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中国科学：数学 2022年第7期52卷 823-844页

作者：陈志敏熊向明深圳大学数学与统计学院深圳518060 The Program in Applied and Computational Mathematics Princeton UniversityrincetonJ08544USA

Charney和DeVore(1979)发现,由于地形的影响,β平面槽道中正压大气的准地转流动有多个平衡态.本文证明了这种流动的基本流在某个参数范围内(包括平坦地表的特殊情形)是渐近稳定的,因此排除了这一参数范围内存在多个平衡态的可能性.本文... 详细信息

Charney和DeVore(1979)发现,由于地形的影响,β平面槽道中正压大气的准地转流动有多个平衡态.本文证明了这种流动的基本流在某个参数范围内(包括平坦地表的特殊情形)是渐近稳定的,因此排除了这一参数范围内存在多个平衡态的可能性.本文作者还发现,为了使Charney-DeVore准地转方程适定,需要补充一个平均经向力或平均经向速度的条件.本文在不同的地表起伏幅度下,采用拟弧长延拓法证实至少存在3个平衡态;还通过高精度直接数值模拟研究了这些平衡态的稳定性.

关键词：准地转方程正压流动 β平面槽道模型地形效应非线性稳定性拟弧长延拓法

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On the particle approximation of lagged Feynman–Kac formulae

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Stochastic Processes and their Applications 2025年 188卷

作者： Awadelkarim, Elsiddig Caffarel, Michel Del Moral, Pierre Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal 23955-6900 Saudi Arabia University of Toulouse CNRS-Lab. de Chimie et Physique Quantiques Toulouse 31062 France Centre de Recherche Inria Bordeaux Sud-Ouest Talence 33405 France School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen China

In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman–Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operators as found, for example, in physics or particle simulation of rare-events. We focus on a novel lagged approximation of this invariant measure, based upon the introduction of a ratio of time-averaged Feynman–Kac marginals associated with a positive operator iterated l∈N times;a lagged Feynman–Kac formula. This estimator and its approximation using Diffusion Monte Carlo (DMC) are commonly used in the physics literature. In short, DMC is an iterative algorithm involving N∈N particles or walkers simulated in parallel, that undergo sampling and resampling operations. In this work, it is shown that for the DMC approximation of the lagged Feynman–Kac formula, one has an almost sure characterization of the L1-error as the time parameter (iteration) goes to infinity and this is at most of O(exp{−κl}/N), for κ>0. In addition a non-asymptotic in time, and time uniform L1−bound is proved which is O(l/N). We also prove a novel central limit theorem to give a characterization of the exact asymptotic in time variance. This analysis demonstrates that the strategy used in physics, namely, to run DMC with N and l small and, for long time enough, is mathematically justified. Our results also suggest how one should choose N and l in practice. We emphasize that these results are not restricted to physical applications;they have broad relevance to the general problem of particle simulation of the Feynman–Kac formula, which is utilized in a great variety of scientific and engineering fields. © 2025 Elsevier B.V.

关键词： Diffusion Monte Carlo Eigenvalue approximation Feynman–Kac formula

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Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations

arXiv

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arXiv 2019年

作者： Jiequn, H.A.N. Chao, M.A. Zheng, M.A. Weinan, E. Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Department of Mathematics Purdue University Beijing Institute of Big Data Research

A new framework is introduced for constructing interpretable and truly reliable reduced models for multi-scale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first through an autoencoder to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active learning procedure is introduced to help ensure that the data set used is representative enough. The reduced system takes the form of the conventional moment systems and works regardless of the numerical discretization used. Numerical results are presented for the BGK model. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow. Copyright © 2019, The Authors. All rights reserved.

关键词： Hydrodynamics

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Constructions and existence of tight fusion frames

Constructions and existence of tight fusion frames

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Wavelets XIII

作者： Casazza, Peter G. Fickus, Matthew Mixon, Dustin G. Wang, Yang Zhou, Zhengfang Department of Mathematics and Statistics Air Force Institute of Technology Wright-Patterson Air Force Base OH 45433 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Mathematics Michigan State University East Lansing MI 48824 United States Department of Mathematics University of Missouri Columbia MO 65211 United States

ISBN: (纸本)9780819477361

Fusion frames are an emerging topic of frame theory, with applications to communications and distributed processing. However, until recently, little was known about the existence of tight fusion frames, much less how to construct them. We discuss a new method for constructing tight fusion frames which is akin to playing Tetris with the spectrum of the frame operator. When combined with some easily obtained necessary conditions, these Spectral Tetris constructions provide a near complete characterization of the existence of tight fusion frames. © 2009 SPIE.

