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Statistical Mechanics of Thermostatically Controlled Multi-Zone Buildings

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arXiv 2022年

作者： Valenzuela, Lucas Fuentes Williams, Lindell Chertkov, Michael Department of Electrical Engineering Stanford University United States Department of Physics Cornell University United States Program in Applied Mathematics & Department of Mathematics University of Arizona United States

We study the collective phenomena and constraints associated with the aggregation of individual cooling units from a statistical mechanics perspective. These units are modelled as Thermostatically Controlled Loads (TCLs) and represent zones in a large commercial or residential building. Their energy input is centralized and controlled by a collective unit – the Air Handling Unit (AHU) — delivering cool air to all TCLs, thereby coupling them together. Aiming to identify representative qualitative features of the AHU-to-TCL coupling, we build a realistic but also sufficiently simple model and analyze it in two distinct regimes: the Constant Supply Temperature (CST) and the Constant Power Input (CPI) regimes. In both cases, we center our analysis on the relaxation dynamics of individual TCL temperatures to a statistically steady state. We observe that while the dynamics are relatively fast in the CST regime, resulting in all TCLs evolving around the control setpoint, the CPI regime reveals emergence of a bi-modal probability distribution and two, possibly strongly separated, time scales. We observe that the two modes in the CPI regime are associated with all TCLs being in the same low and high-temperature states, respectively, with occasional (and therefore possibly rare) collective transition between the modes akin in the Kramer’s phenomenon of statistical physics. To the best of our knowledge, this phenomenon was overlooked in the context of the multi-zone energy building engineering, even thought it has direct implications on the operations of centralized cooling systems in buildings. It teaches us that a balance needs to be struck between occupational comfort — related to variations in the individual temperatures — and power output predictability — the main focus of the DR schemes. © 2022, CC BY.

关键词： Statistical mechanics

ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE

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arXiv 2022年

作者： Greengard, Philip Hoskins, Jeremy G. Marshall, Nicholas F. Singer, Amit Department of Statistics Columbia University United States Department of Statistics University of Chicago United States Program in Applied and Computational Mathematics The Department of Mathematics Princeton University United States

This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W2. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = -∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1]2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented. Copyright © 2022, The Authors. All rights reserved.

关键词： Laplace transforms

Interval graphs for Genome-Fragment combinations

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Interval graphs for Genome-Fragment combinations

2023 International Conference on Mathematical and Statistical Physics, Computational Science, Education and Communication, ICMSCE 2023

作者： Kao, Shin-Shin Wu, Tzong-Yuan Department of Applied Mathematics Program of International Master of Business Administration Chung Yuan Christian University No. 200 Jong-Bei Road Chung-Li District Tao-Yuan320314 Taiwan Department of Bioscience Technology Chung Yuan Christian University No. 200 Jong-Bei Road Chung-Li District Tao-Yuan320314 Taiwan

ISBN: (纸本)9781510671768

The main purpose of this research work is to apply the theory of "interval graphs" to a specific viral genome, reconstruct the sequence from the unknown-ordered fragments for possible combinations for further studies. We use the restriction digestion method to cut the chromosome into many segments with several restriction enzymes. These cut segments can be separated by agarose gel electrophoresis according to their molecular size and are not in order of the original sequence. Besides, the segments cut out by each enzyme are different, so there will be intersections (sub-segments). The method of interval graph, suggested by14, attempts to use the information at the intersection to reorganize the original sequence. We selected our own patented insect viral gene sequence and conducted research on the above-mentioned interval graph method. More specifically, the isolated recombinant baculovirus, tentatively named ABM, and the widely-used prototype of baculovirus species, which is Autographa californica multiple nucleopolyhedrovirus (AcMNPV), were used as the model system. Although the length of the insect viral genome sequence we used is relatively short as compared with the genome of organisms, this study failed to achieve the purpose of sorting by the method of interval graph. The two main reasons can be summarized as follows: First, the enzyme experiment detected multiple short segments that did not intersect with other segments, and the interval graph method could not determine their positions in the original sequence. Secondly, any "graph" is made up of vertices and edges. Representing gene sequences with interval graphs means translating segments and intersecting segments with vertices and edges. Even if the improved method was adopted, the number of vertices and edges that can be loaded by the existing hardware makes this research unable to complete all the steps of the basic interval graph construction. This paper illustrates the experimental methods, mathematic

