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An Iterative Thresholding Method for the Minimum Compliance Problem

Communications in computational Physics 2023年第4期33卷 1189-1216页

作者： Luyu Cen Wei Hu Dong Wang Xiaoping Wang Department of Mathematics Hong Kong University of Science and TechnologyClear Water BayKowloonHong KongChina The Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA School of Science and Engineering The Chinese University of Hong KongShenzhenGuangdong 518172China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big DataGuangdong 518172China

In this paper,we propose a simple energy decaying iterative thresholding algorithm to solve the two-phase minimum compliance *** material domain is implicitly represented by its characteristic function,and the problem is formulated into a minimization problem by the principle of minimum complementary *** prove that the energy is decreasing in each *** effective continuation schemes are proposed to avoid trapping into the local *** results on 2D isotropic linear material demonstrate the effectiveness of the proposed methods.

关键词： Minimum compliance convolution thresholding linear elasticity

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Correlations in interacting electron liquids: Many-body statistics and hyperuniformity

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Physical Review B 2024年第10期110卷 104201页

作者： Haina Wang Rhine Samajdar Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Materials Institute Program in Applied and Computational Mathematics

Disordered hyperuniform many-body systems are exotic states of matter with novel optical, transport, and mechanical properties. These systems are characterized by an anomalous suppression of large-scale density fluctuations compared to ordinary liquids. The structure factor of disordered hyperuniform systems often obeys the scaling relation S(k)∼Bkα with B,α>0 in the limit k→0. Ground states of d-dimensional free fermionic gases, which are fundamental models for many metals and semiconductors, are key examples of quantum disordered hyperuniform states with important connections to random matrix theory. However, the effects of electron-electron interactions as well as the polarization of the electron liquid on hyperuniformity have not been explored thus far. In this paper, we systematically address these questions by deriving the analytical small-k behaviors (and, associated, α and B) of the total and spin-resolved structure factors of quasi-one-dimensional, two-dimensional, and three-dimensional electron liquids for varying polarizations and interaction parameters. We validate that these equilibrium disordered ground states are hyperuniform, as dictated by the fluctuation-compressibility relation. Interestingly, free fermions, partially polarized interacting fermions, and fully polarized interacting fermions are characterized by different values of the small-k scaling exponent α and coefficient B. In particular, partially polarized fermionic liquids exhibit a unique form of multihyperuniformity, in which the net configuration exhibits a stronger form of hyperuniformity (i.e., larger α) than each individual spin component. The detailed theoretical analysis of such small-k behaviors enables the construction of corresponding equilibrium classical systems under effective one- and two-body interactions that mimic the pair statistics of quantum electron liquids. Our paper thus reveals that highly unusual hyperuniform and multihyperuniform states can be achieved in simple

关键词： Electronic structure Quantum many-body systems Two-dimensional electron gas

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Local order metrics for many-particle systems across length scales

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Physical Review Research 2024年第3期6卷 033262页

作者： Charles Emmett Maher Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in d-dimensional Euclidean space Rd across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of n-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance σN2(R) associated with a spherical sampling window of radius R (which encodes pair correlations) and an integral measure derived from it ΣN(Ri,Rj) that depends on two specified radial distances Ri and Rj. Across the first three space dimensions (d=1,2,3), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale R. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of R. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius R [S. Torquato et al., Phys. Rev. X 11, 021028 (2021)] to devise even more sensitive order metrics.

关键词： Classical statistical mechanics

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Semi-analytical finite element method applied for characterizing micropolar fibrous composites

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applied mathematics and Mechanics(English Edition) 2024年第12期45卷 2147-2164页

作者： J.A.OTERO Y.ESPINOSA-ALMEYDA R.RODRIGUEZ-RAMOS J.MERODIO School of Engineering and Sciences Monterrey Institute of TechnologyEstado de Mexico 52926Mexico Institute of Engineering and Technology Autonomous University of Ciudad JuarezCiudad JuarezChihuahua 32310Mexico Postgraduate Program in Computational Modeling in Science and Technology Federal Fluminense UniversityRıo de Janeiro 27255-125Brazil Faculty of Mathematics and Computer Science University of HavanaLa Habana 10400Cuba Department of Mathematics Applied to ICT School of Informatics Systems EngineeringTechnical University of MadridMadrid 28031Spain

A semi-analytical finite element method(SAFEM),based on the two-scale asymptotic homogenization method(AHM)and the finite element method(FEM),is implemented to obtain the effective properties of two-phase fiber-reinforced composites(FRCs).The fibers are periodically distributed and unidirectionally aligned in a homogeneous *** framework addresses the static linear elastic micropolar problem through partial differential equations,subject to boundary conditions and perfect interface contact *** mathematical formulation of the local problems and the effective coefficients are presented by the *** local problems obtained from the AHM are solved by the FEM,which is denoted as the *** numerical results are provided,and the accuracy of the solutions is analyzed,indicating that the formulas and results obtained with the SAFEM may serve as the reference points for validating the outcomes of experimental and numerical computations.

