second order finite volume

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Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution

Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution

R + or a sub-interval of R + , it belongs to a space strictly included in R N + , where N is the number of moments [23, 24, 25]. This space is called the moment space. The numerical methods have to ensure that the variables stay in this moment space, i.e. that the moments stay realizable. This issue is not always considered, thus leading to unphysical results (e.g. invalid moment sets). Indeed, the classical schemes for high-order transport in physical space can lead to invalid moment sets [26, 2, 27], as well as for the source terms [13, 12], even if the closure itself ensures the realizability at the continuous level. This happens all the more easily when some moment sets are at the boundary of the moment space, thus corresponding to a sum of a few weighted Dirac delta functions, as obtained through nucleation. To circumvent this issue, some authors resort to moment correction algorithms [28, 26] based on a necessary but eventually not sufficient condition for realizability in order to obtain a valid moment set. The cost of the method then increases and the correction spoils the overall accuracy.
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A Centered Second-Order Finite Volume Scheme for the Heterogeneous Maxwell Equations in Three Dimensions on Arbitrary Unstructured Meshes

A Centered Second-Order Finite Volume Scheme for the Heterogeneous Maxwell Equations in Three Dimensions on Arbitrary Unstructured Meshes

A centered second-order finite volume scheme for the heterogeneous Maxwell equations in three dimensions on arbitrary unstructured meshes Serge Piperno — Malika Remaki — Loula Fezoui.. a[r]

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High-order Finite Volume WENO schemes for non-local multi-class traffic flow models

High-order Finite Volume WENO schemes for non-local multi-class traffic flow models

The computation of numerical solutions for (1.1) is challenging due to the high non- linearity of the system and the dependence of the flux function on integral terms. First and second order finite volume schemes for (1.1) were proposed and analyzed in [5, 6]. In this paper, a high-order finite-volume WENO (FV-WENO) scheme is proposed to solve the non- local multi-class system (1.1). The procedure proposed in [4] is used and extended to the multi-class cases in order to evaluate the non-local term that appears in the flux functions.

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First- and second-order finite volume methods for the one-dimensional nonconservative Euler system

First- and second-order finite volume methods for the one-dimensional nonconservative Euler system

7. Numerical Results Numerical investigations have been carried out to test the numerical scheme based on the Rusanov flux and the nonconservative flux given in section 3. Two sets of tests are proposed: the first set of tests aims to check the ability of the numerical method to solve the Riemann problem for sev- eral characteristics situations (rarefaction and resonant configurations) while a second set of tests is dedicated to the comparison between the first- and second-order schemes with a regular porosity function to check the perfor- mance of the decomposition into nonconservative flux and source term. Com- putations are performed using the OFELI library of Touzani (1998-2003) to handle the mesh.
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Functional renormalisation group in a finite volume

Functional renormalisation group in a finite volume

a finite volume and the integrated flow of ∆ ˙p k . The findings of the present and the last section carry over to the theory in the presence of three- and four- dimensional sharp regulators, (6). Indeed, in these cases the ultraviolet growth is very similar. Below (6) it has been discussed that the theory in the presence of sharp regulators is directly related to respective ultraviolet mo- mentum cutoff regularisations of DSEs in a finite volume and at finite temperature, [11–14]. Consequently the re- spective correlation functions miss the exponential de- cays unless one applies a volume and/or temperature de- pendent renormalisation procedure. Note that only for the present thermodynamical observables, pressure and free energy density, this shows up as a growth with the ultraviolet cutoff scale Λ, in higher order correlation func- tions the missing exponential decay is hidden in polyno- mially decaying terms.
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Second-Order Type Isomorphisms Through Game Semantics

