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Reducing bias and variance for CTF estimation in single particle cryo-EM

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arXiv 2019年

作者： Heimowitz, Ayelet Andén, Joakim Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Program in Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ08544 United States

When using an electron microscope for imaging of particles embedded in vitreous ice, the recorded image, or micrograph, is a significantly degraded version of the tomographic projection of the sample. Apart from noise, the image is affected by the optical configuration of the microscope. This transformation is typically modeled as a convolution with a point spread function. The Fourier transform of this function, known as the contrast transfer function (CTF), is oscillatory, attenuating and amplifying different frequency bands, and sometimes flipping their signs. High-resolution reconstruction requires this CTF to be accounted for, but as its form depends on experimental parameters, it must first be estimated from the micrograph. We present a new method for CTF estimation based on multitaper methods, which reduces bias and variance in the estimate. We also use known properties of the CTF and the background noise power spectrum to further reduce the variance through background subtraction and steerable basis projection. We show that the resulting power spectrum estimates better capture the zero-crossings of the CTF and yield accurate CTF estimates on several experimental micrographs. Copyright © 2019, The Authors. All rights reserved.

关键词： Linear programming

Semidefinite programming approach for the quadratic assignment problem with a sparse graph

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arXiv 2017年

作者： Bravo Ferreira, José F.S. Khoo, Yuehaw Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Stanford University StanfordCA94305 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension n2 where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension O(n) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Augmented Direction Method of Multipliers (ADMM) in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension O(n). The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy (NMR) to tackle the assignment problem. Copyright © 2017, The Authors. All rights reserved.

关键词： Combinatorial optimization

Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles

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arXiv 2019年

作者： Sharon, Nir Kileel, Joe Khoo, Yuehaw Landa, Boris Singer, Amit School of Mathematical Sciences Tel Aviv University Israel Program in Applied and Computational Mathematics Princeton University United States Department of Statistics and the College University of Chicago United States Applied Mathematics Program Yale University United States Program in Applied and Computational Mathematics Department of Mathematics Princeton University United States

Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an increasingly popular technique for determining the 3-D structure of a molecule from several noisy 2-D projections images taken at unknown viewing angles. Most reconstruction algorithms require a low-resolution initialization for the 3-D structure, which is the goal of ab initio modeling. Suggested by Zvi Kam in 1980, the method of moments (MoM) offers one approach, wherein low-order statistics of the 2-D images are computed and a 3-D structure is estimated by solving a system of polynomial equations. Unfortunately, Kam’s method suffers from restrictive assumptions, most notably that viewing angles should be distributed uniformly. Often unrealistic, uniformity entails the computation of higher-order correlations, as in this case first and second moments fail to determine the 3-D structure. In the present paper, we remove this hypothesis, by permitting an unknown, non-uniform distribution of viewing angles in MoM. Perhaps surprisingly, we show that this case is statistically easier than the uniform case, as now first and second moments generically suffice to determine low-resolution expansions of the molecule. In the idealized setting of a known, non-uniform distribution, we find an efficient provable algorithm inverting first and second moments. For unknown, non-uniform distributions, we use non-convex optimization methods to solve for both the molecule and distribution. Copyright © 2019, The Authors. All rights reserved.

关键词： Method of moments

Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations

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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

CHAOTIC TRANSPORT IN 2-DIMENSIONAL AND 3-DIMENSIONAL FLOW PAST A CYLINDER

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PHYSICS OF FLUIDS A-FLUID DYNAMICS 1991年第5期3卷 1051-1062页

作者： BATCHO, P KARNIADAKIS, GE Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

The response of transport measures (Nusselt number, drag and lift force) for two- and three-dimensional flow past a heated cylinder reaching a chaotic state is investigated numerically using a spectral element discretization at a Reynolds number Re = 500. The undisturbed two-dimensional flow remains periodic at this Reynolds number, unless a suitable forcing is applied on the naturally produced system. Three-dimensional simulations establish that three-dimensionality sets in at Re almost-equal-to 200. Successive supercritical states are established through a series of period-doublings, before a chaotic state is reached at a Re almost-equal-to 500. For the two-dimensional forced flow, all transport measures oscillate aperiodically in time and undergo a "crisis," i.e., a sudden and dramatic increase in their amplitude. The corresponding three-dimensional, naturally produced chaotic state corresponds to a less drastic change of the transport quantities with both rms and mean values lower than their two-dimensional counterparts.

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Probabilistic Theory of Mean Field Games with Applications I 1

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丛书名： Probability Theory and Stochastic Modelling

2018年

作者： René Carmona François Delarue

ISBN: (数字)9783319589206

ISBN: (纸本)9783319564371;9783030132606

Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems.

关键词： Probability Theory Calculus of Variations and Optimization Differential Equations Quantitative Economics

AN OPTIMAL SCHEDULED LEARNING RATE FOR A RANDOMIZED KACZMARZ ALGORITHM

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arXiv 2022年

作者： Marshall, Nicholas F. Mickelin, Oscar Department of Mathematics Oregon State University United States Program in Applied and Computational Mathematics Princeton University United States

We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving Ax ≈ b + Ε, where Ax = b is a consistent linear system and Ε has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases;in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-W function of an exponential. Copyright © 2022, The Authors. All rights reserved.

关键词： Linear systems

A Priori Estimates of the Population Risk for Residual Networks

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arXiv 2019年

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

Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted path norm, as the regularization term. The weighted path norm treats the skip connections and the nonlinearities differently so that paths with more nonlinearities are regularized by larger weights. The error estimates are a priori in the sense that the estimates depend only on the target function, not on the parameters obtained in the training process. The estimates are optimal, in a high dimensional setting, in the sense that both the bound for the approximation and estimation errors are comparable to the Monte Carlo error rates. A crucial step in the proof is to establish an optimal bound for the Rademacher complexity of the residual networks. Comparisons are made with existing norm-based generalization error bounds. Copyright © 2019, The Authors. All rights reserved.

关键词： Error analysis

Ordered and disordered stealthy hyperuniform point patterns across spatial dimensions

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Physical Review Research 2024年第3期6卷 033260页

作者： Peter K. Morse Paul J. Steinhardt Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute of Materials Princeton Center for Theoretical Science Program in Applied and Computational Mathematics

In previous work [Phys. Rev. X 5, 021020 (2015)] it was shown that stealthy hyperuniform systems can be regarded as hard spheres in Fourier space in the sense that the structure factor is exactly zero in a spherical region around the origin in analogy with the pair-correlation function of real-space hard spheres. While this earlier work focused on spatial dimensions d=1–4, here we extend the analysis to higher dimensions in order to make connections to high-dimensional sphere packings and the mean-field theory of glasses. We exploit this correspondence to confirm that the densest Fourier-space hard-sphere system is that of a Bravais lattice in contrast to real-space hard spheres, whose densest configuration is conjectured to be disordered. In passing, we give a concise form for the position of the first Bragg peak. We also extend the virial series previously suggested for disordered stealthy hyperuniform systems to higher dimensions in order to predict spatial decorrelation as a function of dimension. This prediction is then borne out by numerical simulations of disordered stealthy hyperuniform ground states in dimensions d=2–8, which have only recently been made possible due to a highly parallelized algorithm.

关键词： Classical statistical mechanics Disordered systems