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Bayesian Parameter Inference for Partially Observed Stochastic Volterra Equations

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arXiv 2023年

作者： Jasra, Ajay Ruzayqat, Hamza Wu, Amin Applied Mathematics & Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia

In this article we consider Bayesian parameter inference for a type of partially observed stochastic Volterra equation (SVE). SVEs are found in many areas such as physics and mathematical finance. In the latter field they can be used to represent long memory in unobserved volatility processes. In many cases of practical interest, SVEs must be time-discretized and then parameter inference is based upon the posterior associated to this time-discretized process. Based upon recent studies on time-discretization of SVEs (e.g. [24]) we use Euler-Maruyama methods for the afore-mentioned discretization. We then show how multilevel Markov chain Monte Carlo (MCMC) methods [19] can be applied in this context. In the examples we study, we give a proof that shows that the cost to achieve a 4 mean square error (MSE) of O(ϵ2), ϵ > 0, is O(ϵ− 4/2H+1 ), where H is the Hurst parameter. If one uses a single level MCMC method then the cost is O(ϵ−2(2H+3)/ 2H+1 ) to achieve the same MSE. We illustrate these results in the context of state-space and stochastic volatility models, with the latter applied to real *** Codes 62F15, 62M20, 60J10, 60J22, 65C40 © 2023, CC BY.

关键词： Mean square error

Photonic band gap and x-ray optics in warm dense matter

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Photonic band gap and x-ray optics in warm dense matter

作者： Ku, S. Son, S. Moon, Sung Joon Courant Institute of Mathematical Sciences New York University New York NY 10012 United States 18 Caleb Lane Princeton NJ 08540 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States 28 Benjamin Rush Lane Princeton NJ 08540 United States

Photonic band gaps for the soft x rays, formed in the periodic structures of solids or dense plasmas, are theoretically investigated. Optical manipulation mechanisms for the soft x rays, which are based on these band gaps, are computationally demonstrated. The reflection and amplification of the soft x rays and the compression and stretching of chirped soft x-ray pulses are discussed. A scheme for lasing with atoms with two energy levels, utilizing the band gap, is also studied. © 2010 American Institute of Physics.

关键词： Energy gap

Data-driven model reduction and transfer operator approximation

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arXiv 2017年

作者： Klus, Stefan Nüske, Feliks Koltai, Péter Wu, Hao Kevrekidis, Ioannis Schütte, Christof Noé, Frank Department of Mathematics and Computer Science Freie Universität Berlin Germany Department of Chemical and Biological Engineering & Program in Applied and Computational Mathematics Princeton University United States Zuse Institute Berlin Germany

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis (TICA), dynamic mode decomposition (DMD), and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods. Copyright © 2017, The Authors. All rights reserved.

关键词： Dynamical systems

Manifold learning with arbitrary norms

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arXiv 2020年

作者： Kileel, Joe Moscovich, Amit Zelesko, Nathan Singer, Amit Department of Mathematics Oden Institute University of Texas at Austin United States Department of Statistics and Operations Research Tel-Aviv University Israel Department of Mathematics Brown University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using any norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry. Copyright © 2020, The Authors. All rights reserved.

关键词： Mathematical operators

Isotope effects in molecular structures and electronic properties of liquid water via deep potential molecular dynamics based on the SCAN functional

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Physical Review B 2020年第21期102卷 214113-214113页

作者： Jianhang Xu Chunyi Zhang Linfeng Zhang Mohan Chen Biswajit Santra Xifan Wu Department of Physics Temple University Philadelphia Pennsylvania 19122 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA CAPT HEDPS College of Engineering Peking University Beijing 100871 China Institute for Computational Molecular Science Temple University Philadelphia Pennsylvania 19122 USA

Feynman path-integral deep potential molecular dynamics (PI-DPMD) calculations have been employed to study both light (H2O) and heavy water (D2O) within the isothermalisobaric ensemble. In particular, the deep neural network is trained based on ab initio data obtained from the strongly constrained and appropriately normed (SCAN) exchange-correlation functional. Because of the lighter mass of hydrogen than deuteron, the properties of light water are more influenced by nuclear quantum effect than those of heavy water. Clear isotope effects are observed and analyzed in terms of hydrogen-bond structure and electronic properties of water that are closely associated with experimental observables. The molecular structures of both liquid H2O and D2O agree well with the data extracted from scattering experiments. The delicate isotope effects on radial distribution functions and angular distribution functions are well reproduced as well. Our approach demonstrates that deep neural network combined with SCAN functional based ab initio molecular dynamics provides an accurate theoretical tool for modeling water and its isotope effects.

