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arXiv 2022年

作者： Xu, Limin Mao, Jiahao Gao, Xingyu Liu, Zheng Department of Mathematical Sciences Tsinghua University Beijing100084 China State Key Laboratory of Low Dimensional Quantum Physics Department of Physics Tsinghua University Beijing100084 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China Institute for Advanced Study Tsinghua University Beijing100084 China

Hohenberg-Kohn (HK) theorem is a cornerstone of modern electronic structure calculations. For interacting electrons, given that the internal part of the Hamiltonian (Ĥint), containing the kinetic energy and Couloumb interaction of electrons, has a fixed form, the theorem states that when the electrons are subject to an external electrostatic field, the ground-state density can inversely determine the field, and thus the full Hamiltonian completely. For a general quantum system, a HK-type Hamiltonian in the form of Ĥhk{gi} = Ĥint+Pi giÔi can always be defined, by grouping those terms with fixed or preknown coefficients into Ĥint, and factorizing the remaining as superposition of a set of Hermitian operators {Ôi}. We ask whether the HK theorem can be extended, so that the ground-state expectation values of {Ôi} as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. We show that the question can be addressed by introducing the concept of generalized density correlation matrix (GDCM) defined with respect to the {Ôi} operators. The invertibility of the GDCM represents a mathematically rigorous and practically useful criterion for the extension of HK theorem to be valid. We apply this criterion to several representative systems, including the quantum Ising dimer, the frustration-free systems, N-level quantum systems with fixed inter-level transition amplitude and tunable level energies, and a fermionic Hubbard chain with inhomogeneous on-site interactions. We suggest that for a finite-size system, finding an invertible GDCM under one single {gi} configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space. Copyright © 2022, The Authors. All rights reserved.

关键词： Hamiltonians

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Functional law of large numbers and PDEs for epidemic models with infection-age dependent infectivity

arXiv

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arXiv 2021年

作者： Pang, Guodong Pardoux, Étienne Department of Computational Applied Mathematics and Operations Research George R. Brown College of Engineering Rice University HoustonTX77005 United States Aix–Marseille Univ CNRS I2M Marseille France

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time of infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic Volterra-type integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures. © 2021, CC0.

关键词： Random processes

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Quadruped robot obstacle negotiation via reinforcement learning

Quadruped robot obstacle negotiation via reinforcement learn...

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IEEE International Conference on Robotics and Automation (ICRA)

作者： Honglak Lee Yirong Shen Chih-Han Yu G. Singh A.Y. Ng Stanford University Stanford CA US Department of Electrical Engineering University of Stanford Stanford CA USA Scientific Computing and Computational Mathematics Program University of Stanford Stanford CA USA Computer Science Department University of Stanford Stanford CA USA

Legged robots can, in principle, traverse a large variety of obstacles and terrains. In this paper, we describe a successful application of reinforcement learning to the problem of negotiating obstacles with a quadruped robot. Our algorithm is based on a two-level hierarchical decomposition of the task, in which the high-level controller selects the sequence of foot-placement positions, and the low-level controller generates the continuous motions to move each foot to the specified positions. The high-level controller uses an estimate of the value function to guide its search; this estimate is learned partially from supervised data. The low-level controller is obtained via policy search. We demonstrate that our robot can successfully climb over a variety of obstacles which were not seen at training time

关键词： Learning Robot kinematics Legged locomotion Foot Mobile robots Leg Motion control Robotics and automation Path planning Computer science

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Effects of size polydispersity on the extinction spectra of colloidal nanoparticle aggregates

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Physical Review B 2012年第4期85卷 045421-045421页

作者： Alexander E. Ershov Ivan L. Isaev Polina N. Semina Vadim A. Markel Sergei V. Karpov L. V. Kirenskiy Institute of Physics Russian Academy of Sciences Kransoyarsk 660036 Russian Federation Siberian Federal University Krasnoyarsk 660041 Russian Federation Department of Radiology Department of Bioengineering and the Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania Philadelphia Pennsylvania 19104 USA