关键词： Optical engineering

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Sigma-delta quantization and finite frames

Sigma-delta quantization and finite frames

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International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： J.J. Benedetto O. Yilmaz A.M. Powell Department of Mathematics University of Maryland College Park MD USA Program in Applied and Computational Mathematics Princeton University Princeton NJ USA

It is shown that sigma-delta (/spl Sigma//spl Delta/) algorithms can be used effectively to quantize finite frame expansions for R/sup d/. Error estimates for various quantized frame expansions are derived, and, in pa... 详细信息

关键词： Delta-sigma modulation Quantization Hilbert space mathematics Signal processing algorithms Phase change materials Signal processing Dictionaries Educational institutions Digital signal processing

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Level sets of quantum control landscapes

Level sets of quantum control landscapes

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International Symposium on Communications Control and Signal Processing (ISCCSP)

作者： Vincent Beltrani Jason Dominy Tak-San Ho Herschel Rabitz Department of Chemistry Princeton University NJ USA Program in Applied and Computational Mathematics Princeton University NJ USA

A controlled quantum system possesses a search landscape defined by the observable value as a functional of the control field. Within the search landscape, there exist level sets of controls giving the same observable value. This paper focuses on level sets of the transition probability P i¿f . For transition probabilities 0 < P i¿f < 1, a first order diffeomorphic modulation observable response preserving homotopy (D-MORPH) algorithm is utilized to investigate level sets. At the top of the control landscape, P i¿f = 1, a second order D-MORPH algorithm is presented that can explore the perfect control level set. D-MORPH is utilized to identify level set members that exhibit certain desirable secondary characteristics, e.g., minimal pulse fluence. Numerical simulations for finite level systems are presented to illustrate the variety of control behavior found across level set members.

关键词： Level set Communication system control Control systems Optimal control Topology Signal processing algorithms Centralized control Relativistic quantum mechanics Process control Signal processing

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A trace bound for the hereditary discrepancy 00

A trace bound for the hereditary discrepancy

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Proceedings of the sixteenth annual symposium on computational geometry

作者： Bernard Chazelle Alexey Lvov Department of Computer Science Princeton University and NEC Research Institute Program in Applied and Computational Mathematics Princeton University

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Covariance estimation using conjugate gradient for 3D classification in CRYO-EM

Covariance estimation using conjugate gradient for 3D classi...

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IEEE International Symposium on Biomedical Imaging

作者： Joakim Andén Eugene Katsevich Amit Singer Program in Applied and Computational Mathematics Princeton University Princeton NJ Department of Statistics Stanford University Stanford CA

Classifying structural variability in noisy projections of biological macromolecules is a central problem in Cryo-EM. In this work, we build on a previous method for estimating the covariance matrix of the three-dimensional structure present in the molecules being imaged. Our proposed method allows for incorporation of contrast transfer function and non-uniform distribution of viewing angles, making it more suitable for real-world data. We evaluate its performance on a synthetic dataset and an experimental dataset obtained by imaging a 70S ribosome complex.

关键词： Covariance matrices Eigenvalues and eigenfunctions Microscopy Image reconstruction Biology Image resolution

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Structural characterization of many-particle systems on approach to hyperuniform states

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Physical Review E 2021年第5期103卷 052126-052126页

作者： Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA.

The study of hyperuniform states of matter is an emerging multidisciplinary field, impinging on topics in the physical sciences, mathematics, and biology. The focus of this work is the exploration of quantitative descriptors that herald when a many-particle system in d-dimensional Euclidean space Rd approaches a hyperuniform state as a function of the relevant control parameter. We establish quantitative criteria to ascertain the extent of hyperuniform and nonhyperuniform distance-scaling regimes as well as the crossover point between them in terms of the “volume” coefficient A and “surface-area” coefficient B associated with the local number variance σ2(R) for a spherical window of radius R. The larger the ratio B/A, the larger the hyperuniform scaling regime, which becomes of infinite extent in the limit B/A→∞. To complement the known direct-space representation of the coefficient B in terms of the total correlation function h(r), we derive its corresponding Fourier representation in terms of the structure factor S(k), which is especially useful when scattering information is available experimentally or theoretically. We also demonstrate that the free-volume theory of the pressure of equilibrium packings of identical hard spheres that approach a strictly jammed state either along the stable crystal or metastable disordered branch dictates that such end states be exactly hyperuniform. Using the ratio B/A, as well as other diagnostic measures of hyperuniformity, including the hyperuniformity index H and the direct-correlation function length scale ξc, we study three different exactly solvable models as a function of the relevant control parameter, either density or temperature, with end states that are perfectly hyperuniform. Specifically, we analyze equilibrium systems of hard rods and “sticky” hard-sphere systems in arbitrary space dimension d as a function of density. We also examine low-temperature excited states of many-particle systems interacting with “stealt

关键词： Fluctuations & noise

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Diffusion spreadability as a probe of the microstructure of complex media across length scales

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Physical Review E 2021年第5期104卷 054102-054102页

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering, and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the “excess” spreadability, S(∞)−S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the “slowest” being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability

关键词： Complex systems

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