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SUBSPACE METHOD OF MOMENTS FOR AB INITIO 3-D SINGLE-PARTICLE CRYO-EM RECONSTRUCTION

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arXiv 2024年

作者： Hoskins, Jeremy Khoo, Yuehaw Mickelin, Oscar Singer, Amit Wang, Yuguan Departments of Statistics and CCAM University of Chicago Chicago IL 60637 USA Departments of Statistics and CCAM University of Chicago Chicago IL 60637 USA Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Program in Applied and Computational Mathematics and Department of Mathematics Princeton University Princeton NJ 08544 USA Departments of Statistics University of Chicago Chicago IL 60637 USA

Cryo-electron microscopy (Cryo-EM) is a widely-used technique for recovering the 3-D structure of biological molecules from a large number of experimentally generated noisy 2-D tomographic projection images of the 3-D structure, taken from unknown viewing angles. Through computationally intensive algorithms, these observed images are processed to reconstruct the 3-D structures. Many popular computational methods rely on estimating the unknown angles as part of the reconstruction process, which becomes particularly challenging at low signal-to-noise ratio. The method of moments (MoM) offers an alternative approach that circumvents the estimation of viewing angles of individual projection images by instead estimating the underlying distribution of the viewing angles, and is robust to noise given sufficiently many images. However, the method of moments typically entails computing high-order moments of the projection images, incurring significant storage and computational costs. To mitigate this, we propose a new approach called the subspace method of moments (subspace MoM), which compresses the first three moments using data-driven low-rank tensor techniques as well as expansion into a suitable function basis. The compressed moments can be efficiently computed from the set of projection images using numerical quadrature and can be employed to jointly recover the 3-D structure and the distribution of viewing angles. We illustrate the practical applicability of the subspace MoM in numerical experiments using up to the third-order moment, which significantly improves the resolution of MoM reconstructions compared to previous approaches. Copyright © 2024, The Authors. All rights reserved.

关键词： Image reconstruction

Thermal disorder and phonon softening in the ferroelectric phase transition of lead titanate

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arXiv 2024年

作者： Xie, Pinchen Chen, Yixiao Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States AI for Science Institute Beijing China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Materials Institute Princeton University PrincetonNJ08544 United States

We report an extensive molecular dynamics study of ab-initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained on density functional theory data with modern machine learning techniques. Our simulations demonstrate that the transition has a strong order-disorder character, in agreement with diffraction experiments, and provide fresh insight into the approach to equilibrium upon crossing the transition temperature. We find that the macroscopic polarization appears/disappears due to dipolar switching at the nanometer scale. We also compute the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. For a long time, the almost ideal displacive character of the soft mode detected by Raman scattering in the paraelectric phase has been contrasted with the order-disorder character of the transition suggested by diffraction. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation due to the thermal disordering of the dipoles. The Debye relaxation feature is centered at zero frequency and appears near the transition temperature in both the ferroelectric and the paraelectric phases. © 2024, CC BY.

关键词： Ferroelectricity

A Probabilistic Approach to Discounted Infinite Horizon and Invariant Mean Field Games

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arXiv 2024年

作者： Carmona, René Tangpi, Ludovic Zhang, Kaiwen Operations Research and Financial Engineering Program in Applied and Computational Mathematics Princeton University United States Operations Research and Financial Engineering Princeton University United States

This paper considers discounted infinite horizon mean field games by extending the probabilistic weak formulation of the game as introduced in [16]. Under similar assumptions as in the finite horizon game, we prove existence and uniqueness of solutions for the extended infinite horizon game. The key idea is to construct local versions of the previously considered stable topologies. Further, we analyze how sequences of finite horizon games approximate the infinite horizon one. Under a weakened Lasry-Lions monotonicity condition, we can quantify the convergence rate of solutions for the finite horizon games to the one for the infinite horizon game using a novel stability result for mean field games. Lastly, applying our results allows to solve the invariant mean field game as well. Copyright © 2024, The Authors. All rights reserved.

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Ab initio multi-scale modeling of ferroelectrics: The case of PbTiO3

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arXiv 2022年

作者： Xie, Pinchen Chen, Yixiao Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States

We report an ab initio multi-scale study of lead titanate using the Deep Potential (DP) models, a family of machine learning-based atomistic models, trained on first-principles density functional theory data, to represent potential and polarization surfaces. Our approach includes anharmonic effects beyond the limitations of reduced models and of the linear approximation for the polarization. The calculated enthalpy, spontaneous polarization, specific heat and dielectric susceptibility agree well with experiments on single crystals. In addition, we study how the free energy depends on the polarization with enhanced sampling methods, further supporting the first-order and order-disorder character of the transition. The latter is evidenced by persistence of local dipoles above the transition temperature. The simulated free energy surface as a function of the global polarization leads to a Landau-Devonshire theory of the single domain crystal. © 2022, CC BY.

