G. Fort, B. Jourdain, E. Kuhn, T. Lelièvre, G. Stoltz
We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.
{"title":"Efficiency of the Wang–Landau Algorithm: A Simple Test Case","authors":"G. Fort, B. Jourdain, E. Kuhn, T. Lelièvre, G. Stoltz","doi":"10.1093/AMRX/ABU003","DOIUrl":"https://doi.org/10.1093/AMRX/ABU003","url":null,"abstract":"We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"82 1","pages":"275-311"},"PeriodicalIF":0.0,"publicationDate":"2013-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79679438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The quantum de Finetti theorem asserts that the k-body density matrices of a N-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In this note we review a construction due to Christandl, Mitchison, Konig and Renner valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella, which gives the density matrices of the constructed state as a function of those of the original state.
{"title":"REMARKS ON THE QUANTUM DE FINETTI THEOREM FOR BOSONIC SYSTEMS","authors":"Mathieu Lewin, P. T. Nam, N. Rougerie","doi":"10.1093/AMRX/ABU006","DOIUrl":"https://doi.org/10.1093/AMRX/ABU006","url":null,"abstract":"The quantum de Finetti theorem asserts that the k-body density matrices of a N-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In this note we review a construction due to Christandl, Mitchison, Konig and Renner valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella, which gives the density matrices of the constructed state as a function of those of the original state.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"54 1","pages":"48-63"},"PeriodicalIF":0.0,"publicationDate":"2013-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90920913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Smooth Periodic Solutions of a 2×2 System of Nonlinear Hyperbolic Conservation Laws","authors":"M. Shearer","doi":"10.1093/AMRX/ABT006","DOIUrl":"https://doi.org/10.1093/AMRX/ABT006","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"3 1","pages":"114-121"},"PeriodicalIF":0.0,"publicationDate":"2013-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86389344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Note on a 3D Haptotaxis Model of Cancer Invasion","authors":"Jishan Fan, Kun Zhao","doi":"10.1093/AMRX/ABT004","DOIUrl":"https://doi.org/10.1093/AMRX/ABT004","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"56 ","pages":"74-86"},"PeriodicalIF":0.0,"publicationDate":"2013-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/AMRX/ABT004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72498259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximate Solutions of a Stochastic Variational Inequality Modeling an Elasto-Plastic Problem with Noise","authors":"Héctor Jasso-Fuentes, L. Mertz, S. Yam","doi":"10.1093/AMRX/ABT003","DOIUrl":"https://doi.org/10.1093/AMRX/ABT003","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"12 1","pages":"52-73"},"PeriodicalIF":0.0,"publicationDate":"2013-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79756642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step $delta t$ and on the mesh size $delta x$ and for $N$ - the number of realizations - reasonably large, we control this error by a term of order $mathcal{O}(sqrt{delta t /N})$. We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.
我们考虑基于半拉格朗日格式和解的概率表示组合的偏微分方程的蒙特卡罗离散化。我们在一个简单的例子中研究了蒙特卡罗误差，并表明在时间步长$delta t$和网格尺寸$delta x$上的反cfl条件下，对于$N$ -实现的数量-相当大，我们通过一个阶项$mathcal{O}(sqrt{delta t /N})$来控制该误差。我们还提供了一些数值实验来证实误差估计，并给出了一些可以用数值方法处理的方程的例子。
{"title":"Analysis of the Monte-Carlo Error in a Hybrid Semi-Lagrangian Scheme","authors":"Charles-Edouard Br'ehier, E. Faou","doi":"10.1093/AMRX/ABV001","DOIUrl":"https://doi.org/10.1093/AMRX/ABV001","url":null,"abstract":"We consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. We study the Monte-Carlo error in a simple case, and show that under an anti-CFL condition on the time-step $delta t$ and on the mesh size $delta x$ and for $N$ - the number of realizations - reasonably large, we control this error by a term of order $mathcal{O}(sqrt{delta t /N})$. We also provide some numerical experiments to confirm the error estimate, and to expose some examples of equations which can be treated by the numerical method.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"12 1","pages":"167-203"},"PeriodicalIF":0.0,"publicationDate":"2013-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90379032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the eff ective electric properties of a composite with high contrast. The energy density is given locally in terms of a convex function of the gradient of the potential. The permittivity may take very large values in a fairly general distribution of parallel fibers of tiny cross sections. For a critical size of the cross sections, we show that a concentration of electric energy may arise in a small region of space surrounding the fi bers. This extra contribution is caused by the discrepancy between the behaviors of the potential in the matrix and in the fibers and is characterized by the density of the cross sections of the bers with respect to the cross section of the body in terms of some suitable notion of capacity. Our results extend those established in [7] in the periodic case for the p-Laplacian to a general nonlinear framework and a non-periodic distribution of fi bers.
