38 research outputs found
Computation of sum of squares polynomials from data points
We propose an iterative algorithm for the numerical computation of sums of
squares of polynomials approximating given data at prescribed interpolation
points. The method is based on the definition of a convex functional
arising from the dualization of a quadratic regression over the Cholesky
factors of the sum of squares decomposition. In order to justify the
construction, the domain of , the boundary of the domain and the behavior at
infinity are analyzed in details. When the data interpolate a positive
univariate polynomial, we show that in the context of the Lukacs sum of squares
representation, is coercive and strictly convex which yields a unique
critical point and a corresponding decomposition in sum of squares. For
multivariate polynomials which admit a decomposition in sum of squares and up
to a small perturbation of size , is always
coercive and so it minimum yields an approximate decomposition in sum of
squares. Various unconstrained descent algorithms are proposed to minimize .
Numerical examples are provided, for univariate and bivariate polynomials
Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations
In this article, we are interested in the asymptotic analysis of a finite
volume scheme for one dimensional linear kinetic equations, with either
Fokker-Planck or linearized BGK collision operator. Thanks to appropriate
uniform estimates, we establish that the proposed scheme is
Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the
discrete framework the hypocoercivity method proposed by [J. Dolbeault, C.
Mouhot and C. Schmeiser, Trans. Amer. Math. Soc., 367, 6 (2015)] to prove the
exponential return to equilibrium of the approximate solution. We obtain decay
rates that are bounded uniformly in the diffusive limit.
Finally, we present an efficient implementation of the proposed numerical
schemes, and perform numerous numerical simulations assessing their accuracy
and efficiency in capturing the correct asymptotic behaviors of the models.Comment: 39 pages, 10 figures, 2 table
Computation of sum of squares polynomials from data points
International audienceWe propose an iterative algorithm for the numerical computation of sums of squares of polynomials approximating given data at prescribed interpolation points. The method is based on the definition of a convex functional arising from the dualization of a quadratic regression over the Cholesky factors of the sum of squares decomposition. In order to justify the construction, the domain of , the boundary of the domain and the behavior at infinity are analyzed in details. When the data interpolate a positive univariate polynomial, we show that in the context of the Lukacs sum of squares representation, is coercive and strictly convex which yields a unique critical point and a corresponding decomposition in sum of squares. For multivariate polynomials which admit a decomposition in sum of squares and up to a small perturbation of size , is always coercive and so it minimum yields an approximate decomposition in sum of squares. Various unconstrained descent algorithms are proposed to minimize . Numerical examples are provided, for univariate and bivariate polynomials
A note on hypocoercivity for kinetic equations with heavy-tailed equilibrium
In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic LĂ©vy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local LĂ©vy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation
Large time behavior of nonlinear finite volume schemes for convection-diffusion equations
In this contribution we analyze the large time behavior of a family of nonlinear finite volume schemes for anisotropic convection-diffusion equations set in a bounded bidimensional domain and endowed with either Dirichlet and / or no-flux boundary conditions. We show that solutions to the two-point flux approximation (TPFA) and discrete duality finite volume (DDFV) schemes under consideration converge exponentially fast toward their steady state. The analysis relies on discrete entropy estimates and discrete functional inequalities. As a biproduct of our analysis, we establish new discrete Poincaré-Wirtinger, Beckner and logarithmic Sobolev inequalities. Our theoretical results are illustrated by numerical simulations
Numerical analysis of a finite volume scheme for charge transport in perovskite solar cells
In this paper, we consider a drift-diffusion charge transport model for
perovskite solar cells, where electrons and holes may diffuse linearly
(Boltzmann approximation) or nonlinearly (e.g. due to Fermi-Dirac statistics).
To incorporate volume exclusion effects, we rely on the Fermi-Dirac integral of
order -1 when modeling moving anionic vacancies within the perovskite layer
which is sandwiched between electron and hole transport layers. After
non-dimensionalization, we first prove a continuous entropy-dissipation
inequality for the model. Then, we formulate a corresponding two-point flux
finite volume scheme on Voronoi meshes and show an analogous discrete
entropy-dissipation inequality. This inequality helps us to show the existence
of a discrete solution of the nonlinear discrete system with the help of a
corollary of Brouwer's fixed point theorem and the minimization of a convex
functional. Finally, we verify our theoretically proven properties numerically,
simulate a realistic device setup and show exponential decay in time with
respect to the L^2 error as well as a physically and analytically meaningful
relative entropy
Large-time behaviour of a family of finite volume schemes for boundary-driven convection-diffusion equations
International audienceWe are interested in the large-time behaviour of solutions to finite volume discretizations of convectionâdiffusion equations or systems endowed with nonhomogeneous Dirichlet- and Neumann-type boundary conditions. Our results concern various linear and nonlinear models such as FokkerâPlanck equations, porous media equations or driftâdiffusion systems for semiconductors. For all of these models, some relative entropy principle is satisfied and implies exponential decay to the stationary state. In this paper we show that in the framework of finite volume schemes on orthogonal meshes, a large class of two-point monotone fluxes preserves this exponential decay of the discrete solution to the discrete steady state of the scheme. This includes for instance upwind and centred convections or ScharfetterâGummel discretizations. We illustrate our theoretical results on several numerical test cases