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Comment on “Jamming at zero temperature and zero applied stress: The epitome of disorder”

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Physical Review E 2004年第4期70卷 043301-043301页

作者： Aleksandar Donev Salvatore Torquato Frank H. Stillinger Robert Connelly Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Materials Institute Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Mathematics Cornell University Ithaca New York 14853 USA

O’Hern, Silbert, Liu, and Nagel [Phys. Rev. E. 68, 011306 (2003)] claim that a special point J of a “jamming phase diagram” (in density, temperature, stress space) is related to random close packing of hard spheres and that it represents, for their suggested definitions of jammed and random, the recently introduced maximally random jammed state. We point out several difficulties with their definitions and question some of their claims. Furthermore, we discuss the connections between their algorithm and other hard-sphere packing algorithms in the literature.

关键词： .RCp Phys Rev Hard Spheres hard-sphere well-de jamming point out maximally claim special point nitions jammed MRJ OSLN

Group synchronization on grids

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arXiv 2017年

作者： Abbe, Emmanuel Massoulie, Laurent Montanari, Andrea Sly, Allan Srivastava, Nikhil Program in Applied and Computational Mathematics EE Department Princeton University Inria MSR-Inria Joint Centre Department of Electrical Engineering Department of Sttistics Stanford University Department of Mathematics Princeton University Department of Mathematics University of California Berkeley United States

Group synchronization requires to estimate unknown elements (θv)v∈V of a compact group G associated to the vertices of a graph G = (V, E), using noisy observations of the group differences associated to the edges. This model is relevant to a variety of applications ranging from structure from motion in computer vision to graph localization and positioning, to certain families of community detection problems. We focus on the case in which the graph G is the d-dimensional grid. Since the unknowns θv are only determined up to a global action of the group, we consider the following weak recovery question. Can we determine the group difference θ−u1θv between far apart vertices u, v better than by random guessing? We prove that weak recovery is possible (provided the noise is small enough) for d ≥ 3 and, for certain finite groups, for d ≥ 2. Viceversa, for some continuous groups, we prove that weak recovery is impossible for d = 2. Finally, for strong enough noise, weak recovery is always impossible. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

On singularity formation in a Hele-Shaw model

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arXiv 2017年

作者： Constantin, Peter Elgindi, Tarek Nguyen, Huy Vicol, Vlad Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Uc San Diego San DiegoCA92093 United States Program In Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in [CDG+93] and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h = 2h(x, t) of a thin neck of fluid, ∂th + ∂x(h ∂x3h) = 0, for x ∈ (−1, 1) and t ≥ 0. The boundary conditions fix the neck height and the pressure jump: h(±1, t) = 1, ∂x2h(±1, t) = P > 0. We prove that starting from smooth and positive h, as long as h(x, t) > 0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., inf[−1,1]×[0,T∗) h = 0, for some T∗ ∈ (0, ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies. August 30, 2017.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

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Optimal estimation of Eddy viscosity and friction coefficients for a Quasi‐three‐dimensional numerical tidal model

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Atmosphere 1995年第3期33卷 581-611页

作者： R.W. Lardner[a] & Y. Song[a][b]* [a] Applied and Computational Mathematics Program Simon Fraser University Burnaby B.C. V5A 1S6 Canada [b] Institute of Marine and Coastal Sciences Rutgers University New Brunswick NJ

It is shown that the parameters in a quasi‐three‐dimensional numerical tidal model can be estimated accurately by assimilation of data from current meters and tide gauges. The tidal model considered is a semi‐linearized one in which advective nonlinearities are neglected but nonlinear bottom friction is included. The parameters estimated are the eddy viscosity, bottom friction coefficient, water depth and wind drag coefficient, the first three of which are allowed to be position‐dependent. The adjoint method is used to construct the gradient of a cost function defined as a certain norm of the difference between computed and observed current and surface elevations. On the basis of a number of tests, it is shown that very effective estimation of the nodal values of the parameters can be achieved using the current data either alone or in combination with elevation data. When random errors are introduced into the data, the estimated parameters are quite sensitive to the magnitude of the errors, and in particular the eddy viscosity is unstably sensitive. The sensitivity of the viscosity can be stabilized by incorporating an appropriate penalty term in the cost function or alternatively by reducing the number of estimated viscosity values via a finite element approximation. Once stabilized, the sensitivity of the estimates to data errors is significantly reduced by assimilating a longer data record.RésuméOn montre que les paramètres d'un modèle numérique quasi trois dimensions de la marée peuvent être estimés avec exactitude en assimilant les données de courantomètres et de marégraphes. Le modèle de la marée examiné est semi linéarisé et les non linéarités advectives y sont négligées mais la friction de fond non linéaire est incluse. On a estimé les coeffecients de viscosité, de friction de fond, de la profondeur de l'eau et de traînée du vent, allouant les trois premiers d'être translatables. La méthode adjointe est utilisée pour construire le gradient d'une fonction de