We investigate the effect of particle polydispersity on the optical extinction spectra of colloidal aggregates of spherical metallic (silver) nanoparticles, taking into account the realistic interparticle gaps caused by layers of stabilizing polymer adsorbed on the metal surface (adlayers). The spectra of computer-generated aggregates are computed using two different methods. The coupled-multipole method is used in the quasistatic approximation and the coupled-dipole method beyond the quasistatics. The latter approach is applicable if the interparticle gaps are sufficiently wide relative to the particle radii. Simulations are performed for two different particle size distribution functions (bimodal and Gaussian), varying the number of particles per aggregate, and different distribution functions of the interparticle gap width. The strong influence of the latter factor on the spectra is demonstrated and investigated in detail.

关键词： DISPERSION METAL nanoparticles COLLOIDS SPECTRUM analysis SIMULATION methods & models CLUSTERING of particles METALLIC surfaces

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Generalized Splay State in Coupled Chaotic Oscillators Induced by Weak Mutual Resonant Interactions

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Physical Review Letters 2001年第8期86卷 1510-1510页

作者： Meng Zhan Gang Hu Ying Zhang Daihai He Department of Physics Beijing Normal University Beijing 100875 China China Center for Advanced Science and Technology (CCAST) (World Laboratory) P.O. Box 8730 Beijing 100080 China LCP Institute of Applied Physics and Computational Mathematics P.O. Box 8009(26) Beijing 100088 China

Dynamic behavior of coupled chaotic oscillators is investigated. A transition from high-dimensional hyperchaos to a generalized periodic splay state is found for extremely weak coupling. Chaotic nature of a single oscillator and mutual resonant interactions are regarded to be responsible for this self-organized ordering. The functional phase distribution of the generalized splay state, which is essentially different from the equal-phase-separation distribution of the conventional splay states, can be well predicted by analyzing a single periodically forced oscillator.

关键词： Oscillators (mechanical)

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Fully Stochastic Trust-Region Sequential Quadratic programming for Equality-Constrained Optimization Problems

arXiv

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arXiv 2022年

作者： Fang, Yuchen Na, Sen Mahoney, Michael W. Kolar, Mladen Committee on Computational and Applied Mathematics The University of Chicago United States ICSI and Department of Statistics University of California Berkeley United States Lawrence Berkeley National Laboratory United States Booth School of Business The University of Chicago United States

We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where at each step a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to utilize indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm must address an infeasibility issue — the linearized equality constraints and trust-region constraints may lead to infeasible SQP subproblems. In this regard, we propose an adaptive relaxation technique to compute the trial step, consisting of a normal step and a tangential step. To control the lengths of these two steps while ensuring a scale-invariant property, we adaptively decompose the trust-region radius into two segments, based on the proportions of the rescaled feasibility and optimality residuals to the rescaled full KKT residual. The normal step has a closed form, while the tangential step is obtained by solving a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish a global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection. © 2022, CC BY.

关键词： Stochastic systems

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What can real information content tell us about compressing climate model data?

What can real information content tell us about compressing ...

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IEEE/ACM International Workshop on Data Analysis and Reduction for Big Scientific Data (DRBSD-5)

作者： Hayden Sather Alexander Pinard Allison H. Baker Dorit M. Hammerling Department of Computer Science Colorado School of Mines Golden CO USA Applied Mathematics and Statistics Colorado School of Mines Golden CO USA Computational and Information Systems Lab National Center for Atmospheric Research Boulder CO USA

ISBN: (纸本)9781665463386

The massive data volumes produced by climate simulation models create an urgent need for data reduction. Lossy compression in particular is a solution that can significantly reduce storage requirements, however, a tradeoff must be made between the amount of compression applied and the scientific integrity of the data. Determining how much compression can be applied is therefore vital for applying lossy compression. One particular metric for gauging the quality of compression is the percentage of real information present in the original data that is preserved in the compressed data. We compute bitwise real information content for several climate variables from the popular Community Earth System Model, and we investigate the amount of compression that can be applied to each of these climate variables using two popular compression algorithms designed for floating-point data while preserving 99% of the real information content. The analysis of the real information content of data after lossy compression has been applied shows a helpful visualization of how compression artifacts have been introduced to the data. Finally, we demonstrate how this real information content can be used in a straightforward manner to determine compressor settings for our data.