Second-Order Type Isomorphisms Through Game Semantics

1 Introduction Denotational semantics Defining a semantic for a language is a funda- mental tool for understanding the way this language works. Thus, semantics is a very active domain of research in theoretical computer science: in par- ticular, there has been an important investigation on semantics which could modelize a language as precisely as possible; this has led to the emergence of game semantics in the early 90s, whose success is due to the deep adequa- tion of its models with the syntax. The present work illustrates the ability of game semantics to modelize a language precisely: consequently, it is possi- ble to extract from the model some properties of the language. So, this work has to be understood as an example of accomplishment of the original goal of denotational semantics: using abstract tools to prove concrete properties on a programming language. In this article, the property we extract con- cerns a non-trivial problem, the characterization of type isomorphisms for second-order languages.
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A second-order geometry-preserving finite volume method for conservation laws on the sphere

A second-order geometry-preserving finite volume method for conservation laws on the sphere

We consider nonlinear hyperbolic conservation laws posed on curved geometries —refered to as “geometric Burgers equations” after Ben-Artzi and LeFloch— when the underlying geometry is the two-dimensional sphere and the flux vector field is determined from a po- tential function. Despite its apparent simplicity, this hyperbolic model exhibits complex wave phenomena that are not observed in absence of geometrical effects. We formulate a second-order accurate, finite volume method which is based on a latitude/longitude trian- gulation of the sphere and on a generalized Riemann solver and a direction splitting based on the sphere geometry. Importantly, this scheme is geometry-preserving in the sense that the discrete form of the scheme respects the divergence free condition for the conservation law on the sphere. A total variation diminishing Runge-Kutta method with an operator splitting approach is used for temporal integration. The quality of the numerical solutions is largely improved using the proposed piecewise linear reconstruction and the method per- forms well for discontinuous solutions with large amplitude and shocks in comparison with the existing schemes. With this method, we numerically investigate the properties of discon- tinuous solutions and numerically demonstrate the contraction, time-variation monotonicity, and entropy monotonicity properties. Next, we study the late-time asymptotic behavior of solutions, and discuss it in terms of the properties of the flux vector field. We thus provide a rigorous validation of the accuracy and efficiency of the proposed finite volume method in presence of nonlinear hyperbolic waves and a curved geometry. The method should be extendable to the shallow water model posed on the sphere.
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Finite volume schemes for constrained conservation laws

Finite volume schemes for constrained conservation laws

In the present paper, we characterize and approximate entropy solutions of ( 1 - 3 ) in the L ∞ setting. The core of the paper is the convergence analysis for finite volume schemes adapted to the constrained problem ( 1 - 3 ). The schemes are constructed as follows. We consider a classical monotone three-point finite volume scheme (see [ EGH00 ]) and denote by g(u, v) the associated numerical flux; at the interface of the mesh which corresponds to the obstacle position {x = 0}, the numerical flux is replaced by min(g(u, v), F ) in order to comply with the constraint ( 3 ) (see Section 4 for more details). Our approach is simpler than the wave-front tracking algorithm devised in [ CG07 ], because we do not need to define explicitly the Riemann solvers at the interface {x = 0} which would fit the constraint at time t. Notice that with our approach, existing finite volume codes for the non-constrained conservation law ( 1 ) are trivially combined with the constraint ( 3 ).
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Second-order cone optimization of the gradostat

Second-order cone optimization of the gradostat

MISTEA, Univ. Montpellier, INRAE, Institut Agro, Montpellier, France Abstract The gradostat consists of multiple chemostats interconnected by mass flows and diffusion. It has been used to model biochemical systems such as wastewater treatment networks and microbial activity in soil. In this paper we maximize the production of biogas in a gradostat at steady state. The physical decision variables are the water, substrate, and biomass entering each tank and the flows through the interconnecting pipes. Our main technical focus is the nonconvex constraint describing microbial growth. We formulate a relaxation and prove that it is exact when the gradostat is outflow connected, its system matrix is irreducible, and the growth rate satisfies a simple condition. The relaxation has second-order cone representations for the Monod and Contois growth rates. We extend the steady state models to the case of multiple time periods by replacing the derivatives with numerical approximations instead of setting them to zero. The resulting optimizations are second-order cone programs, which can be solved at large scales using standard industrial software.
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SECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN

SECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN

a consequence, Q λ 0 ,λ 0 (K) is a conformal symmetry of ∆ Y if and only if Obs(K) = 0. We illustrate our results on two examples in dimension three. In the first one, the space R 3 is endowed with the most general Riemannian metric admitting a Killing 2-tensor K, which is diagonal in orthogonal coordinates [ 28 ]. Then, Obs(K) [ is a non-trivial exact 1-form and, up to our knowledge, the symmetry of ∆ Y that we obtain is new. In the second one, we

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Second-order cone optimization of the gradostat

Second-order cone optimization of the gradostat

Keywords: Gradostat; second-order cone programming; convex relaxation; wastewater treatment; biogas. 1. Introduction The gradostat is a nonlinear dynamical system in which multiple chemostats are interconnected by mass flows and diffusion. In each chemostat, microbial growth converts a substrate to biomass. This also produces biogas, a useful energy source. Our primary motivation for this setup is the design and operation of a network of wastewater treatment plants [1].

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High-order filtered schemes for time-dependent second order HJB equations

High-order filtered schemes for time-dependent second order HJB equations

Abstract In this paper, we present and analyse a class of “filtered” numerical schemes for second order Hamilton-Jacobi-Bellman (HJB) equations. Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Amp` ere partial differential equation, SIAM J. Numer. Anal., 51(1):423–444, 2013, and more recently applied by other authors to stationary or time-dependent first order Hamilton-Jacobi equations. For high order approximation schemes (where “high” stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by “filtering” them with a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions. We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.
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Pour une édition électronique du second volume de Bouvard et Pécuchet : le dossier médical

Pour une édition électronique du second volume de Bouvard et Pécuchet : le dossier médical

f° 24-158) est une entité mixte, qui se compose de 237 pages (soit 136 feuillets) relevant de plusieurs strates génétiques 2 . Il est constitué en majeure partie par les notes de lecture que le romancier a prises en vue de la rédaction de la première moitié du chapitre III. Le nombre des ouvrages médicaux consultés se monte au total à soixante-neuf, sans compter que parmi ces titres se trouve d’ailleurs le Dictionnaire des sciences médicales qui compte à lui seul 60 volumes (1812-1822). Le dossier médical comprend également 14 pages de « notes de notes » (f° 138-146), lesquelles sont comme des scénarios thématiques établis à partir des notes de lecture proprement dites. Il contient aussi 5 pages de récapitulations (f° 154-157 v°) où se trouvent regroupés divers extraits manifestement orientés vers le second volume qu’on appelle communément le
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Factorized second order methods in neural networks

Factorized second order methods in neural networks

INTRODUCTION This thesis presents my work during my master at MILA under the supervision of Pascal Vincent. Artificial neural networks are a powerful machine learning tool for modeling complex functions. Training a neural network for a given task often reduces to minimizing a scalar function of several millions of variables, which are the parameters of the model. While optimization is a full field of research on its own, usual methods do not scale to the order of magnitude of several millions of variables. For this reason neural networks practitioners stick to first order optimization methods, while not benefiting of the acceleration provided by using more powerful methods. Amongst the family of optimization methods, second order methods are a conceptually simple way of accelerating optimization. But practically, they require too much memory and computational power in order to be really useful when scaled to millions of parameters. We circumvent these practical constraints by approximating second order methods, trading off between computational cost, and speed up.
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Entre le premier et le second volume de Bouvard et Pécuchet: Flaubert et Raspail

