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Topological Classification of Insulators: II. Quasi-Two-Dimensional Locality

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arXiv 2024年

作者： Chung, Jui-Hui Shapiro, Jacob Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States

We provide an alternative characterization of two-dimensional locality (necessary e.g. to define the Hall conductivity of a Fermi projection) using the spectral projections of the Laughlin flux operator. Using this abstract characterization, we define generalizations of this locality, which we term quasi-2D. We go on to calculate the path-connected components of spaces of unitaries or orthogonal projections which are quasi-2D-local and find a starkly different behavior compared with the actual 2D column of the Kitaev table, exhibiting e.g., in the unitary chiral case, infinitely many Z-valued indices. Copyright © 2024, The Authors. All rights reserved.

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The quenching-activation behavior of the gradient descent dynamics for two-layer neural network models

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Institute of Big Data Research

A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neurons are divided into two groups: a group of a few "activated" neurons that dominate the dynamics and a group of background (or "quenched") neurons that support the continued activation and deactivation process. This neural network-like behavior is continued into the mildly over-parametrized regime, where it undergoes a transition to a random feature-like behavior. The quenching-activation process seems to provide a clear mechanism for "implicit regularization". This is qualitatively different from the dynamics associated with the "mean-field" scaling where all neurons participate equally and there does not appear to be qualitative changes when the network parameters are changed. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

Ab-initio Contrast Estimation and Denoising of Cryo-EM Images

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arXiv 2022年

作者： Shi, Yunpeng Singer, Amit Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States

Background and Objective: The contrast of cryo-EM images varies from one to another, primarily due to the uneven thickness of the ice layer. This contrast variation can affect the quality of 2-D class averaging, 3-D ab-initio modeling, and 3-D heterogeneity analysis. Contrast estimation is currently performed during 3-D iterative refinement. As a result, the estimates are not available at the earlier computational stages of class averaging and ab-initio modeling. This paper aims to solve the contrast estimation problem directly from the picked particle images in the ab-initio stage, without estimating the 3-D volume, image rotations, or class averages. Methods: The key observation underlying our analysis is that the 2-D covariance matrix of the raw images is related to the covariance of the underlying clean images, the noise variance, and the contrast variability between images. We show that the contrast variability can be derived from the 2-D covariance matrix and we apply the existing Covariance Wiener Filtering (CWF) framework to estimate it. We also demonstrate a modification of CWF to estimate the contrast of individual images. Results: Our method improves the contrast estimation by a large margin, compared to the previous CWF method. Its estimation accuracy is often comparable to that of an oracle that knows the ground truth covariance of the clean images. The more accurate contrast estimation also improves the quality of image restoration as demonstrated in both synthetic and experimental datasets. Conclusions: This paper proposes an effective method for contrast estimation directly from noisy images without using any 3-D volume information. It enables contrast correction in the earlier stage of single particle analysis, and may improve the accuracy of downstream *** Codes 15A29, 65D18, 62H35, 94A08 Copyright © 2022, The Authors. All rights reserved.

关键词： Covariance matrix

Fast, Accurate and Memory-Efficient Partial Permutation Synchronization

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arXiv 2022年

作者： Li, Shaohan Shi, Yunpeng Lerman, Gilad School of Mathematics University of Minnesota United States Program in Applied and Computational Mathematics Princeton University United States

MSC Codes 90C26, 90C10, 90C17, 68Q87, 65C20Previous partial permutation synchronization (PPS) algorithms, which are commonly used for multi-object matching, often involve computation-intensive and memory-demanding matrix operations. These operations become intractable for large scale structure-from-motion datasets. For pure permutation synchronization, the recent Cycle-Edge Message Passing (CEMP) framework suggests a memory-efficient and fast solution. Here we overcome the restriction of CEMP to compact groups and propose an improved algorithm, CEMP-Partial, for estimating the corruption levels of the observed partial permutations. It allows us to subsequently implement a nonconvex weighted projected power method without the need of spectral initialization. The resulting new PPS algorithm, MatchFAME (Fast, Accurate and Memory-Efficient Matching), only involves sparse matrix operations, and thus enjoys lower time and space complexities in comparison to previous PPS algorithms. We prove that under adversarial corruption, though without additive noise and with certain assumptions, CEMP-Partial is able to exactly classify corrupted and clean partial permutations. We demonstrate the state-of-the-art accuracy, speed and memory efficiency of our method on both synthetic and real datasets. Copyright © 2022, The Authors. All rights reserved.

