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作者： Bandeira, A.S. Scheinberg, K. Vicente, L.N. Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Industrial and Systems Engineering Lehigh University Harold S. Mohler Laboratory 200 West Packer Avenue Bethlehem PA 18015-1582 United States CMUC Department of Mathematics University of Coimbra 3001-501 Coimbra Portugal

Interpolation-based trust-region methods are an important class of algorithms forDerivative-Free Optimizationwhich rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse structure in the Hessian matrix. To be able to exploit Hessian sparsity, existing optimization approaches require the knowledge of the sparsity structure. The goal of this paper is to develop and analyze a method where the sparse models are constructed automatically. The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements. Such a recovery is achieved by minimizing the l1-norm of a vector subject to the measurements constraints. We suggest an approach for building sparse quadratic polynomial interpolationmodels byminimizing the l1-norm of the entries of themodel Hessian subject to the interpolation conditions. We show that this procedure recovers accurate models when the function Hessian is sparse, using relatively few randomly selected sample points. Motivated by this result, we developed a practical interpolation-based trust-region method using deterministic sample sets and minimum l1-norm quadratic models. Our computational results show that the new approach exhibits a promising numerical performance both in the general case and in the sparse one. © Springer and Mathematical Optimization Society 2012.

关键词： Interpolation

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Diffusion curvature for estimating local curvature in high dimensional data 22

Diffusion curvature for estimating local curvature in high d...

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Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Dhananjay Bhaskar Kincaid MacDonald Oluwadamilola Fasina Dawson Thomas Bastian Rieck Ian Adelstein Smita Krishnaswamy Department of Genetics Yale University Department of Mathematics Yale University Applied Mathematics Program Yale University Department of Mathematics Department of Physics Yale University Institute of AI for Health Helmholtz Pioneer Campus Helmholtz Munich Department of Computer Science Department of Genetics Applied Mathematics Program Program for Computational Biology and Bioinformatics Yale University

ISBN: (纸本)9781713871088

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes.

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Chaotic behavior of interacting elliptical instability modes

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The Physics of Fluids 1988年第1期31卷 6-8页

作者： Carl S. Hellberg Steven A. Orszag Department of Physics Princeton University Princeton New Jersey 08544 Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

A three‐mode projection of the Navier–Stokes equations for nonlinear perturbations to an elliptical vortex is studied numerically. It is found that, as the Reynolds number increases, the perturbations undergo a sequence of period doublings leading to chaos according to the Feigenbaum scenario [J. Statis. Phys. 19, 25 (1978); Phys. Lett. 74 A, 375 (1979)].

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MOMENT-BASED METRICS FOR MOLECULES COMPUTABLE FROM CRYO-EM IMAGES

arXiv

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arXiv 2024年

作者： Zhang, Andy Mickelin, Oscar Kileel, Joe Verbeke, Eric J. Marshall, Nicholas F. Gilles, Marc Aurèle Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Oden Institute University of Texas At Austin AustinTX78712 United States Department of Mathematics Oregon State University CorvallisOR97331 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity. Copyright © 2024, The Authors. All rights reserved.

关键词： Method of moments

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Domain Formation in the Plasma Membrane: Roles of Nonequilibrium Lipid Transport and Membrane Proteins

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Physical Review Letters 2008年第17期100卷 178102-178102页

作者： Jun Fan Maria Sammalkorpi Mikko Haataja Department of Mechanical and Aerospace Engineering Princeton Institute for the Science and Technology of Materials (PRISM) and Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

Compositional lipid domains (“lipid rafts”) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes and lifetimes of these spatially extended domains are poorly understood at the moment. Here we show that the competition between phase separation in an immiscible lipid system and active cellular lipid transport processes naturally leads to the formation of such domains. Furthermore, we demonstrate that local interactions with immobile membrane proteins can spatially localize the rafts and lead to further clustering.