关键词： Measurement Earth Analytical models Data analysis Conferences computational modeling Data visualization

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Statistical mechanics for networks of real neurons

arXiv

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arXiv 2024年

作者： Meshulam, Leenoy Bialek, William Center for Computational Neuroscience Department of Applied Mathematics University of Washington SeattleWA98195 United States Joseph Henry Laboratories of Physics and Lewis Sigler Institute for Integrative Genomics Princeton University PrincetonNJ08544 United States

Perceptions and actions, thoughts and memories result from coordinated activity in hundreds or even thousands of neurons in the brain. It is an old dream of the physics community to provide a statistical mechanics description for these and other emergent phenomena of life. These aspirations appear in a new light because of developments in our ability to measure the electrical activity of the brain, sampling thousands of individual neurons simultaneously over hours or days. We review the progress that has been made in bringing theory and experiment together, focusing on maximum entropy methods and a phenomenological renormalization group. These approaches have uncovered new, quantitatively reproducible collective behaviors in networks of real neurons, and provide examples of rich parameter–free predictions that agree in detail with experiment. © 2024, CC BY.

关键词： Neurons

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Data-driven prediction and origin identification of epidemics in population networks

arXiv

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arXiv 2017年

作者： Chen, Zhizhong Larson, Karen Bowman, Clark Hadjidoukas, Panagiotis Papadimitriou, Costas Koumoutsakos, Petros Matzavinos, Anastasios Division of Applied Mathematics Brown University ProvidenceRI02912 United States Computational Science and Engineering Laboratory ETH Zürich CH-8092 Switzerland Department of Mechanical Engineering University of Thessaly VolosGR-38334 Greece

While there exist a number of mathematical approaches to modeling the spread of disease on a network, analyzing such systems in the presence of uncertainty introduces significant complexity. In scenarios where system parameters must be inferred from limited observations, general approaches to uncertainty quantification can generate approximate distributions of the unknown parameters, but these methods often become computationally expensive if the underlying disease model is complex. In this paper, we apply the recent massively parallelizable Bayesian uncertainty quantification framework Π4U to a model of a disease spreading on a network of communities, showing that the method can accurately and tractably recover system parameters and select optimal models in this *** Codes 62P10, 92-08 Copyright © 2017, The Authors. All rights reserved.

关键词： Uncertainty analysis

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A PRIORI AND A POSTERIORI ERROR ESTIMATES FOR DISCONTINUOUS GALERKIN TIME-DISCRETE METHODS VIA MAXIMAL REGULARITY

arXiv

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arXiv 2024年

作者： Akrivis, Georgios Larsson, Stig Department of Computer Science and Engineering University of Ioannina Ioannina451 10 Greece Institute of Applied and Computational Mathematics FORTH Crete Heraklion700 13 Greece Department of Mathematical Sciences Chalmers University of Technology University of Göteborg Göteborg Sweden

The maximal regularity property of discontinuous Galerkin methods for linear parabolic equations is used together with variational techniques to establish a priori and a posteriori error estimates of optimal order under optimal regularity assumptions. The analysis is set in the maximal regularity framework of UMD Banach spaces. Similar results were proved in an earlier work, based on the consistency analysis of Radau IIA methods. The present error analysis, which is based on variational techniques, is of independent interest, but the main motivation is that it extends to nonlinear parabolic equations;in contrast to the earlier work. Both autonomous and nonautonomous linear equations are *** Codes 65M12, 65M15 © 2024, CC BY.

关键词： Galerkin methods

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