关键词： Message passing

On the solution of Laplace’s equation in the vicinity of triple-junctions

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arXiv 2019年

作者： Hoskins, Jeremy Rachh, Manas Applied Mathematics Program Yale University United States Center for Computational Mathematics Flatiron Institute United States

In this paper we characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e. points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form tβ, where t is the distance of the point from the junction and the powers β depend only on the material properties of the media and the angles at which their boundaries meet. Moreover, we use this analysis to design efficient discretizations of boundary integral equations for Laplace transmission problems in regions with triple junctions and demonstrate the accuracy and efficiency of this algorithm with a number of examples. Copyright © 2019, The Authors. All rights reserved.

关键词： Laplace transforms

ZERO KINETIC UNDERCOOLING LIMIT IN THE SUPERCOOLED STEFAN PROBLEM

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arXiv 2020年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in [DNS19], [LS18b]. The key tools in the proof are a Feynman-Kac formula, which expresses the free boundary in the problem with kinetic undercooling through a local time of a reflected process, and a resulting comparison principle for the free boundaries with different kinetic undercooling *** Codes 80A22, 35B44, 60H30 Copyright © 2020, The Authors. All rights reserved.

关键词： Supercooling

An Ant Colony Algorithm Approach to the Vehicle Routing Problem and Variants

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An Ant Colony Algorithm Approach to the Vehicle Routing Prob...

IEEE Symposium on Systems and Information Engineering Design, SIEDS

作者： Tulio Tablada Cleon Davis Neil Palumbo Applied and Computational Mathematics Program Johns Hopkins University Electrical and Computer Engineering Program Johns Hopkins University

ISBN: (数字)9798350385144

ISBN: (纸本)9798350385151

In the basic vehicle routing problem (VRP), a vehicle must deliver goods from one centralized warehouse to multiple customers efficiently. Several VRP variants and constraints exist, including different product types, specific delivery times, multiple warehouses, vehicle fuel constraints, and pick-up from one location and delivery to another. This work proposes and demonstrates a flexible algorithm for solving the VRP via ant colony optimization (ACO) that can address many of the variants discussed above. ACO algorithms mimic the behavior of ants, learning optimal paths to a food source and back to the nest based on stigmergic behavior. This work compares a proposed, more flexible ACO algorithm to a traditional optimization algorithm that is implemented in the Google VRP solver. Readily available data were used to test and demonstrate results. Several VRP variants were implemented in both the (proposed) flexible ACO algorithm and the Google VRP solver. The flexible ACO algorithm performed better in terms of distance traveled versus the Google VRP solver for two variants and worse for three other cases. However, the Google VRP solver was not able to solve some of the VRP variant combinations considered here and failed when solving some backhaul datasets. Because the flexible ACO algorithm was able to better handle many case variations, it may be an attractive alternative optimization tool.

关键词： Ant colony optimization Vehicle routing Symbols Internet Fuels Optimization Backhaul networks

On the discretization of Laplace's equation with Neumann boundary conditions on polygonal domains

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arXiv 2020年

作者： Hoskins, Jeremy Rachh, Manas Applied Mathematics Program Yale University United States Center for Computational Mathematics Flatiron Institute United States

In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which poses a challenge for many existing methods. If the boundary data is smooth on each edge of the polygon, then in the vicinity of each corner the solution to the corresponding boundary integral equation has an expansion in terms of certain (analytically available) singular powers. Using the known behavior of the solution, universal discretizations have been constructed for the solution of the Dirichlet problem. However, the leading order behavior of solutions to the Neumann problem is O(tμ) for μ ∈ (-1/2;0) depending on the angle at the corner (compared to O(C + tμ) with μ > 1=2 for the Dirichlet problem);this presents a significant challenge in the design of universal discretizations. Our approach is based on using the discretization for the Dirichlet problem in order to compute a solution in the "weak sense" by solving an adjoint linear system;namely, it can be used to compute inner products with smooth functions accurately, but it cannot be interpolated. Furthermore we present a procedure to obtain accurate solutions arbitrarily close to the corner, by solving a sequence of small local subproblems in the vicinity of that corner. The results are illustrated with several numerical examples. Copyright © 2020, The Authors. All rights reserved.

关键词： Boundary value problems

A SINGULAR TWO-PHASE STEFAN PROBLEM AND PARTICLES INTERACTING THROUGH THEIR HITTING TIMES

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arXiv 2022年

作者： Baker, Graeme Shkolnikov, Mykhaylo Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States ORFE Department Bendheim Center for Finance and Program in Applied & Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel *** Codes 82C22, 60H30, 35K10, 35K60, 91G80 Copyright © 2022, The Authors. All rights reserved.

关键词： Stochastic systems