关键词： Lipids

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Densest local sphere-packing diversity. II. Application to three dimensions

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Physical Review E 2011年第1期83卷 011304-011304页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry and Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N=1054. In the predecessor to this paper [A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N=348. Here we analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12–6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the fcc Barlow packing. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance Rmin(N). In particular, we find two “maracas” packings at N=77 and N=93, each consisting of a few unjammed sphere

关键词： DENSITY functionals SPHERES MATHEMATICAL models EUCLIDEAN metric CORRELATION (Statistics) SYMMETRY

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Influence of nonequilibrium lipid transport, membrane compartmentalization, and membrane proteins on the lateral organization of the plasma membrane

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Physical Review E 2010年第1期81卷 011908-011908页

Compositional lipid domains (lipid rafts) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes, lifetimes, and spatial localization of these domains are rather poorly understood at the moment. We propose a robust mechanism for the formation of finite-sized lipid raft domains in plasma membranes, the competition between phase separation in an immiscible lipid system and active cellular lipid transport processes naturally leads to the formation of such domains. Simulations of a continuum model reveal that the raft size distribution is broad and the average raft size is strongly dependent on the rates of cellular and interlayer lipid transport processes. We demonstrate that spatiotemporal variations in the recycling may enable the cell to localize larger raft aggregates at specific parts along the membrane. Moreover, we show that membrane compartmentalization may further facilitate spatial localization of the raft domains. Finally, we demonstrate that local interactions with immobile membrane proteins can spatially localize the rafts and lead to further clustering.

关键词： Cell membranes

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Geometrical ambiguity of pair statistics. II. Heterogeneous media

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Physical Review E 2010年第1期82卷 011106-011106页

作者： Yang Jiao Frank H. Stillinger Salvatore Torquato Department of Chemistry and Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics and Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA

In the first part of this series of two papers [Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 011105 (2010)], we considered the geometrical ambiguity of pair statistics associated with point configurations. Here we focus on the analogous problem for heterogeneous media (materials). Heterogeneous media are ubiquitous in a host of contexts, including composites and granular media, biological tissues, ecological patterns, and astrophysical structures. The complex structures of heterogeneous media are usually characterized via statistical descriptors, such as the n-point correlation function Sn. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function S2 of a target medium. Recently, general claims of the uniqueness of reconstructions using S2 have been made based on numerical studies, which implies that S2 suffices to uniquely determine the structure of a medium within certain numerical accuracy. In this paper, we provide a systematic approach to characterize the geometrical ambiguity of S2 for both continuous two-phase heterogeneous media and their digitized representations in a mathematically precise way. In particular, we derive the exact conditions for the case where two distinct media possess identical S2, i.e., they form a degenerate pair. The degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations, respectively. By examining these equations and constructing their rigorous solutions for specific examples, we conclusively show that in general S2 is indeed not sufficient information to uniquely determine the structure of the medium, which is consistent with the results of our recent study on heterogeneous-media reconstruction [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. U.S.A. 106, 17634 (2009)]. The analytical examples include complex patterns composed of building blo

关键词： Inverse problems

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Disordered strictly jammed binary sphere packings attain an anomalously large range of densities

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Physical Review E 2013年第2期88卷 022205-022205页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton Institute for the Science and Technology of Materials Department of Physics Princeton Center for Theoretical Science Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions 0.634≤ϕ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain α and small sphere relative number concentrations x approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about 0.624 over the majority of the α-x plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the α-x plane where the backbone is not roughly constant, we find that backbone packing fractions range from about 0.606 to 0.829, with the volume of rattler spheres comprising between 1.6% and 26.9% of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such hig

关键词： PACKING fractions SHEAR (Mechanics) ALGORITHMS SOLID propellants CONCRETE CERAMICS

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Organizing principles for dense packings of nonspherical hard particles: Not all shapes are created equal

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Physical Review E 2012年第1期86卷 011102-011102页

作者： Salvatore Torquato Yang Jiao []Department of Chemistry Department of Physics Princeton Center for Theoretical Science Program of Applied and Computational Mathematics Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, “lens-shaped” particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.

关键词： PARTICLES (Nuclear physics) PACKINGS (Chromatography) CONCAVE functions CATALANS POLYHEDRA

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