are especially interested in studying the decay onto numerical steady-states for two reasons: 1/ conventional upwi...

An original well-balanced (WB) Godunov scheme relying on an exact Riemann solver involving a nonconservative (NC) product is developed in order to solve accurately the time-dependent one-dimensional radiative transfer equation in the discrete-ordinates approximation with an arbitrary even number of velocities. The collision term is thus concentrate...

We derive a sufficient condition by means of which one can recover a scale-limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by Donoho and Stark \cite{DS}. The proof follows from simple computations involving the Zak transform, well-known in solid-state physics. Geometric harmonics (in...

We consider a rather simple algorithm to address the fascinating field of numerical extrapolation of (analytic) band-limited functions. It relies on two main elements: namely, the lower frequencies are treated by projecting the known part of the signal to be extended onto the space generated by “Prolate Spheroidal Wave Functions” (PSWF, as original...

We consider the problem of short-time extrapolation of blue chips’ stocks indexes in the context of wavelet subspaces following the theory proposed by X.-G. Xia and co-workers in a series of articles , , and . The idea is first to approximate the oscillations of the corresponding stock index at some scale by means of the scaling function which is p...

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 eas...

We prove Ole\u \i nik-type decay estimates for entropy solutions of $n\times n$ strictly hyperbolic systems of balance laws built out of a wave-front tracking procedure inside which the source term is treated as a nonconservative product localized on a discrete lattice.

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 amplitu...

This paper investigates a simple one-dimensional model of incommensurate “harmonic crystal” in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional...

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 algori...

## Questions

Questions (4)

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Any clue would be welcome! Happy new 2019 to everyone!

--

Laurent

## Projects

Projects (9)

This project aims at taking advantage of "filtering methods", well-established in the signal processing community, in order to stabilize inverse design problems in the presence of shocks. Such filtering is practically mandatory in order to restore a bit of uniqueness despite an infinity of inverse designs can be shown to bring an entropy shock wave.