Beniamin Bogosel, Giuseppe Buttazzo, Edouard Oudet
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.
{"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}
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.
{"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}
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.
{"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}
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.
设计并分析了可压缩正压欧拉系统的半隐式时间熵稳定有限体积格式，并将其弱收敛到耗散测度值解[E]。Feireisl et al.，可压缩Navier-Stokes系统的耗散测度值解，Calc. Var.偏微分方程，2016]的Euler系统。熵的稳定性是通过在质量和动量平衡的对流通量中引入一个位移速度来实现的，只要满足一些类似cfl的条件来保证稳定性。在一些物理上合理的有界性假设下，利用Lax等价定理的精神进行了一致性分析。k -收敛的概念[E]。fereiisl等，数值分析中的k收敛新工具，[j]。分析的，[2020]是为了获得一些强收敛结果，然后通过严格的数值案例研究来说明。利用弱-强唯一性原理和相对熵，给出了该方案收敛于欧拉系统的DMV解、弱解和强解。
{"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}
Ashot Aleksian, Pierre Del Moral, Aline Kurtzmann, Julian Tugaut
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.
{"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}
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.
{"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}
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.
{"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}
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.
{"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}
Martin Chak, Nikolas Kantas, Tony Lelièvre, Grigorios Pavliotis
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.
{"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}
Sining Gong, Jay Gopalakrishnan, Johnny Guzmán, Michael Neilan
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.
{"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}
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