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Multilevel Particle Filters for Partially Observed McKean-Vlasov Stochastic Differential Equations

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arXiv 2024年

作者： Awadelkarim, Elsiddig Jasra, Ajay 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 School of Data Science The Chinese University of Hong Kong Shenzhen China

In this paper we consider the filtering problem associated to partially observed McKean-Vlasov stochastic differential equations (SDEs). The model consists of data that are observed at regular and discrete times and the objective is to compute the conditional expectation of (functionals) of the solutions of the SDE at the current time. This problem, even the ordinary SDE case is challenging and requires numerical approximations. Based upon the ideas in [3, 12] we develop a new particle filter (PF) and multilevel particle filter (MLPF) to approximate the afore-mentioned expectations. We prove under assumptions that, for ϵ > 0, to obtain a mean square error of O(ϵ2) the PF has a cost per-observation time of O(ϵ−5) and the MLPF costs O(ϵ−4) (best case) or O(ϵ−4 log(ϵ)2) (worst case). Our theoretical results are supported by numerical experiments. © 2024, CC BY.

关键词： Mean square error

Dimension reduction of dynamical systems on networks with leading and non-leading eigenvectors of adjacency matrices

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Physical Review Research 2022年第2期4卷 023257-023257页

作者： Naoki Masuda Prosenjit Kundu Department of Mathematics State University of New York at Buffalo New York 14260-2900 USA Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo Buffalo New York 14260-5030 USA

Dimension reduction techniques for dynamical systems on networks are considered to promote our understanding of the original high-dimensional dynamics. One strategy of dimension reduction is to derive a low-dimensional dynamical system whose behavior approximates the observables of the original dynamical system that are weighted linear summations of the state variables at the different nodes. Recently proposed methods use the leading eigenvector of the adjacency matrix of the network as the mixture weights to obtain such observables. In the present study, we explore performances of this type of one-dimensional reductions of dynamical systems on networks when we use non-leading eigenvectors of the adjacency matrix as the mixture weights. Our theory predicts that non-leading eigenvectors can be more efficient than the leading eigenvector and enables us to select the eigenvector minimizing the error. We numerically verify that the optimal non-leading eigenvector outperforms the leading eigenvector for some dynamical systems and networks. We also argue that, despite our theory, it is practically better to use the leading eigenvector as the mixture weights to avoid misplacing the bifurcation point too distantly and to be resistant against dynamical noise.

关键词： Network resilience

The phase diagram of a deep potential water model

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arXiv 2021年

作者： Zhang, Linfeng Wang, Han Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

Using the Deep Potential methodology, we construct a model that reproduces accurately the potential energy surface of the SCAN approximation of density functional theory for water, from low temperature and pressure to about 2400 K and 50 GPa, excluding the vapor stability region. The computational efficiency of the model makes it possible to predict its phase diagram using molecular dynamics. Satisfactory overall agreement with experimental results is obtained. The fluid phases, molecular and ionic, and all the stable ice polymorphs, ordered and disordered, are predicted correctly, with the exception of ice III and XV that are stable in experiments, but metastable in the model. The evolution of the atomic dynamics upon heating, as ice VII transforms first into ice VII′′ and then into an ionic fluid, reveals that molecular dissociation and breaking of the ice rules coexist with strong covalent fluctuations, explaining why only partial ionization was inferred in experiments. © 2021, CC BY-NC-ND.

关键词： Phase diagrams

Hyper-molecules: On the representation and recovery of dynamical structures, with application to flexible macro-molecular structures in cryo-EM

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arXiv 2019年

作者： Lederman, Roy R. Andén, Joakim Singer, Amit Department of Statistics and Data Science Yale University New HavenCT United States Center for Computational Mathematics Flatiron Institute New YorkNY United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States

Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for determining the 3-D structure of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. The molecular structures are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori;one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in determining rigid structures from homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the "hyper-molecule" framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for recovering such maps of heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a prototype applied to synthetic data. Copyright © 2019, The Authors. All rights reserved.

关键词： Medical imaging

Pairwise Learning for Autism Spectrum Disorder Imbalanced Classification

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Pairwise Learning for Autism Spectrum Disorder Imbalanced Cl...

IEEE Eurasia Conference on Biomedical Engineering, Healthcare and Sustainability (ECBIOS)

作者： Shu Liu Qiang Wu Xin Yang Computational and Data Science PhD Program Middle Tennessee State University Murfreesboro TN USA Department of Mathematics University of Tennessee Knoxville TN USA Department of Computer Science Middle Tennessee State University Murfreesboro TN USA

ISBN: (数字)9798350396133

ISBN: (纸本)9798350396140

This paper performs a classification task on data obtained from the Autism Brain Imaging data Exchange (ABIDE) repository. In real-world case analysis, the number of autism spectrum disorder (ASD) patients is much smaller than typically developed people. To address this issue, this paper proposes the utilization of pairwise robust support vector machine (PRSVM) algorithms to classify autism spectrum disorder (ASD) patients. In this project's experiments, the correlation matrix derived from functional magnetic resonance imaging (fMRI) data was employed as a classification feature. A comprehensive evaluation was conducted to compare the classification performance of PRSVM with various machine learning methods. The comparative analysis encompassed various aspects, including different data dimensions, imbalanced ratios, and sample sizes, providing valuable insights into the relative performance of the algorithms under different experimental conditions. The experimental results demonstrate that PRSVM can detect autistic patients more accurately when the data is imbalanced. Moreover, the results indicate that PRSVM outperforms or achieves comparable performance to other conventional classification methods in a variety of situations. Furthermore, our approach can be further improved by augmenting the training set with either exclusively normal person samples or by incorporating patient samples and normal people samples in a proportionate manner. This augmentation strategy holds promising application value, as it contributes to improving the performance and robustness of our method.

