检索结果-内蒙古大学图书馆

Manifold learning for organizing unstructured sets of process observations

学校读者我要写书评

暂无评论

arXiv 2018年

作者： Dietrich, Felix Kooshkbaghi, Mahdi Bollt, Erik M. Kevrekidis, Ioannis G. Department of Chemical and Biomolecular Engineering Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Department of Electrical and Computer Engineering Clarkson Center for Complex Systems Science Clarkson University PotsdamNY13699-5815 United States

Data mining is routinely used to organize ensembles of short temporal observations so as to reconstruct useful, low-dimensional realizations of an underlying dynamical system. In this paper, we use manifold learning to organize unstructured ensembles of observations ("trials") of a system's response surface. We have no control over where every trial starts;and during each trial operating conditions are varied by turning "agnostic" knobs, which change system parameters in a systematic but unknown way. As one (or more) knobs "turn" we record (possibly partial) observations of the system response. We demonstrate how such partial and disorganized observation ensembles can be integrated into coherent response surfaces whose dimension and parametrization can be systematically recovered in a data-driven fashion. The approach can be justified through the Whitney and Takens embedding theorems, allowing reconstruction of manifolds/attractors through different types of observations. We demonstrate our approach by organizing unstructured observations of response surfaces, including the reconstruction of a cusp bifurcation surface for Hydrogen combustion in a Continuous Stirred Tank Reactor. Finally, we demonstrate how this observation-based reconstruction naturally leads to informative transport maps between input parameter space and output/state variable *** codes 37M20 Copyright © 2018, The Authors. All rights reserved.

关键词： Dynamical systems

Fourier phase retrieval: Uniqueness and algorithms

学校读者我要写书评

暂无评论

arXiv 2017年

作者： Bendory, Tamir Beinerty, Robert Eldarz, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Graz8010 Austria Andrew and Erna Viterbi Faculty of Electrical Engineering Technion - Israel Institute of Technology Haifa Israel

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field. Copyright © 2017, The Authors. All rights reserved.

关键词： Fourier transforms

Stochastic Dynamics and Probability Analysis for a Generalized Epidemic Model with Environmental Noise

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Boukanjime, Brahim Maama, Mohamed Laboratory of Mathematical Modeling and Economic Calculations Hassan First University of Settat Morocco Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Saudi Arabia

In this paper we consider a stochastic SEIQR (susceptible-exposed-infected-quarantined-recovered) epidemic model with a generalized incidence function. Using the Lyapunov method, we establish the existence and uniqueness of a global positive solution to the model, ensuring that it remains well-defined over time. Through the application of Young’s inequality and Chebyshev’s inequality, we demonstrate the concepts of stochastic ultimate boundedness and stochastic permanence, providing insights into the long-term behavior of the epidemic dynamics under random perturbations. Furthermore, we derive conditions for stochastic extinction, which describes scenarios where the epidemic may eventually die out, and V-geometric ergodicity, which indicates the rate at which the system’s state converges to its equilibrium. Finally, we perform numerical simulations to verify our theoretical results and assess the model’s behavior under different parameters. Copyright © 2024, The Authors. All rights reserved.

关键词： Stochastic models

Heat transport in liquid water from first-principles and deep-neural-network simulations

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Tisi, Davide Zhang, Linfeng Bertossa, Riccardo Wang, Han Car, Roberto Baroni, Stefano SISSA - Scuola Internazionale Superiore di Studi Avanzati Trieste34136 Italy Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States CNR Istituto Officina Dei Materiali SISSA Unit Trieste34136 Italy

We compute the thermal conductivity of water within linear response theory from equilibrium molecular dynamics simulations, by adopting two different approaches. In one, the potential energy surface (PES) is derived on the fly from the electronic ground state of density functional theory (DFT) and the corresponding analytical expression is used for the energy flux. In the other, the PES is represented by a deep neural network (DNN) trained on DFT data, whereby the PES has an explicit local decomposition and the energy flux takes a particularly simple expression. By virtue of a gauge invariance principle, established by Marcolongo, Umari, and Baroni, the two approaches should be equivalent if the PES were reproduced accurately by the DNN model. We test this hypothesis by calculating the thermal conductivity, at the GGA (PBE) level of theory, using the direct formulation and its DNN proxy, finding that both approaches yield the same conductivity, in excess of the experimental value by approximately 60%. Besides being numerically much more efficient than its direct DFT counterpart, the DNN scheme has the advantage of being easily applicable to more sophisticated DFT approximations, such as meta-GGA and hybrid functionals, for which it would be hard to derive analytically the expression of the energy flux. We find in this way, that a DNN model, trained on meta-GGA (SCAN) data, reduce the deviation from experiment of the predicted thermal conductivity by about 50%, leaving the question open as to whether the residual error is due to deficiencies of the functional, to a neglect of nuclear quantum effects in the atomic dynamics, or, likely, to a combination of the two. Copyright © 2021, The Authors. All rights reserved.