关键词： Free energy

Local Sequential MCMC for Data Assimilation with Applications in Geoscience

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arXiv 2024年

作者： Ruzayqat, Hamza Knio, Omar 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

This paper presents a new data assimilation (DA) scheme based on a sequential Markov Chain Monte Carlo (SMCMC) DA technique [36] which is provably convergent and has been recently used for filtering, particularly for high-dimensional non-linear, and potentially, non-Gaussian state-space models. Unlike particle filters, which can be considered exact methods and can be used for filtering non-linear, non-Gaussian models, SMCMC does not assign weights to the samples/particles, and therefore, the method does not suffer from the issue of weight-degeneracy when a relatively small number of samples is used. We design a localization approach within the SMCMC framework that focuses on regions where observations are located and restricts the transition densities included in the filtering distribution of the state to these regions. This results in immensely reducing the effective degrees of freedom and thus improving the efficiency. We test the new technique on high-dimensional (d ∼ 104 − 105) linear Gaussian model and non-linear shallow water models with Gaussian noise with real and synthetic observations. For two of the numerical examples, the observations mimic the data generated by the Surface Water and Ocean Topography (SWOT) mission led by NASA, which is a swath of ocean height observations that changes location at every assimilation time step. We also use a set of ocean drifters' real observations in which the drifters are moving according the ocean kinematics and assumed to have uncertain locations at the time of assimilation. We show that when higher accuracy is required, the proposed algorithm is superior in terms of efficiency and accuracy over competing ensemble methods and the original SMCMC *** Codes 62M20, 60G35, 60J20, 94A12, 93E11, 65C40 © 2024, CC BY.

关键词： Markov chains

Bayesian Perspective for Orientation Estimation in Cryo-EM and Cryo-ET

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arXiv 2024年

作者： Xu, Sheng Balanov, Amnon Bendory, Tamir Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Electrical Engineering Tel Aviv University Tel Aviv69978 Israel

Accurate orientation estimation is a crucial component of 3D molecular structure reconstruction, both in single-particle cryo-electron microscopy (cryo-EM) and in the increasingly popular field of cryo-electron tomography (cryo-ET). The dominant method, which involves searching for an orientation with maximum cross-correlation relative to given templates, falls short, particularly in low signal-to-noise environments. In this work, we propose a Bayesian framework to develop a more accurate and flexible orientation estimation approach, with the minimum mean square error (MMSE) estimator as a key example. This method effectively accommodates varying structural conformations and arbitrary rotational distributions. Through simulations, we demonstrate that our estimator consistently outperforms the cross-correlation-based method, especially in challenging conditions with low signal-to-noise ratios, and offer a theoretical framework to support these improvements. We further show that integrating our estimator into the iterative refinement in the 3D reconstruction pipeline markedly enhances overall accuracy, revealing substantial benefits across the algorithmic workflow. Finally, we show empirically that the proposed Bayesian approach enhances robustness against the "Einstein from Noise" phenomenon, reducing model bias and improving reconstruction reliability. These findings indicate that the proposed Bayesian framework could substantially advance cryo-EM and cryo-ET by enhancing the accuracy, robustness, and reliability of 3D molecular structure reconstruction, thereby facilitating deeper insights into complex biological systems. © 2024, CC BY.

关键词： Electron microscopy

GLOBAL SOLUTIONS TO THE SUPERCOOLED STEFAN PROBLEM WITH BLOW-UPS: REGULARITY AND UNIQUENESS

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Probability and Mathematical Physics

Probability and Mathematical Physics 2022年第1期3卷 171-213页

作者： Delarue, François Nadtochiy, Sergey Shkolnikov, Mykhaylo Laboratoire J.A.Dieudonné Université de Nice Sophia-Antipolis UMR CNRS-UNS No. 7351 Nice France Department of Applied Mathematics Illinois Institute of Technology Rettaliata Engineering Center Chicago IL United States ORFE Department Bendheim Center for Finance Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows us to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hölder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal 1889 paper for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. © 2022 Mathematical Sciences Publishers.

关键词： blow-ups free boundary problem heat equation interacting particle systems mean-field interaction physical solutions probabilistic reformulation self-excitation supercooled Stefan problem zero set

Robust Group Synchronization via Quadratic programming

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arXiv 2022年

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

We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation *** Codes 90C26, 90C17, 68Q87, 65C20, 90-08, 60-08 Copyright © 2022, The Authors. All rights reserved.

关键词： Quadratic programming

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