{"title":"Nonlinear Capacitary Problems for a General Distribution of Fibers","authors":"Michel Bellieud, C. Licht, S. Orankitjaroen","doi":"10.1093/AMRX/ABT002","DOIUrl":"https://doi.org/10.1093/AMRX/ABT002","url":null,"abstract":"We determine the eff ective electric properties of a composite with high contrast. The energy density is given locally in terms of a convex function of the gradient of the potential. The permittivity may take very large values in a fairly general distribution of parallel fibers of tiny cross sections. For a critical size of the cross sections, we show that a concentration of electric energy may arise in a small region of space surrounding the fi bers. This extra contribution is caused by the discrepancy between the behaviors of the potential in the matrix and in the fibers and is characterized by the density of the cross sections of the bers with respect to the cross section of the body in terms of some suitable notion of capacity. Our results extend those established in [7] in the periodic case for the p-Laplacian to a general nonlinear framework and a non-periodic distribution of fi bers.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"46 1","pages":"1-51"},"PeriodicalIF":0.0,"publicationDate":"2013-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90350795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Armstrong-Frederick model for nonlinear kinematic hardening is regarded as a benchmark model in contemporary elastoplasticity. This work presents an existence result to an appropriately time-rescaled evolution for that model. To do so, we have to resort to a regularization of the dependence of the convex of plasticity upon the back stress. Such a regularization process seems to be the unfortunate price one has to pay for a successful mathematical analysis.
{"title":"Quasi-Static Evolution for the Armstrong-Frederick Hardening-Plasticity Model","authors":"G. Francfort, U. Stefanelli","doi":"10.1093/AMRX/ABT001","DOIUrl":"https://doi.org/10.1093/AMRX/ABT001","url":null,"abstract":"The Armstrong-Frederick model for nonlinear kinematic hardening is regarded as a benchmark model in contemporary elastoplasticity. This work presents an existence result to an appropriately time-rescaled evolution for that model. To do so, we have to resort to a regularization of the dependence of the convex of plasticity upon the back stress. Such a regularization process seems to be the unfortunate price one has to pay for a successful mathematical analysis.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"39 1","pages":"297-344"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77954380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study a model of Landau–Lifshitz equations of ferromagnetism that does not contain the regularizing term of exchange energy. Without the exchange energy, due to the lack of certain derivative estimates and compactness, such an equation becomes degenerate and cannot be studied by the usual Galerkin method based on the elliptic equation theory. For such a degenerate model, it is known that the weak solutions can be obtained through the quasi-stationary limits of certain coupled Landau– Lifshitz–Maxwell systems as the dielectric permittivity tends to zero. In this paper, we use a simplified Landau–Lifshitz–Maxwell system with constant permittivity to present a different but more direct proof of this quasi-stationary limit result. We also establish a finite-time local L2-stability result for weak solutions of the degenerate Landau–Lifshitz equation, which yields the new uniqueness result on weak solution with bounded initial data.
{"title":"Quasi-Stationary Limit and a Degenerate Landau–Lifshitz Equation of Ferromagnetism","authors":"Wei-Qi Deng, Baisheng Yan","doi":"10.1093/AMRX/ABS019","DOIUrl":"https://doi.org/10.1093/AMRX/ABS019","url":null,"abstract":"In this paper, we study a model of Landau–Lifshitz equations of ferromagnetism that does not contain the regularizing term of exchange energy. Without the exchange energy, due to the lack of certain derivative estimates and compactness, such an equation becomes degenerate and cannot be studied by the usual Galerkin method based on the elliptic equation theory. For such a degenerate model, it is known that the weak solutions can be obtained through the quasi-stationary limits of certain coupled Landau– Lifshitz–Maxwell systems as the dielectric permittivity tends to zero. In this paper, we use a simplified Landau–Lifshitz–Maxwell system with constant permittivity to present a different but more direct proof of this quasi-stationary limit result. We also establish a finite-time local L2-stability result for weak solutions of the degenerate Landau–Lifshitz equation, which yields the new uniqueness result on weak solution with bounded initial data.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"29 1","pages":"277-296"},"PeriodicalIF":0.0,"publicationDate":"2012-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86988998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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