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A Stochastic Approximate Expectation-Maximization for Structure Determination Directly from Cryo-EM Micrographs

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arXiv 2023年

作者： Kreymer, Shay Singer, Amit Bendory, Tamir The School of Electrical Engineering Tel Aviv University Tel Aviv Israel The Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

A single-particle cryo-electron microscopy (cryoEM) measurement, called a micrograph, consists of multiple two-dimensional tomographic projections of a three-dimensional molecular structure at unknown locations, taken under unknown viewing directions. All existing cryo-EM algorithmic pipelines first locate and extract the projection images, and then reconstruct the structure from the extracted images. However, if the molecular structure is small, the signal-to-noise ratio (SNR) of the data is very low, and thus accurate detection of projection images within the micrograph is challenging. Consequently, all standard techniques fail in low-SNR regimes. To recover molecular structures from measurements of low SNR, and in particular small molecular structures, we devise a stochastic approximate expectation-maximization algorithm to estimate the three-dimensional structure directly from the micrograph, bypassing locating the projection images. We corroborate our computational scheme with numerical experiments, and present successful structure recoveries from simulated noisy measurements. © 2023, CC BY.

关键词： Molecular structure

NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

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Journal of computational mathematics 2019年第4期37卷 506-523页

作者： Weimin Han Ziping Huang Cheng Wang Wei Xu Program in Applied Mathematical and Computational Sciences (A MCS)& Department of Mathematics University of Iowa Iowa City IA 522/2 USA School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College J taxing 314051 China School of Mathematical Sciences Tojigji University Shanghai 200092 China School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College Jiaxing 314051 China

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

关键词： Hemivariational inequality Interior semipermeability Boundary semipermeability Finite element method Error estimate

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

Direct minimization of the optimized effective problem based on efficient finite differences

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Physical Review B 2011年第16期84卷 165122-165122页

作者： Chen Huang Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and the Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

To avoid the difficult-to-solve optimized effective potential (OEP) integral equation, we introduce an efficient direct minimization scheme for performing OEP calculations within Kohn–Sham density functional theory (KS-DFT). We reformulated the functional derivative of the total energy with respect to the KS effective potential in terms of efficient finite differences. Our method only uses the orbitals involved in the construction of the KS exchange-correlation functionals. We demonstrate our scheme by performing exact-exchange OEP for sodium clusters, in which only occupied KS orbitals are needed to obtain the OEP. Our efficient direct minimization scheme should aid future development of orbital-dependent density functionals and render OEP to be a practical choice for various applications.

关键词： INTEGRAL equations DENSITY functionals TOTAL energy systems (On-site electric power production) FINITE differences PHYSICS

IMPROVED EFFICIENCY OF MULTILEVEL MONTE CARLO FOR STOCHASTIC PDE THROUGH STRONG PAIRWISE COUPLING

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arXiv 2021年

作者： Chada, Neil K. Hoel, Håkon Jasra, Ajay Zouraris, Georgios E. Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Mathematics for Uncertainty Quantification RWTH Aachen University Aachen Germany Department of Mathematics and Applied Mathematics Voutes Campus Crete HeraklionGR-700 13 Greece

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when MLMC is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the MLMC method with strong pairwise coupling that was developed and studied numerically on filtering problems in [Chernov et al., Numer. Math., 147 (2021), 71-125], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this *** Codes 65C05, 65N35, 60H35, 65C30 © 2021, CC BY.

关键词： Monte Carlo methods

Angular momentum dependent orbital-free density functional theory: Formulation and implementation

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Physical Review B 2014年第15期89卷 155112-155112页

作者： Youqi Ke Florian Libisch Junchao Xia Emily A. Carter []Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

