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8th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2012

作者： Kiss, Gábor Giannelli, Carlotta Juẗtler, Bert Doctoral Program Computational Mathematics Johannes Kepler University Linz Altenberger Str. 69 4040 Linz Austria Institute of Applied Geometry Johannes Kepler University Linz Altenberger Str. 69 4040 Linz Austria

ISBN: (纸本)9783642543814

Tensor-product B-spline surfaces are commonly used as standard modeling tool in Computer Aided Geometric Design and for numerical simulation in Isogeometric Analysis. However, when considering tensor-product grids, there is no possibility of a localized mesh refinement without propagation of the refinement outside the region of interest. The recently introduced truncated hierarchical B-splines (THB-splines) [5] provide the possibility of a local and adaptive refinement procedure, while simultaneously preserving the partition of unity property. We present an effective implementation of the fundamental algorithms needed for the manipulation of THB-spline representations based on standard data structures. By combining a quadtree data structure - which is used to represent the nested sequence of subdomains - with a suitable data structure for sparse matrices, we obtain an efficient technique for the construction and evaluation of THB-splines. © Springer-Verlag 2014.

关键词： Data structures

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Geometry based data generation 32

Geometry based data generation

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32nd Conference on Neural Information Processing Systems, NeurIPS 2018

作者： Lindenbaum, Ofir Wolf, Guy Stanley, Jay S. Krishnaswamy, Smita Applied Mathematics Program Yale University New HavenCT06511 United States Computational Biology and Bioinformatics Program Yale University New HavenCT06510 United States Departments of Genetics and Computer Science Yale University New HavenCT06510 United States

We propose a new type of generative model for high-dimensional data that learns a manifold geometry of the data, rather than density, and can generate points evenly along this manifold. This is in contrast to existing generative models that represent data density, and are strongly affected by noise and other artifacts of data collection. We demonstrate how this approach corrects sampling biases and artifacts, thus improves several downstream data analysis tasks, such as clustering and classification. Finally, we demonstrate that this approach is especially useful in biology where, despite the advent of single-cell technologies, rare subpopulations and gene-interaction relationships are affected by biased sampling. We show that SUGAR can generate hypothetical populations, and it is able to reveal intrinsic patterns and mutual-information relationships between genes on a single-cell RNA sequencing dataset of hematopoiesis. © 2018 Curran Associates Inc..All rights reserved.

关键词： Genes

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Simultaneous tracking of movement and gene expression in multiple Drosophila melanogaster flies using GFP and DsRED fluorescent reporter transgenes

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BMC Research Notes 2009年第1期2卷 1-11页

作者： Grover, Dhruv Yang, Junsheng Ford, Daniel Tavaré, Simon Tower, John Molecular and Computational Biology Program Department of Biological Sciences University of Southern California Los Angeles CA 90089-2910 United States Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge CB3 0WA United Kingdom

Background. Fluorescent proteins such as GFP (Green Fluorescent Protein) and DsRED (Discosoma *** Fluorescent Protein) are often used as reporter molecules for transgene expression in Drosophila and other species. We have recently reported methods that allow simultaneous tracking of animal movement and GFP expression in real time, however the assay was limited to single animals and a single transgene. Numerous studies would be facilitated by methods that allow for assay of multiple animals and multiple transgenes. Findings. Here we report an improved fly video tracking system that allows multiple transgenic flies to be tracked simultaneously using visible light, GFP fluorescence and DsRED fluorescence. The movement of multiple flies could be accurately tracked at real time rates, while simultaneously assaying the expression level of two different transgenes marked with GFP and DsRED. The individual flies could be accurately tracked and distinguished even during periods when transgene fluorescence was undetected. For example, characteristic patterns of hsp70 and hsp22 transgene induction could be simultaneously quantified and correlated with animal movement in aging flies, and as groups of flies died due to dessication/starvation. Conclusion. The improved methods allow for more efficient assay of the correlation between gene expression, behavior, aging and mortality: multiple animals can be assayed with simultaneous quantification of multiple transgenes using GFP and DsRED fluorescence. These methods should allow for increased flexibility in experimental designs. For example, in the future it should be possible to use gene expression levels to predict remaining life span more accurately, and to quantify gene expression changes caused by interactions between animals in real time. © 2009 Tower et al.

