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On the numerical approximation of Blaschke-Santalo diagrams using Centroidal Voronoi Tessellations 用质心Voronoi镶嵌的Blaschke-Santalo图的数值逼近
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-15 DOI: 10.1051/m2an/2023092
Beniamin Bogosel, Giuseppe Buttazzo, Edouard Oudet
{"title":"On the numerical approximation of Blaschke-Santalo diagrams using Centroidal Voronoi Tessellations ","authors":"Beniamin Bogosel, Giuseppe Buttazzo, Edouard Oudet","doi":"10.1051/m2an/2023092","DOIUrl":"https://doi.org/10.1051/m2an/2023092","url":null,"abstract":"Identifying Blaschke-Santal'o diagrams is an important topic that essentially consists in determining the image $Y=F(X)$ of a map $F:Xto{mathbb{R}}^d$, where the dimension of the source space $X$ is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, $X$ can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number $N$ of points $x_1,dots,x_N$ in $X$ and plotting them in the target space ${mathbb{R}}^d$, produces in many cases areas in $Y$ of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points $x_i$ in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set $Y$ by a relatively small number $N$ of samples which is very useful when the dimension of the source space $X$ is large (or even infinite) and the evaluation of $F(x_i)$ is costly. Our method consists in a suitable use of {it Centroidal Voronoi Tessellations} which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"8 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Error analysis of Fourier-Legendre and Fourier-Hermite spectral-Galerkin methods for the Vlasov-Poisson system Vlasov-Poisson系统Fourier-Legendre和Fourier-Hermite谱伽辽金方法的误差分析
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-15 DOI: 10.1051/m2an/2023091
XIAOLONG ZHANG, Li-Lian Wang, HONGLI JIA
{"title":"Error analysis of Fourier-Legendre and Fourier-Hermite spectral-Galerkin methods for the Vlasov-Poisson system","authors":"XIAOLONG ZHANG, Li-Lian Wang, HONGLI JIA","doi":"10.1051/m2an/2023091","DOIUrl":"https://doi.org/10.1051/m2an/2023091","url":null,"abstract":"We conduct a rigorous error analysis of the spectral-Galerkin methods for the 1D1V Vlasov-Poisson system with the velocity variable in both a finite or an infinite domain. The estimates significantly improve the very limited existing results. We also provide numerical results to demonstrate the effectiveness of the analysed methods.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"4 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Two approaches to compute unsteady compressible two-phase flow models with stiff relaxation terms 含刚性松弛项的非定常可压缩两相流模型的两种计算方法
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-15 DOI: 10.1051/m2an/2023090
Jean-Marc Hérard, Guillaume Jomée
{"title":"Two approaches to compute unsteady compressible two-phase flow models with stiff relaxation terms","authors":"Jean-Marc Hérard, Guillaume Jomée","doi":"10.1051/m2an/2023090","DOIUrl":"https://doi.org/10.1051/m2an/2023090","url":null,"abstract":"The paper deals with the numerical modeling of two-phase flows while using Baer-Nunziato type models. Focus is given here on the numerical treatment of source terms that involve three (or four) relaxation time scales. A new coupled approach relying on the continuous analysis of the system of ODEs is compared with a more widely used strategy grounded on the fractional step approach. Properties of schemes are given in both cases. Several numerical applications show that the coupled approach should be prefered for both stability and accuracy reasons.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"2 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system 正压欧拉系统耗散测度值解的半隐式有限体积格式
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-15 DOI: 10.1051/m2an/2023093
Amogh Krishnamurthy, K.R. Arun
{"title":"A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system","authors":"Amogh Krishnamurthy, K.R. Arun","doi":"10.1051/m2an/2023093","DOIUrl":"https://doi.org/10.1051/m2an/2023093","url":null,"abstract":"A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax's equivalence theorem under some physically reasonable boundedness assumptions. The concept of K-convergence [E. Feireisl et al., K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak-strong uniqueness principle and relative entropy are presented.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"4 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Self-interacting diffusions: long-time behaviour and exit-problem in the uniformly convex case 自相互作用扩散:均匀凸情况下的长时间行为和退出问题
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-15 DOI: 10.1051/ps/2023020
Ashot Aleksian, Pierre Del Moral, Aline Kurtzmann, Julian Tugaut