Orbital-free density functional theory (OFDFT) directly solves for the ground-state electron density. It scales linearly with respect to system size, providing a promising tool for large-scale material simulations. Removal of the orbitals requires use of approximate noninteracting kinetic energy density functionals. If replacing ionic cores with pseudopotentials, removal of the orbitals also requires these pseudopotentials to be local. These are two severe challenges to the capabilities of conventional OFDFT. While main group elements are often well described within conventional OFDFT, transition metals remain intractable due to their localized d electrons. To advance the accuracy and general applicability of OFDFT, we have recently reported a general angular momentum dependent formulation as a next-generation OFDFT. In this formalism, we incorporate the angular momenta of electrons by devising a hybrid scheme based on a muffin tin geometry: inside spheres centered at the ionic cores, the electron density is expanded in a set of atom-centered basis functions combined with an onsite density matrix. The explicit treatment of the angular momenta of electrons provides an important basis for accurately describing the important ionic core region, which is not possible in conventional OFDFT. In addition to the conventional OFDFT total energy functional, we introduce a nonlocal energy term containing a set of angular momentum dependent energies to correct the errors due to the approximate kinetic energy density functional and local pseudopotentials. Our approach greatly increases the accuracy of OFDFT while largely preserving its numerical simplicity. Here, we provide details of the theoretical formulation and practical implementation, including the hybrid scheme, the derivation of the nonlocal energy term, the choice of basis functions, the direct minimization of the total energy, the procedure to determine the angular momentum dependent energies, the force formula with Pu

关键词： ANGULAR momentum (Nuclear physics) DENSITY functional theory GROUND state (Quantum mechanics) ELECTRON density SIMULATION methods & models DENSITY functionals

Modeling of multilayer ion etching processes

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Journal of Vacuum Science & Technology B: Microelectronics a...

Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena 1993年第4期11卷 1310-1313页

作者： S. J. Sherwin E. Barouch G. E. Karniadakis S. A. Orszag Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08540 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08540

Numerical simulation is used to model ion etching in trilayer lithography. The simulations are capable of capturing the evolution of the boundary between two materials as well as the physically observed phonemena reactive ion etching lag and undercutting. Numerical results are compared with experimental data and a good agreement is found except close to the material interface where the slope of the surface is large. This error is attributed to a purely energy dependent yield used in the simulations.

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Possible origin of the discrepancy in Peierls stresses of fcc metals: First-principles simulations of dislocation mobility in aluminum

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Physical Review B 2013年第6期88卷 064106-064106页

作者： Ilgyou Shin Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

Dislocation motion governs the strength and ductility of metals, and the Peierls stress (σp) quantifies dislocation mobility. σp measurements carry substantial uncertainty in face-centered cubic (fcc) metals, and σp values can differ by up to two orders of magnitude. We perform first-principles simulations based on orbital-free density functional theory (OFDFT) to calculate the most accurate currently possible σp for the motion of 12〈110〉111 dislocations in fcc Al. We predict the σps of screw and edge dislocations (dissociated in their equilibrium state) to be 1.9×10−4G and 4.9×10−5G, respectively (G is the shear modulus). These values fall within the range of measurements from mechanical deformation tests (10−4–10−5G). OFDFT also finds a new metastable structure for a screw dislocation not seen in earlier simulations, in which a dislocation core on the glide plane does not dissociate into partials. The corresponding σp for this undissociated dislocation is predicted to be 1.1×10−2G, which agrees with typical Bordoni peak measurements (10−2–10−3G). The calculated σps for dissociated and undissociated screw dislocations differ by two orders of magnitude. The presence of undissociated, as well as dissociated, screw dislocations may resolve the decades-long mystery in fcc metals regarding the two orders of magnitude discrepancy in σp measurements.

关键词： ALUMINUM DUCTILITY STRENGTH of materials STRAINS & stresses (Mechanics) DISLOCATIONS in metals FACE centered cubic structure

Recursive projection-aggregation decoding of Reed-Muller codes

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arXiv 2019年

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

We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding and code concatenation techniques. We run our main algorithm for AWGN channels and Binary Symmetric Channels at the short code length (≤ 1024) and low code rate (≤ 0.5) regime. Simulation results show that the new algorithm not only outperforms the previous decoding algorithms for RM codes, it also outperforms the optimal decoder for polar codes (SCL+CRC) with the same parameters by a wide margin. The performance of the new algorithm for RM codes in those regimes is in fact close to that of the maximal likelihood decoder. Finally, the new decoder naturally allows for parallel implementations. Copyright © 2019, The Authors. All rights reserved.

关键词： Decoding