关键词： Enzymes

Fractional Integrable Nonlinear Soliton Equations

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Physical Review Letters 2022年第18期128卷 184101-184101页

作者： Mark J. Ablowitz Joel B. Been Lincoln D. Carr Department of Applied Mathematics University of Colorado Boulder Colorado 80309 USA Department of Applied Mathematics and Statistics Colorado School of Mines Golden Colorado 80401 USA Department of Physics Colorado School of Mines Golden Colorado 80401 USA Quantum Engineering Program Colorado School of Mines Golden Colorado 80401 USA

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques. As examples, this general method is used to characterize fractional extensions to two physically relevant, pervasive integrable nonlinear equations: the Korteweg–deVries and nonlinear Schrödinger equations. These equations are shown to predict superdispersive transport of nondissipative solitons in fractional media.

关键词： Solitons

Deformations of acid-mediated invasive tumors in a model with Allee effect

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arXiv 2024年

作者： Carter, Paul Doelman, Arjen van Heijster, Peter Levy, Daniel Maini, Philip Okey, Erin Yeung, Paige Department of Mathematics University of California Irvine United States Mathematical Institute Leiden University Leiden Netherlands Mathematical and Statistical Methods—Biometris Wageningen University & Research Wageningen Netherlands Program in Applied and Computational Mathematics Princeton University Princeton United States Mathematical Institute University of Oxford Oxford United Kingdom Division of Applied Mathematics Brown University ProvidenceRI United States Massachusetts Institute of Technology Cambridge United States

We consider a Gatenby–Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and/or different invasion speeds. © 2024, CC BY.

关键词： Perturbation techniques

Multilevel Monte Carlo Methods for the Grad-Shafranov Free Boundary Problem

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SSRN

SSRN 2023年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Applied Mathematics & Statistics and Scientific Computing Program University of Maryland College Park United States Department of Mathematics The University of Arizona United States

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costsby performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductionstypically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model. © 2023, The Authors. All rights reserved.

关键词： Monte Carlo methods

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

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arXiv 2023年

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 © 2023, CC BY.

关键词： Monte Carlo methods

An expectation-maximization algorithm for continuous-time hidden Markov models

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arXiv 2021年

作者： Wang, Qingcan Weinan, E. Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States

We propose a unified framework that extends the inference methods for classical hidden Markov models to continuous settings, where both the hidden states and observations occur in continuous time. Two different settings are analyzed: hidden jump process with a finite state space, and hidden diffusion process with a continuous state space. For each setting, we first estimate the hidden states given the observations and model parameters, showing that the posterior distribution of the hidden states can be described by differential equations in continuous time. We then consider the estimation of unknown model parameters, deriving the continuous-time formulas for the expectation-maximization algorithm. We also propose a Monte Carlo method based on the continuous formulation, sampling the posterior distribution of the hidden states and updating the parameter *** Codes 60J25, 65C40 Copyright © 2021, The Authors. All rights reserved.

关键词： Continuous time systems

A comparative analysis of optimization and generalization properties of two-layer neural network and random feature models under gradient descent dynamics

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Science China mathematics 2020年第7期63卷 1235-1258页

作者： Weinan E Chao Ma Lei Wu Department of Mathematics Princeton UniversityPrincetonNJ 08544USA The Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA Beijing Institute of Big Data Research Beijing 100871China

A fairly comprehensive analysis is presented for the gradient descent dynamics for training two-layer neural network models in the situation when the parameters in both layers are *** initialization schemes as well as general regimes for the network width and training data size are *** the overparametrized regime,it is shown that gradient descent dynamics can achieve zero training loss exponentially fast regardless of the quality of the *** addition,it is proved that throughout the training process the functions represented by the neural network model are uniformly close to those of a kernel *** general values of the network width and training data size,sharp estimates of the generalization error are established for target functions in the appropriate reproducing kernel Hilbert space.

关键词： two-layer neural network random feature model Gram matrix generalization error