关键词： Atomic & molecular structure Electronic structure Isotope effect Water Ab initio calculations Density functional theory Machine learning Molecular dynamics Path-integral methods

Classical many-particle systems with unique disordered ground states

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Physical Review E 2017年第4期96卷 042146-042146页

作者： G. Zhang F. H. Stillinger S. Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Classical ground states (global energy-minimizing configurations) of many-particle systems are typically unique crystalline structures, implying zero enumeration entropy of distinct patterns (aside from trivial symmetry operations). By contrast, the few previously known disordered classical ground states of many-particle systems are all high-entropy (highly degenerate) states. Here we show computationally that our recently proposed “perfect-glass” many-particle model [Sci. Rep. 6, 36963 (2016)] possesses disordered classical ground states with a zero entropy: a highly counterintuitive situation . For all of the system sizes, parameters, and space dimensions that we have numerically investigated, the disordered ground states are unique such that they can always be superposed onto each other or their mirror image. At low energies, the density of states obtained from simulations matches those calculated from the harmonic approximation near a single ground state, further confirming ground-state uniqueness. Our discovery provides singular examples in which entropy and disorder are at odds with one another. The zero-entropy ground states provide a unique perspective on the celebrated Kauzmann-entropy crisis in which the extrapolated entropy of a supercooled liquid drops below that of the crystal. We expect that our disordered unique patterns to be of value in fields beyond glass physics, including applications in cryptography as pseudorandom functions with tunable computational complexity.

关键词： Nonequilibrium statistical mechanics Pattern formation

Antithetic Multilevel Particle Filters

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arXiv 2023年

作者： Jasra, Ajay Maama, Mohamed Ombao, Hernando Applied Mathematics and Computational Science Program United States Statistics Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of [12]. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of [10]. We show that for a class of diffusion problems, for ϵ > 0 given, that the cost to produce a mean square error (MSE) in estimation of the filter, of O(ϵ2) is O(ϵ−2 log(ϵ)2). In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of [12] has a cost of O(ϵ−2.5) to achieve the same MSE. We support our theory with numerical results in several examples. Copyright © 2023, The Authors. All rights reserved.

关键词： Mean square error

Interior electroneutrality in nernst-planck-navier-stokes systems

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arXiv 2020年

作者： Constantin, Peter Ignatova, Mihaela Lee, Fizay-Noah DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States DEPARTMENT OF MATHEMATICS TEMPLE UNIVERSITY PHILADELPHIAPA19122 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider the limit of vanishing Debye length for ionic diffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. In the asymptotically stable cases of blocking (vanishing normal flux) and uniform selective (special Dirichlet) boundary conditions for the ionic concentrations, we prove that the ionic charge density ρ converges in time to zero in the interior of the domain, in the limit of vanishing Debye length (ǫ → 0). For the unstable regime of Dirichlet boundary conditions for the ionic concentrations, we prove bounds that are uniform in time and ǫ. We also consider electroneutral boundary conditions, for which we prove that electroneutrality ρ → 0 is achieved at any fixed ǫ > 0, exponentially fast in time in Lp, for all 1 ≤ p MSC Codes 35Q30, 35Q35, 35Q92 Copyright © 2020, The Authors. All rights reserved.

关键词： Boundary conditions

Ab Initio Generalized Langevin Equation

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arXiv 2022年

作者： Xie, Pinchen Car, Roberto Weinan, E. Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States 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 AI for Science Institute Beijing China Center for Machine Learning Research School of Mathematical Sciences Peking University Beijing China

We introduce a machine learning-based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori-Zwanzig formalism, under the constraint of the fluctuation-dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multi-scale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of collective variables by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude, and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency. © 2022, CC BY.

关键词： Molecular dynamics