{"title":"Self-interacting diffusions: long-time behaviour and exit-problem in the uniformly convex case","authors":"Ashot Aleksian, Pierre Del Moral, Aline Kurtzmann, Julian Tugaut","doi":"10.1051/ps/2023020","DOIUrl":"https://doi.org/10.1051/ps/2023020","url":null,"abstract":"We study a class of time-inhomogeneous diffusion: the self-interacting one. We show a convergence result with a rate of convergence that does not depend on the diffusion coefficient. Finally, we establish a so-called Kramers' type law for the first exit-time of the process from domain of attractions when the landscapes are uniformly convex.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"5 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136227409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Third-order exponential integrator for linear Klein-Gordon equations with time and space-dependant mass. 质量随时间和空间变化的线性Klein-Gordon方程的三阶指数积分器。
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-11-03 DOI: 10.1051/m2an/2023087
Karolina Kropielnicka, Karolina Lademann
{"title":"Third-order exponential integrator for linear Klein-Gordon equations with time and space-dependant mass.","authors":"Karolina Kropielnicka, Karolina Lademann","doi":"10.1051/m2an/2023087","DOIUrl":"https://doi.org/10.1051/m2an/2023087","url":null,"abstract":"Allowing for space- and time-dependence of mass in Klein–Gordon equations re- solves the problem of negative probability density and of violation of Lorenz covariance of interaction in quantum mechanics. Moreover it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper we propose a third-order exponential integrator, where the main idea lies in embed- ding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"151 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135777685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A convergent finite volume scheme for the stochastic barotropic compressible Euler equations. 随机正压可压缩欧拉方程的收敛有限体积格式。
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-10-23 DOI: 10.1051/m2an/2023085
Abhishek Chaudhary, Ujjwal Koley
{"title":"A convergent finite volume scheme for the stochastic barotropic compressible Euler equations.","authors":"Abhishek Chaudhary, Ujjwal Koley","doi":"10.1051/m2an/2023085","DOIUrl":"https://doi.org/10.1051/m2an/2023085","url":null,"abstract":"In this paper, we analyze a semi-discrete finite volume scheme for the three dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)--strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
A strongly conservative hybridizable discontinuous Galerkin method for the coupled time-dependent Navier--Stokes and Darcy problem 耦合时相关Navier—Stokes和Darcy问题的一种强保守杂交不连续Galerkin方法
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-10-23 DOI: 10.1051/m2an/2023086
Aycil Cesmelioglu, Jeonghun Lee, Sander Rhebergen
{"title":"A strongly conservative hybridizable discontinuous Galerkin method for the coupled time-dependent Navier--Stokes and Darcy problem","authors":"Aycil Cesmelioglu, Jeonghun Lee, Sander Rhebergen","doi":"10.1051/m2an/2023086","DOIUrl":"https://doi.org/10.1051/m2an/2023086","url":null,"abstract":"We present a strongly conservative and pressure-robust hybridizable discontinuous Galerkin method for the coupled time-dependent Navier–Stokes and Darcy problem. We show existence and uniqueness of a solution and present an optimal a priori error analysis for the fully discrete problem when using Backward Euler time stepping. The theoretical results are verified by numerical examples.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"11 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135365768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimal friction matrix for underdamped Langevin sampling 欠阻尼朗格万采样的最优摩擦矩阵
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-10-18 DOI: 10.1051/m2an/2023083
Martin Chak, Nikolas Kantas, Tony Lelièvre, Grigorios Pavliotis
{"title":"Optimal friction matrix for underdamped Langevin sampling","authors":"Martin Chak, Nikolas Kantas, Tony Lelièvre, Grigorios Pavliotis","doi":"10.1051/m2an/2023083","DOIUrl":"https://doi.org/10.1051/m2an/2023083","url":null,"abstract":"We propose a procedure for optimising the friction matrix of underdamped Langevin dynamics when used for continuous time Markov Chain Monte Carlo. Starting from a central limit theorem for the ergodic average, we present a new expression of the gradient of the asymptotic variance with respect to friction matrix. In addition, we present an approximation method that uses simulations of the associated first variation/tangent process. Our algorithm is applied to a variety of numerical examples such as toy problems with tractable asymptotic variance, diffusion bridge sampling and Bayesian inference problem for high dimensional logistic regression.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Discrete elasticity exact sequences on Worsey-Farin splits Worsey-Farin分裂上的离散弹性精确序列
4区 数学
Esaim-Probability and Statistics Pub Date : 2023-10-17 DOI: 10.1051/m2an/2023084
Sining Gong, Jay Gopalakrishnan, Johnny Guzmán, Michael Neilan
{"title":"Discrete elasticity exact sequences on Worsey-Farin splits","authors":"Sining Gong, Jay Gopalakrishnan, Johnny Guzmán, Michael Neilan","doi":"10.1051/m2an/2023084","DOIUrl":"https://doi.org/10.1051/m2an/2023084","url":null,"abstract":"We construct conforming finite element elasticity complexes on Worsey-Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey-Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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