关键词： Green Fluorescent Protein Extended Kalman Filter Green Fluorescent Protein Expression Green Fluorescent Protein Fluorescence Daily Activity Pattern

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Characterization of maximally random jammed sphere packings: Voronoi correlation functions

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Physical Review E 2014年第5期90卷 052120-052120页

作者： Michael A. Klatt Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in th

关键词： Probability density function

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An L2 Analysis of Reinforcement Learning in High Dimensions with Kernel and Neural Network Approximation

arXiv

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arXiv 2021年

作者： Long, Jihao Han, Jiequn Weinan, E. Program of Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States Center for Computational Mathematics Flatiron Institute School of Mathematical Sciences Peking University

Reinforcement learning (RL) algorithms based on high-dimensional function approximation have achieved tremendous empirical success in large-scale problems with an enormous number of states. However, most analysis of such algorithms gives rise to error bounds that involve either the number of states or the number of features. This paper considers the situation where the function approximation is made either using the kernel method or the two-layer neural network model, in the context of a fitted Q-iteration algorithm with explicit regularization. We establish an Õ(H3|A|1/4 n- 1/4) bound for the optimal policy with Hn samples, where H is the length of each episode and |A| is the size of action space. Our analysis hinges on analyzing the L2 error of the approximated Q-function using n data points. Even though this result still requires a finite-sized action space, the error bound is independent of the dimensionality of the state space. Copyright © 2021, The Authors. All rights reserved.

关键词： Error analysis

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Comment on ‘‘Bifurcation structure and the Eckhaus instability’’

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Physical Review Letters 1991年第8期67卷 1051-1051页

作者： Laurette S. Tuckerman Dwight Barkley Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

A Comment on the Letter by Tsiveriotis, and Brown, Phys. Rev. Lett. 63, 2048 (1989).

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Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

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Physical Review E 2018年第1期97卷 012118-012118页

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

关键词： Classical statistical mechanics

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Confined disordered strictly jammed binary sphere packings

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Physical Review E 2015年第6期92卷 062207-062207页

作者： D. Chen S. Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Disordered jammed packings under confinement have received considerably less attention than their bulk counterparts and yet arise in a variety of practical situations. In this work, we study binary sphere packings that are confined between two parallel hard planes and generalize the Torquato-Jiao (TJ) sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)] to obtain putative maximally random jammed (MRJ) packings that are exactly isostatic with high fidelity over a large range of plane separation distances H, small to large sphere radius ratio α, and small sphere relative concentration x. We find that packing characteristics can be substantially different from their bulk analogs, which is due to what we term “confinement frustration.” Rattlers in confined packings are generally more prevalent than those in their bulk counterparts. We observe that packing fraction, rattler fraction, and degree of disorder of MRJ packings generally increase with H, though exceptions exist. Discontinuities in the packing characteristics as H varies in the vicinity of certain values of H are due to associated discontinuous transitions between different jammed states. When the plane separation distance is on the order of two large-sphere diameters or less, the packings exhibit salient two-dimensional features; when the plane separation distance exceeds about 30 large-sphere diameters, the packings approach three-dimensional bulk packings. As the size contrast increases (as α decreases), the rattler fraction dramatically increases due to what we call “size-disparity” frustration. We find that at intermediate α and when x is about 0.5 (50-50 mixture), the disorder of packings is maximized, as measured by an order metric ψ that is based on the number density fluctuations in the direction perpendicular to the hard walls. We also apply the local volume-fraction variance στ2(R) to characterize confined packings and find that these packings possess essentially the same level of

关键词： Packing

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Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

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Physical Review E 2018年第1期98卷 013307-013307页

作者： Zheng Ma Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The quantitative characterization of the microstructure of random heterogeneous media in d-dimensional Euclidean space Rd via a variety of n-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface Fss and surface-void Fsv correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-r behaviors of these functions, which are linked to the mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the d-dimensional statistically homogeneous and isotropic system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated “stealthy” patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.

关键词： Structural properties

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Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions

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Physical Review Letters 2020年第6期125卷 068002-068002页

作者： Zheng Ma Enrique Lomba Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

关键词： Applications of soft matter Colloids Disordered systems

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