Entre le premier et le second volume de Bouvard et Pécuchet: Flaubert et Raspail

Commençons  par  le  premier  volume.  Dans  le  chapitre  III,  un  épisode entier  est  consacré  à  la  médecine  Raspail,  dont  on  sait  qu’elle  a  connu  une vogue bien réelle autour des années 1840 [5] . Dans la fiction, la manière dont Bouvard  et  Pécuchet  en  prennent  connaissance  mérite  d’abord  d’être commentée.  Un  jour,  un  colporteur  accoste  Bouvard  et  lui  propose  entre autres choses « le Manuel de la santé, par François Raspail ». Charmé par la lecture  de  cette  brochure,  il  se  procure  «  le  grand  ouvrage  »,  c’est­à­dire l’Histoire naturelle de la santé et de la maladie, dont « la clarté de la doctrine » séduit tout de suite les deux bonshommes (p. 118­119 [6] ). On voit bien que l’initiation  se  fait  ici  en  deux  temps,  ce  qui  correspond  exactement  à  la stratégie  que  Raspail  mettait  lui­même  en  œuvre  en  vue  de  propager  son système.  Ce  savant  démocrate  distinguait  en  effet  deux  niveaux  dans  la production  scientifique,  qui  visaient  chacun  un  public  différent  :  les  grands traités théoriques d’un côté, les publications pour un large public de l’autre. Si dans les premiers il s’appliquait à développer les principes et les méthodes de sa médecine, les secondes étaient d’une importance capitale pour vulgariser et démocratiser le savoir, dont il n’a cessé de condamner la monopolisation par les agents de la médecine officielle. Lancé à un prix réduit (1 fr. 25) en 1845, le  Manuel  annuaire  de  la  santé  a  eu  un  succès  énorme [7] ,  comme  en témoigne,  entre  autres,  le  fait  qu’il  a  continué  à  paraître  régulièrement,  au­ delà  de  la  mort  de  son  fondateur,  jusqu’en  1935.  Bouvard  et  Pécuchet,  en commençant  par  le  manuel  d’apprentissage  de  cette  médecine  pour approfondir  ensuite  la  matière  dans  l’ouvrage  érudit,  ne  font  donc  que respecter l’ordre canonique indiqué par Raspail.
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Second order backward SDE with random terminal time

Second order backward SDE with random terminal time

Our main interest in this paper is on the extension to the fully nonlinear second order parabolic equations, as initiated in the finite horizon setting by Soner, Touzi & Zhang [STZ12], and further developed by Possama¨ı, Tan & Zhou [PTZ17], see also the first attempt by Cheridito, Soner, Touzi & Victoir [CSTV07], and the closely connected BSDEs in a nonlinear expectation framework of Hu, Ji, Peng & Song [HJPS14a, HJPS14b] (called GBSDEs). This extension is performed on the canonical space of continuous paths with canonical process denoted by X. The key idea is to reduce the fully nonlinear representation to a semilinear representation which is required to hold simultaneously under an appropriate family P of singular semimartingale mea- sures on the canonical space. Namely, an F T − random variable ξ with appropriate integrability
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Perturbation analysis of second-order cone programming problems

Perturbation analysis of second-order cone programming problems

Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex France Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois -[r]

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On the second order spatiochromatic structure of natural images

On the second order spatiochromatic structure of natural images

2.1. Spatial redundancy in natural images There is a large body of works dealing with spatial statistics in natural images, as e.g. reviewed in Srivastava et al. (2003). In the present work, we will focus on relatively simple second order property of natural images, and mostly on their covariance. Our motivation is that such simple struc- tures are, to the best of our knowledge, not fully understood in the case of spatiochromatic dependency and will be addressed in the remaining of this paper. In particular, we will not consider in this work the non-gaussianity of natural images, although it is related to the most geometric aspects of image structure, see e.g. Mumford and Gidas (2001).
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Theoretical analysis of the second-order synchrosqueezing transform

Theoretical analysis of the second-order synchrosqueezing transform

In the VSST case, the window must still be chosen so as to satisfy the separation condition between modes. But, since it uses a second-order approximation of the phase, VSST does not require a small window to produce a concentrated representation. This phenomenon is illustrated in the following Fig. 4 , which displays the EMD with respect to the input SNR for the same signal as in Fig. 3 . It compares FSST and VSST, computed with a Gaussian window with 4 significant values of σ ranked increasingly, meaning that to choose σ outside [σ 1 , σ 4 ] leads to worse results. It is clear that for a given σ, VSST always considerably
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Second Order BSDEs with Jumps: Formulation and Uniqueness

Second Order BSDEs with Jumps: Formulation and Uniqueness

In this paper, we define a notion of second-order backward stochas- tic differential equations with jumps (2BSDEJs for short), which gen- eralizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149–190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our set- ting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the Y component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the so- lution.
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