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arXiv 2023年

作者： Del Moral, Pierre Hu, Shulan Jasra, Ajay Ruzayqat, Hamza Wang, Xinyu Institut de Mathematiques de Bordeaux Bordeaux33405 France School of Statistics and Mathematics Zhongnan University of Economics and Law China 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 Wenlan School of Business Zhongnan University of Economics and Law China

We consider the long time behavior of Wong-Zakai approximations of stochastic differential equations. These piecewise smooth diffusion approximations are of great importance in many areas, such as those with ordinary differential equations associated to random smooth fluctuations;e.g. robust filtering problems. In many examples, the mean error estimate bounds that have been derived in the literature can grow exponentially with respect to the time horizon. We show in a simple example that indeed mean error estimates do explode exponentially in the time parameter, i.e. in that case a Wong-Zakai approximation is only useful for extremely short time intervals. Under spectral conditions, we present some quantitative time-uniform convergence theorems, i.e. time-uniform mean error bounds, yielding what seems to be the first results of this type for Wong-Zakai diffusion *** Codes 60H35(Primary), 65C30(Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Brownian movement

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Jammed lattice sphere packings

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Physical Review E 2013年第6期88卷 062151-062151页

作者： Yoav Kallus Étienne Marcotte Salvatore Torquato Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.

关键词： LATTICE theory SPHERE packings ISOSTATIC pressing STABILITY (Mechanics) MATHEMATICAL models GLASS transitions

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POINT SPREAD FUNCTION APPROXIMATION OF HIGH RANK HESSIANS WITH LOCALLY SUPPORTED NON-NEGATIVE INTEGRAL KERNELS

arXiv

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arXiv 2023年

作者： Alger, Nick Hartland, Tucker Petra, Noemi Ghattas, Omar Oden Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX United States Department of Applied Mathematics University of California Merced MercedCA United States Oden Institute for Computational Engineering and Sciences Walker Department of Mechanical Engineering The University of Texas at Austin AustinTX United States

We present an efficient matrix-free point spread function (PSF) method for approximating operators that have locally supported non-negative integral kernels. The PSF-based method computes impulse responses of the operator at scattered points, and interpolates these impulse responses to approximate entries of the integral kernel. To compute impulse responses efficiently, we apply the operator to Dirac combs associated with batches of point sources, which are chosen by solving an ellipsoid packing problem. The ability to rapidly evaluate kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are then performed with fast H-matrix methods. This end-to-end procedure is illustrated on a blur problem. We demonstrate the PSF-based method’s effectiveness by using it to build preconditioners for the Hessian operator arising in two inverse problems governed by partial differential equations (PDEs): inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low rank structure, and hence a low rank approximation is suitable, for many problems of practical interest the numerical rank of the Hessian is still large. The Hessian impulse responses on the other hand typically become more local as the numerical rank increases, which benefits the PSF-based method. Numerical results reveal that the preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly 5×–10× reductions in the required number of PDE solves, as compared to classical regularization-based preconditioning and no preconditioning. We also present a comprehensive numerical study for the influence of various parameters (that control the shape of the impulse responses and the rank of the Hessian) on the effectiveness of the advection-diffusio

关键词： Method of moments

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Optimal Packings of Superdisks and the Role of Symmetry

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Physical Review Letters 2008年第24期100卷 245504-245504页

作者： Y. Jiao F. H. Stillinger S. Torquato Department of Mechanical and Aerospace Engineering Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA School of Natural Sciences Institute for Advanced Study Princeton New Jersey 08540 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA

Almost all studies of the densest particle packings consider convex particles. Here, we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by |x1|2p+|x2|2... 详细信息

关键词： Condensed matter physics

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Reformulation of the covering and quantizer problems as ground states of interacting particles

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Physical Review E 2010年第5期82卷 056109-056109页

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

It is known that the sphere-packing problem and the number-variance problem (closely related to an optimization problem in number theory) can be posed as energy minimizations associated with an infinite number of point particles in d-dimensional Euclidean space Rd interacting via certain repulsive pair potentials. We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in Rd that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the “void” nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplify the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems. The covering and quantizer problems have relevance in numerous applications, including wireless communication network layouts, the search of high-dimensional data parameter spaces, stereotactic radiation therapy, data compression, digital communications, meshing of space for numerical analysis, and coding and cryptography, among other examples. In the first three space dimensions, the best known solutions of the sphere-packing and number-variance problems (or their “dual” solutions) are directly related to those of the covering and quantizer problems, but such relationships may or may not exist for d≥4, depending on the peculiarities of the dimensions involved. Our reformulation sheds light on the reasons for these similarities and differences. We also show that disordered saturated sphere packings provide relatively thin (economical) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions. In the case of the quantizer problem, we derive improved upper bounds on the q

关键词： Ground state

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Collective coordinate control of density distributions

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Physical Review E 2006年第3期74卷 031104-031104页

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

Real collective density variables C(k) [cf. Eq. (1.3)] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S(k), where k denotes the wave vector. Our objective is to prescribe C(k) and then to find many-particle configurations that correspond to such a target C(k) using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control S(k) in the neighborhood of ∣k∣=0. The optimization method employed generates multiparticle configurations for which S(k)∝∣k∣α, ∣k∣⩽K, and α=1, 2, 4, 6, 8, and 10. The case α=1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid He4, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.