关键词： Training Support vector machines Neuroimaging Autism Machine learning algorithms Functional magnetic resonance imaging Vectors Robustness Classification algorithms Sustainable development

Accuracy of a one-dimensional reduction of dynamical systems on networks

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Physical Review E 2022年第2期105卷 024305-024305页

作者： Prosenjit Kundu Hiroshi Kori Naoki Masuda Department of Mathematics State University of New York at Buffalo Buffalo New York 14260-2900 USA Department of Complexity Science and Engineering The University of Tokyo Chiba 277-8561 Japan Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo Buffalo New York 14260-5030 USA

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

关键词： Complex systems Dynamics of networks Network resilience

Metapopulation models imply non-Poissonian statistics of interevent times

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arXiv 2021年

作者： dos Reis, Elohim Fonseca Masuda, Naoki Department of Mathematics State University of New York at Buffalo BuffaloNY United States Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo BuffaloNY United States

Interevent times in temporal contact data from humans and animals typically obey heavy-tailed distributions, and this property impacts contagion and other dynamical processes on networks. We theoretically show that distributions of interevent times heavier-tailed than exponential distributions are a consequence of the most basic metapopulation model used in epidemiology and ecology, in which individuals move from a patch to another according to the simple random walk. Our results hold true irrespectively of the network structure and also for more realistic mobility rules such as high-order random walks and the recurrent mobility patterns used for modeling human dynamics. Copyright © 2021, The Authors. All rights reserved.

关键词： Random processes

Stability of stochastic gradient descent on nonsmooth convex losses 20

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Stability of stochastic gradient descent on nonsmooth convex...

Proceedings of the 34th International Conference on Neural Information Processing Systems

作者： Raef Bassily Vitaly Feldman Cristóbal Guzmán Kunal Talwar Department of Computer Science & Engineering The Ohio State University Apple Pontificia Universidad Católica de Chile Institute for Mathematical and Computational Engineering ANID – Millennium Science Initiative Program Millennium Nucleus Center for the Discovery of Structures in Complex Data

ISBN: (纸本)9781713829546

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. [20] provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth *** work is the first to address uniform stability of SGD on nonsmooth convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case [16].

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Automated, high throughput exploration of process–structure–property relationships using the MapReduce paradigm

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Materials Discovery 2015年 1卷 21-28页

作者： Wodo, Olga Zola, Jaroslaw Sarath Pokuri, Balaji Sesha Du, Pengfei Ganapathysubramanian, Baskar University at Buffalo Materials Design and Innovation Department Buffalo United States University at Buffalo Mechanical and Aerospace Engineering Department Buffalo United States University at Buffalo Computational and Data-Enabled Science and Engineering Program Buffalo United States University at Buffalo Computer Science and Engineering Department Buffalo United States University at Buffalo Biomedical Informatics Department Buffalo United States Iowa State University Mechanical Engineering Department Ames United States Iowa State University Electrical and Computer Engineering Department Ames United States

The microstructure of a material intimately affects the performance of a device made from this material. The microstructure, in turn, is affected by the processing pathway used to fabricate the device. This forms the process–structure–property triangle that is central to material science. There has been increasing interest to comprehensively understand and subsequently exploit process–structure–property (PSP) relationships to design processing pathways that result in tailored microstructures exhibiting optimal properties. However, unraveling process–structure–property relationships usually requires systematic and tedious combinatorial search of process and system variables to identify the microstructures that are produced. This is further complicated by the necessity to interrogate the properties of the huge set of corresponding microstructures. Motivated by this challenge, we focus on developing a generic methodology to establish and explore PSP pathways. We leverage recent advances in high performance computing (HPC) and high throughput computing (HTC) with the premise that a domain expert should be able to focus on domain specific PSP problems while the highly specialized HPC/HTC knowledge needed to approach such problems should be hidden from the domain expert. Our hypothesis is that PSP exploration can be naturally formulated in terms of a standard paradigm in cloud computing, namely the MapReduce programming model. We show how reformulating PSP exploration into a MapReduce workflow enables us to take advantage of advances in cloud computing while requiring minimal specialized knowledge of HPC. We illustrate this generic approach by exploring PSP relationships relevant to organic photovoltaics. We focus on identifying microstructural traits that correlate with specific properties of the photovoltaic process: exciton generation, exciton dissociation and charge generation. We integrate a graph-based microstructure characterization tool, and a microstructure-aware

关键词： Drift diffusion model High throughput analysis MapReduce paradigm Organic solar cells Process–structure–processing Thermal annealing