关键词： Deep neural networks

Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Law, Kody J.H. Walton, Neil Yang, Shangda Jasra, Ajay School of Mathematics University of Manchester ManchesterM13 9PL United Kingdom 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

We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE−1, while single level methods require MSE−ξ for ξ > 1. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where ξ = 5/4 and ξ = 3/2, respectively. It is also illustrated on more challenging log Gaussian process models, where single level complexity is approximately ξ = 9/4 and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives ξ = 5/4 + ω, for any ω > 0, whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the spectral domain, which facilitates acceleration with fast Fourier transform methods via a cumulant embedding strategy, and may be of independent interest in the context of spatial statistics and machine learning. Copyright © 2022, The Authors. All rights reserved.

关键词： Inverse problems

Meshless Hermite-HDMR finite difference method for high-dimensional Dirichlet problems

学校读者我要写书评

暂无评论

arXiv 2019年

作者： Luo, Xiaopeng Xu, Xin Rabitz, Herschel Department of Chemistry Princeton University PrincetonNJ08544 United States School of Managment and Engineering Nanjing University Nanjing210008 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In this paper, a meshless Hermite-HDMR finite difference method is proposed to solve high-dimensional Dirichlet problems. The approach is based on the local Hermite-HDMR expansion with an additional smoothing technique. First, we introduce the HDMR decomposition combined with the multiple Hermite series to construct a class of Hermite-HDMR approximations, and the relevant error estimate is theoretically built in a class of Hermite spaces. It can not only provide high order convergence but also retain good scaling with increasing dimensions. Then the Hermite-HDMR based finite difference method is particularly proposed for solving high-dimensional Dirichlet problems. By applying a smoothing process to the Hermite-HDMR approximations, numerical stability can be guaranteed even with a small number of nodes. Numerical experiments in dimensions up to 30 show that resulting approximations are of very high quality. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite difference method

Accelerating GMRES with deep learning in real-time

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Luna, Kevin Klymko, Katherine Blaschke, Johannes P. Program in Applied Mathematics The University of Arizona United States Computational Research Division Lawrence Berkeley National Laboratory United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory United States

GMRES is a powerful numerical solver used to find solutions to extremely large systems of linear equations. These systems of equations appear in many applications in science and engineering. Here we demonstrate a real-time machine learning algorithm that can be used to accelerate the time-to-solution for GMRES. Our framework is novel in that is integrates the deep learning algorithm in an in situ fashion: the AI-accelerator gradually learns how to optimizes the time to solution without requiring user input (such as a pre-trained data set). We describe how our algorithm collects data and optimizes GMRES. We demonstrate our algorithm by implementing an accelerated (MLGMRES) solver in Python. We then use MLGMRES to accelerate a solver for the Poisson equation - a class of linear problems that appears in may applications. Informed by the properties of formal solutions to the Poisson equation, we test the performance of different neural networks. Our key takeaway is that networks which are capable of learning non-local relationships perform well, without needing to be scaled with the input problem size, making them good candidates for the extremely large problems encountered in high-performance computing. For the inputs studied, our method provides a roughly 2× acceleration. Copyright © 2021, The Authors. All rights reserved.

关键词： Poisson equation

CAUCHY MARKOV RANDOM FIELD PRIORS FOR BAYESIAN INVERSION

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Suuronen, Jarkko Chada, Neil K. Roininen, Lassi School of Engineering Science Lappeenranta-Lahti University of Technology PO Box 20 LappeenrantaFI-53851 Finland Applied Mathematics and Computational Science Program King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia

The use of Cauchy Markov random field priors in statistical inverse problems can potentially lead to posterior distributions which are non-Gaussian, high-dimensional, multimodal and heavy-tailed In order to use such priors successfully, sophisticated optimization and Markov chain Monte Carlo (MCMC) methods are usually required In this paper, our focus is largely on reviewing recently developed Cauchy difference priors, while introducing interesting new variants, whilst providing a comparison We firstly propose a one-dimensional second order Cauchy difference prior, and construct new first and second order two-dimensional isotropic Cauchy difference priors Another new Cauchy prior is based on the stochastic partial differential equation approach, derived from Matérn type Gaussian presentation The comparison also includes Cauchy sheets Our numerical computations are based on both maximum a posteriori and conditional mean estimation We exploit state-of-the-art MCMC methodologies such as Metropolis-within-Gibbs, Repelling-Attracting Metropolis, and No-U-Turn sampler variant of Hamiltonian Monte Carlo We demonstrate the models and methods constructed for one-dimensional and two-dimensional deconvolution problems Thorough MCMC statistics are provided for all test cases, including potential scale reduction factors. Copyright © 2021, The Authors. All rights reserved.

关键词： Bayesian networks

Numerical meshless solution of high-dimensional sine-gordon equations via fourier HDMR-HC approximation

学校读者我要写书评

暂无评论

arXiv 2019年

作者： Xu, Xin Luo, Xiaopeng Rabitz, Herschel Department of Chemistry Princeton University PrincetonNJ08544 United States School of Managment and Engineering Nanjing University Nanjing210008 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs. Copyright © 2019, The Authors. All rights reserved.

关键词： Numerical methods