Communications on Pure and Applied Mathematics最新文献

{"title":"A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution","authors":"Shidong Jiang, Leslie Greengard","doi":"10.1002/cpa.22240","DOIUrl":"https://doi.org/10.1002/cpa.22240","url":null,"abstract":"We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (<jats:italic>dual‐space multilevel kernel‐splitting</jats:italic>) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"42 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142815825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the isoperimetric Riemannian Penrose inequality","authors":"Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri","doi":"10.1002/cpa.22239","DOIUrl":"https://doi.org/10.1002/cpa.22239","url":null,"abstract":"We prove that the Riemannian Penrose inequality holds for asymptotically flat 3‐manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the mass being a well‐defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential‐theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3‐manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well‐posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"220 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142788421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations","authors":"Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber","doi":"10.1002/cpa.22234","DOIUrl":"https://doi.org/10.1002/cpa.22234","url":null,"abstract":"The randomly pivoted Cholesky algorithm (<jats:sc>RPCholesky</jats:sc>) computes a factorized rank‐ approximation of an positive‐semidefinite (psd) matrix. <jats:sc>RPCholesky</jats:sc> requires only entry evaluations and additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical and theoretical behavior of this fundamental algorithm. For matrix approximation problems that arise in scientific machine learning, experiments show that <jats:sc>RPCholesky</jats:sc> matches or beats the performance of alternative algorithms. Moreover, <jats:sc>RPCholesky</jats:sc> provably returns low‐rank approximations that are nearly optimal. The simplicity, effectiveness, and robustness of <jats:sc>RPCholesky</jats:sc> strongly support its use in scientific computing and machine learning applications.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"27 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hydrodynamic large deviations of TASEP","authors":"Jeremy Quastel, Li‐Cheng Tsai","doi":"10.1002/cpa.22233","DOIUrl":"https://doi.org/10.1002/cpa.22233","url":null,"abstract":"We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP). This problem was studied by Jensen and Varadhan and was shown to be related to entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik for the transition probabilities of the TASEP.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"115 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142690704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the derivation of the homogeneous kinetic wave equation","authors":"Charles Collot, Pierre Germain","doi":"10.1002/cpa.22232","DOIUrl":"https://doi.org/10.1002/cpa.22232","url":null,"abstract":"The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so‐called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearized dynamics then involves a linear Schrödinger equation with a corresponding random potential whose operator norm in Bourgain spaces we are able to estimate on average. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142684236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the stability of Runge–Kutta methods for arbitrarily large systems of ODEs","authors":"Eitan Tadmor","doi":"10.1002/cpa.22238","DOIUrl":"https://doi.org/10.1002/cpa.22238","url":null,"abstract":"We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based on the numerical range of the underlying large matrices involved in such systems, and demonstrate its application with concrete examples of RK stability for hyperbolic methods of lines.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"4 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The α$alpha$‐SQG patch problem is illposed in C2,β$C^{2,beta }$ and W2,p$W^{2,p}$","authors":"Alexander Kiselev, Xiaoyutao Luo","doi":"10.1002/cpa.22236","DOIUrl":"https://doi.org/10.1002/cpa.22236","url":null,"abstract":"We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in <jats:italic>every</jats:italic> Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for <jats:italic>every</jats:italic> Sobolev space unless .","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"197 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mean‐field limit of non‐exchangeable systems","authors":"Pierre‐Emmanuel Jabin, David Poyato, Juan Soler","doi":"10.1002/cpa.22235","DOIUrl":"https://doi.org/10.1002/cpa.22235","url":null,"abstract":"This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect the structure of the connectivities in the network and has critical effects on the collective dynamics. In this article some of the main restrictive hypothesis in the previous literature on the connectivities between the agents (dense graphs) and the cooperation between them (symmetric interactions) are removed.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"248 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Semiconvexity estimates for nonlinear integro‐differential equations","authors":"Xavier Ros‐Oton, Clara Torres‐Latorre, Marvin Weidner","doi":"10.1002/cpa.22237","DOIUrl":"https://doi.org/10.1002/cpa.22237","url":null,"abstract":"In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci. As an application of our result, we establish optimal regularity estimates and smoothness of the free boundary near regular points for the nonlocal obstacle problem on domains. Finally, we also extend the Bernstein technique to parabolic equations and nonsymmetric operators.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"165 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142642581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries","authors":"Dennis Kriventsov, Georg S. Weiss","doi":"10.1002/cpa.22226","DOIUrl":"https://doi.org/10.1002/cpa.22226","url":null,"abstract":"While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about <jats:italic>critical points</jats:italic> of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a <jats:italic>frequency formula</jats:italic> for the Bernoulli problem as well as the celebrated <jats:italic>Naber–Valtorta procedure</jats:italic> to answer this more than 20 year old question in an affirmative way: For a closed class we call <jats:italic>variational solutions</jats:italic> of the Bernoulli problem, we show that the topological free boundary (including <jats:italic>degenerate</jats:italic> singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"5 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142439720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}