关键词： semi-analytical approach fiber-reinforced composite(FRC) effective property finite element method(FEM) asymptotic homogenization method(AHM) micropolar elasticity

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Ordered and disordered stealthy hyperuniform point patterns across spatial dimensions

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Physical Review Research 2024年第3期6卷 033260页

作者： Peter K. Morse Paul J. Steinhardt Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute of Materials Princeton Center for Theoretical Science Program in Applied and Computational Mathematics

In previous work [Phys. Rev. X 5, 021020 (2015)] it was shown that stealthy hyperuniform systems can be regarded as hard spheres in Fourier space in the sense that the structure factor is exactly zero in a spherical region around the origin in analogy with the pair-correlation function of real-space hard spheres. While this earlier work focused on spatial dimensions d=1–4, here we extend the analysis to higher dimensions in order to make connections to high-dimensional sphere packings and the mean-field theory of glasses. We exploit this correspondence to confirm that the densest Fourier-space hard-sphere system is that of a Bravais lattice in contrast to real-space hard spheres, whose densest configuration is conjectured to be disordered. In passing, we give a concise form for the position of the first Bragg peak. We also extend the virial series previously suggested for disordered stealthy hyperuniform systems to higher dimensions in order to predict spatial decorrelation as a function of dimension. This prediction is then borne out by numerical simulations of disordered stealthy hyperuniform ground states in dimensions d=2–8, which have only recently been made possible due to a highly parallelized algorithm.

关键词： Classical statistical mechanics Disordered systems

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LATTICE-VALUED BOTTLENECK DUALITY

arXiv

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arXiv 2024年

作者： Ghrist, Robert Gould, Julian Lopez, Miguel Department of Mathematics Department of Electrical & Systems Engineering and Program in Applied Mathematics & Computational Science University of Pennsylvania United States

This note reformulates certain classical combinatorial duality theorems in the context of order lattices. For source-target networks, we generalize bottleneck path-cut and flow-cut duality results to edges with capacities in a distributive lattice. For posets, we generalize a bottleneck version of Dilworth’s theorem, again weighted in a distributive lattice. These results are applicable to a wide array of non-numerical network flow problems, as shown. All results, proofs, and applications were created in collaboration with AI language models. An appendix documents their role and *** Codes 06D05, 90C35, 06A07 © 2024, CC BY.

关键词： Combinatorial optimization

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DeePKS + ABACUS as a Bridge between Expensive Quantum Mechanical Models and Machine Learning Potentials

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Journal of Physical Chemistry A 2022年第49期126卷 9154-9164页

作者： Li, Wenfei Ou, Qi Chen, Yixiao Cao, Yu Liu, Renxi Zhang, Chunyi Zheng, Daye Cai, Chun Wu, Xifan Wang, Han Chen, Mohan Zhang, Linfeng AI for Science Institute Beijing100080 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States HEDPS CAPT College of Engineering School of Physics Peking University Beijing100871 China Department of Physics Temple University PhiladelphiaPA19122 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China DP Technology Beijing100080 China

Recently, the development of machine learning (ML) potentials has made it possible to perform large-scale and long-time molecular simulations with the accuracy of quantum mechanical (QM) models. However, for different levels of QM methods, such as density functional theory (DFT) at the meta-GGA level and/or with exact exchange, quantum Monte Carlo, etc., generating a sufficient amount of data for training an ML potential has remained computationally challenging due to their high cost. In this work, we demonstrate that this issue can be largely alleviated with Deep Kohn-Sham (DeePKS), an ML-based DFT model. DeePKS employs a computationally efficient neural network-based functional model to construct a correction term added upon a cheap DFT model. Upon training, DeePKS offers closely matched energies and forces compared with high-level QM method, but the number of training data required is orders of magnitude less than that required for training a reliable ML potential. As such, DeePKS can serve as a bridge between expensive QM models and ML potentials: one can generate a decent amount of high-accuracy QM data to train a DeePKS model and then use the DeePKS model to label a much larger amount of configurations to train an ML potential. This scheme for periodic systems is implemented in a DFT package ABACUS, which is open source and ready for use in various applications. © 2022 The Journal of Physical Chemistry A. All right reserved.