关键词： STATISTICAL MECHANICS REALIZABILITY SYSTEM

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Improved reconstructions of random media using dilation and erosion processes

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Physical Review E 2011年第5期84卷 056102-056102页

作者： Chase E. Zachary Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

By using the most sensitive two-point correlation functions introduced to date, we reconstruct the microstructures of two-phase random media with heretofore unattained accuracy. Such media arise in a host of contexts, including porous and composite media, ecological structures, biological media, and astrophysical structures. The aforementioned correlation functions are special cases of the so-called canonical n-point correlation function Hn and generalize the ones that have been recently employed to advance our ability to reconstruct complex microstructures [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. 106, 17634 (2009)]. The use of these generalized correlation functions is tantamount to dilating or eroding a reference phase of the target medium and incorporating the additional topological information of the modified media, thereby providing more accurate reconstructions of percolating, filamentary, and other topologically complex microstructures. We apply our methods to a multiply connected “donut” medium and a dilute distribution of “cracks” (a set of essentially zero measure), demonstrating improved accuracy in both cases with implications for higher-dimensional and biconnected two-phase systems. The high information content of the generalized two-point correlation functions suggests that it would be profitable to explore their use to characterize the structural and physical properties of not only random media, but also molecular systems, including structural glasses.

关键词： FUNCTIONS (mathematics) MICROSTRUCTURE TOPOLOGY COMPOSITE materials PROBABILITY theory STATISTICS

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Anatomically adapted wavelets for integrated statistical analysis of fMRI data

Anatomically adapted wavelets for integrated statistical ana...

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IEEE International Symposium on Biomedical Imaging

作者： S. Görkem Özkaya Dimitri Van De Ville Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Medical Image Processing Laboratory Universita de Genève Geneve Switzerland Medical Image Processing Laboratory Ecole Polytechnique Fédérale de Lausanne Lausanne Switzerland

Wavelets have been successfully used in statistical analysis of fMRI data as a spatial transform providing a compact representation of brain activation maps. However, conventional (tensor-product) wavelet transforms assume a rectangular domain, while the essential brain activity takes place in the convoluted gray-matter layer. We use the lifting scheme to design wavelet bases for more arbitrary domains which do not have a group structure. In particular, we have considered the grey-matter cortical layer as the domain. We then applied the new transform to fMRI data using the wavelet-based SPM (WSPM) framework. Preliminary results show that the adapted wavelets have superior performance in terms of sensitivity than the standard tensor-product wavelets, while having the same control over type-I error rate (specificity).

关键词： Wavelet transforms Wavelet domain Wavelet analysis Statistical analysis Sensitivity Vectors

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Light Localization in Local Isomorphism Classes of Quasicrystals

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Physical Review Letters 2018年第24期120卷 247401-247401页

作者： Chaney Lin Paul J. Steinhardt Salvatore Torquato Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program of Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We study a continuum of photonic quasicrystal heterostructures derived from local isomorphism (LI) classes of pentagonal quasicrystal tilings. These tilings are obtained by direct projection from a five-dimensional hypercubic lattice. We demonstrate that, with the sole exception of the Penrose LI class, all other LI classes result in degenerate, effectively localized states, with precisely predictable and tunable properties (frequencies, frequency splittings, and densities). We show that localization and tunability are related to a mathematical property of the pattern known as “restorability,” i.e., whether the tiling can be uniquely specified given only a set of rules that fix all allowed clusters smaller than some bound.

关键词： Photonic crystals Photonics Quasicrystals

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Three-Dimensional Wound Reconstruction using Point Descriptors: A Comparative Study

Three-Dimensional Wound Reconstruction using Point Descripto...

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Annual IEEE Symposium on Computer-Based Medical Systems

作者： José R. A. Souto Fellipe M. C. Barbosa Bruno M. Carvalho Graduate Program in Systems and Computing Federal University of Rio Grande do Norte Natal Brazil Department of Informatics and Applied Mathematics Federal University of Rio Grande do Norte Natal Brazil

Precise measurements of chronic wound areas are very important for measuring the efficacy of different treatments, since it provides a clear picture of the chosen treatment evolution. The most used techniques still in use for measuring the wound areas involve manual measuring, paper scales and/or acetate tracing, with each health professional possibly measuring the wounds in slightly different ways, specially when dealing with the wound's depth. Thus, it is common to observe inconsistent wound area estimations by different health workers on the same wounds. Moreover, the measuring process requires direct contact with the wound, increasing the contamination and infection risks, besides possibly causing discomfort for the patient. An alternative that has already been proved successful is the use of Structure from Motion (SfM) to recover three-dimensional representations of a wound's surface and estimate that area. In this work, we analyze the effectiveness of several widely used point descriptors (BRIEF, DRINK, FREAK, ORB, SIFT and SURF) in the SfM wound reconstruction task when applied to real images of realistic synthetic wounds made of latex placed on different areas of the body. The results showed that SIFT, SURF and DRINK produced better area estimates. Since the execution time is a factor, we selected DRINK as the standard descriptor for our system, since it is approximately 9.3 and 4.9 times faster than SIFT and SURF, respectively.

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