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Communication-computation efficient gradient coding

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arXiv 2018年

作者： Ye, Min Abbe, Emmanuel Department of Electrical Engineering Princeton University PrincetonNJ United States Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University School of Mathematics Institute for Advanced Study PrincetonNJ08544 United States

This paper develops coding techniques to reduce the running time of distributed learning tasks. It characterizes the fundamental tradeoff to compute gradients (and more generally vector summations) in terms of three parameters: computation load, straggler tolerance and communication cost. It further gives an explicit coding scheme that achieves the optimal tradeoff based on recursive polynomial constructions, coding both across data subsets and vector components. As a result, the proposed scheme allows to minimize the running time for gradient computations. Implementations are made on Amazon EC2 clusters using Python with mpi4py package. Results show that the proposed scheme maintains the same generalization error while reducing the running time by 32% compared to uncoded schemes and 23% compared to prior coded schemes focusing only on stragglers (Tandon et al., ICML 2017). Copyright © 2018, The Authors. All rights reserved.

关键词： Python

First-principles characterization of Ni diffusion kinetics in β-NiAl

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Physical Review B 2008年第18期78卷 184105-184105页

作者： Kristen A. Marino Emily A. Carter Department of Chemical Engineering Princeton University Princeton New Jersey 08544-5263 USA Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544-5263 USA

First-principles density functional theory calculations are performed to examine five postulated diffusion mechanisms for Ni in NiAl: next-nearest-neighbor (NNN) jumps, the triple defect mechanism, and three variants of the six-jump cycle. In contrast to most previous theoretical work, which employed empirical interatomic potentials, we provide a more accurate nonempirical description of the mechanisms. For each pathway, we calculate the activation energy and the pre-exponential factor for the diffusion constant. Although our quantum mechanics calculations are performed at 0 K, we show that it is critical to include the effect of temperature on the pre-exponential factor. We predict that the triple defect mechanism and [110] six-jump cycle both are likely contributors to Ni diffusion in NiAl since their activation energies and pre-exponential factors are in very good agreement with experimental data. Although the activation energy and pre-exponential factor of NNN jumps agree well with experiment, experimental evidence suggests that this is not a dominant contributor to Ni diffusion. Lastly, the activation energies of the [100] bent and straight six-jump cycles are 1 eV higher than the experimental value, allowing us to exclude both [100] cycle mechanisms.

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Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes

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arXiv 2023年

作者： Jasra, Ajay Kamatani, Kengo Maama, Mohamed Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Institute of Statistical Mathematics Tokyo190-0014 Japan

In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes (PDMPs). In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in [20], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameter of that algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples. Copyright © 2023, The Authors. All rights reserved.

关键词： Ordinary differential equations

Solving many-electron Schrödinger equation using deep neural networks

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arXiv 2018年

作者： Han, Jiequn Zhang, Linfeng Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

We introduce a new family of trial wave-functions based on deep neural networks to solve the many-electron Schrödinger equation. The Pauli exclusion principle is dealt with explicitly to ensure that the trial wave-functions are physical. The optimal trial wave-function is obtained through variational Monte Carlo and the computational cost scales quadratically with the number of electrons. The algorithm does not make use of any prior knowledge such as atomic orbitals. Yet it is able to represent accurately the ground-states of the tested systems, including He, H2, Be, B, LiH, and a chain of 10 hydrogen atoms. This opens up new possibilities for solving large-scale many-electron Schrödinger equation. Copyright © 2018, The Authors. All rights reserved.

关键词： Wave functions

Reed-Muller codes polarize

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arXiv 2019年

作者： Abbe, Emmanuel Ye, Min Mathematics Institute School of Computer and Communication Sciences Epfl Switzerland Program in Applied and Computational Mathematics Department of Electrical Engineering Princeton University United States Department of Electrical Engineering Princeton University PrincetonNJ United States

Reed-Muller (RM) codes and polar codes are generated by the same matrix Gm= [1 0/1 1]⊗mbut using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in Gm;while this is a more elaborate and channel-dependent rule, the top-to-down ordering has the advantage of making the conditional mutual information polarize, giving directly a capacity-achieving code on any binary memoryless symmetric channel (BMSC). RM codes are yet to be proved to have such property. In this paper, we reconnect RM codes to polarization theory. It is shown that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in Gm, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that Gmcontains another code, different than the polar code and called here the twin code, that is provably capacity-achieving on any BMSC. This proves a necessary condition for RM codes to achieve capacity on BMSCs. It further gives a sufficient condition if the rows with largest conditional mutual information correspond to the heaviest rows, i.e., if the twin code is the RM code. We show here that the two codes bare similarity with each other and give further evidence that they are likely the same. Copyright © 2019, The Authors. All rights reserved.

关键词： Forward error correction