关键词： Density functional theory

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A Flow Artist for High-Dimensional Cellular Data 33

A Flow Artist for High-Dimensional Cellular Data

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33rd IEEE International Workshop on Machine Learning for Signal Processing, MLSP 2023

作者： MacDonald, Kincaid Bhaskar, Dhananjay Thampakkul, Guy Nguyen, Nhi Zhang, Joia Perlmutter, Michael Adelstein, Ian Krishnaswamy, Smita Yale University Department of Mathematics United States Pomona College Department of Mathematics United States University of Washington Department of Statistics United States Boise State University Department of Mathematics United States Yale University Department of Computer Science United States Yale University Applied Mathematics Program United States Yale University Computational Biology and Bioinformatics Program United States

ISBN: (纸本)9798350324112

We consider the problem of embedding point cloud data sampled from an underlying manifold with an associated flow or velocity. Such data arises in many contexts where static snapshots of dynamic entities are measured, including in high-throughput biology such as single-cell transcriptomics. Existing embedding techniques either do not utilize velocity information or embed the coordinates and velocities independently, i.e., they either impose velocities on top of an existing point embedding or embed points within a prescribed vector field. Here we present FlowArtist, a neural network that embeds points while jointly learning a vector field around the points. The combination allows FlowArtist to better separate and visualize velocity-informed structures. Our results, on toy datasets and single-cell RNA velocity data, illustrate the value of utilizing coordinate and velocity information in tandem for embedding and visualizing high-dimensional data. © 2023 IEEE.

关键词： RNA

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GPU-Accelerated Modified Bessel Function of the Second Kind for Gaussian Processes

GPU-Accelerated Modified Bessel Function of the Second Kind ...

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ISC High Performance 2025 Research Paper Proceedings (40th International Conference)

作者： Zipei Geng Sameh Abdulah Ying Sun Hatem Ltaief David E. Keyes Marc G. Genton Statistics Program King Abdullah University of Science and Technology Thuwal Saudi Arabia Applied Mathematics and Computational Sciences (AMCS) Program King Abdullah University of Science and Technology Thuwal Saudi Arabia

ISBN: (数字)9783982633619

Modified Bessel functions of the second kind are widely used in physics, engineering, spatial statistics, and machine learning. Since contemporary scientific applications, including machine learning, rely on GPUs for acceleration, providing robust GPU-hosted implementations of special functions, such as the modified Bessel function, is crucial for performance. Existing implementations of the modified Bessel function of the second kind rely on CPUs and have limited coverage of the full range of values needed in some applications. In this work, we present a robust implementation of the modified Bessel function of the second kind on GPUs, eliminating the dependence on the CPU host. We cover a range of values commonly used in real applications, providing high accuracy compared to common libraries like the GNU Scientific Library (GSL) when referenced to Mathematica as the authority. Our GPU-accelerated approach also demonstrates a 2.68X performance improvement using a single A100 GPU compared to the GSL on 40-core Intel Cascade Lake CPUs. Our implementation is integrated into ExaGeoStat, the HPC framework for Gaussian process modeling, where the modified Bessel function of the second kind is required by the Matérn covariance function in generating covariance matrices. We accelerate the matrix generation process in ExaGeoStat by up to 12.62X with four A100 GPUs while maintaining almost the same accuracy for modeling and prediction operations using synthetic and real datasets.

关键词： Accuracy computational modeling Graphics processing units Gaussian processes Machine learning Libraries computational efficiency Covariance matrices Optimization Synthetic data

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

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Physical Review B 2025年第9期111卷 094113-094113页

作者： Pinchen Xie Yixiao Chen Weinan E Roberto Car Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA AI for Science Institute Beijing China and Center for Machine Learning Research and School of Mathematical Sciences Peking University Beijing China

We report a 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 across the phase transition. We find that the emergence and disappearance of the macroscopic polarization is driven by dipolar switching at the nanometer scale. We also computed the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. Often, 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 experiments. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation associated with 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.

关键词： Ferroelectricity

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