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Unbiased Estimation using a Class of Diffusion Processes

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arXiv 2022年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 KSA Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (d, B(d)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process {Xt}tϵ[0,1]on d, d ϵ 0. This latter process is constructed such that, starting at X0= 0, one has X1∼ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, [10, 16] consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t. π. We show that for this methodology to achieve a mean square error of O(ϵ2), for arbitrary ϵ > 0, the associated cost is O(ϵ-5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t. π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ϵ2), the associated cost (which is random) is, with high probability, O(ϵ-2| log(ϵ)|2+δ), for any δ > 0. We implement our method on several examples including Bayesian inverse *** Codes 60J60, 62D05, 65C40 © 2022, CC BY.

关键词： Diffusion

Splitting Guarantees for Prophet Inequalities via Nonlinear Systems

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arXiv 2024年

作者： Brustle, Johannes Perez-Salazar, Sebastian Verdugo, Victor Department of Computer and Systems Sciences Sapienza University of Rome Italy Department of Computational Applied Mathematics and Operations Research Rice University United States Institute for Mathematical and Computational Engineering Pontificia Universidad Católica de Chile Chile Department of Industrial and Systems Engineering Pontificia Universidad Católica de Chile Chile

The prophet inequality is one of the cornerstone problems in optimal stopping theory and has become a crucial tool for designing sequential algorithms in Bayesian settings. In the i.i.d. k-selection prophet inequality problem, we sequentially observe n non-negative random values sampled from a known distribution. Each time, a decision is made to accept or reject the value, and under the constraint of accepting at most k. For k = 1, Hill and Kertz [Ann. Probab. 1982] provided an upper bound on the worst-case approximation ratio that was later matched by an algorithm of Correa et al. [Math. Oper. Res. 2021]. The worst-case tight approximation ratio for k = 1 is computed by studying a differential equation that naturally appears when analyzing the optimal dynamic programming policy. A similar result for k > 1 has remained elusive. In this work, we introduce a nonlinear system of differential equations for the i.i.d. k-selection prophet inequality that generalizes Hill and Kertz's equation when k = 1. Our nonlinear system is defined by k constants that determine its functional structure, and their summation provides a lower bound on the optimal policy's asymptotic approximation ratio for the i.i.d. k-selection prophet inequality. To obtain this result, we introduce for every k an infinite-dimensional linear programming formulation that fully characterizes the worst-case tight approximation ratio of the k-selection prophet inequality problem for every n, and then we follow a dual-fitting approach to link with our nonlinear system for sufficiently large values of n. As a corollary, we use our provable lower bounds to establish a tight approximation ratio for the stochastic sequential assignment problem in the i.i.d. non-negative regime. Copyright © 2024, The Authors. All rights reserved.

关键词： Nonlinear systems

ANALYSIS OF A DILUTE POLYMER MODEL WITH A TIME-FRACTIONAL DERIVATIVE

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arXiv 2023年

作者： Fritz, Marvin Süli, Endre Wohlmuth, Barbara Computational Methods for PDEs Johann Radon Institute for Computational and Applied Mathematics Altenberger Str. 69 Linz4040 Austria Mathematical Institute University of Oxford Andrew Wiles Building Woodstock Road OxfordOX2 6GG United Kingdom Department of Mathematics Technical University of Munich Boltzmannstr. 3 Garching bei München85748 Germany

We investigate the well-posedness of a coupled Navier–Stokes–Fokker–Planck system with a time-fractional derivative. Such systems arise in the kinetic theory of dilute solutions of polymeric liquids, where the motion of noninteracting polymer chains in a Newtonian solvent is modelled by a stochastic process exhibiting power-law waiting time, in order to capture subdiffusive processes associated with non-Fickian diffusion. We outline the derivation of the model from a subordinated Langevin equation. The elastic properties of the polymer molecules immersed in the solvent are modelled by a finitely extensible nonlinear elastic (FENE) dumbbell model, and the drag term in the Fokker–Planck equation is assumed to be corotational. We prove the global-in-time existence of large-data weak solutions to this time-fractional model of order α ∈ (12 , 1), and derive an energy inequality satisfied by weak *** Codes 35Q30, 35Q84, 35R11, 60G22, 82C31, 82D60 © 2023, CC BY.

关键词： Fokker Planck equation

Cross sections for valence-shell excitations of H2O in the 9.85–12.15-eV energy-loss range studied by high-energy inelastic electron scattering

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Physical Review A 2022年第3期105卷 032801-032801页

作者： Wei-Qing Xu Yi-Geng Peng Jian-Hui Zhu Xiao-Jiao Du Yuan-Chen Xu Li-Han Wang Bo Li Hao-Ran Zhang Bei-Yuan Zhang Zi-Ru Ma Shu-Xing Wang Yong Wu Jian-Guo Wang Lin-Fan Zhu Center for Transformative Science ShanghaiTech University Shanghai 201210 China Department of Applied Physics Nanjing University of Science and Technology Nanjing 210094 China Institute of Applied Physics and Computational Mathematics Beijing 100088 China Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics University of Science and Technology of China Hefei Anhui 230026 China HEDPS Center for Applied Physics and Technology and College of Engineering Peking University Beijing 100871 China

The recent measurement of the generalized oscillator strengths (GOSs) for ÃB11 and B̃A11 of H2O exhibits the apparent deviations from the first Born approximation calculations in the large squared momentum transfer region [W. Q. Xu et al., Phys. Rev. A 103, 032808 (2021)]. Here a simplified theoretical model was proposed to calculate the GOSs of the two excitations at large momentum transfer within the second Born approximation. The accord of the experimental GOSs and the theoretical ones is improved, but the differences still exist. In addition, the GOSs and the integral cross sections for the higher excitations of H2O in 9.85–12.15 eV are presented, which could be used to complete the database of electron-impact excitation cross sections of H2O and be beneficial for modeling interaction processes between electrons and H2O in the earth's atmosphere, plasmas, radiation physics, biology, etc.

关键词： Electron & positron scattering Electronic excitation & ionization Electronic structure of atoms & molecules Molecular spectra Molecules Configuration interaction High-resolution electron energy loss spectroscopy

Analysis of Max-Product via Local Maxifiers

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Analysis of Max-Product via Local Maxifiers

IEEE International Symposium on Information Theory

作者： Stephan Winkler Sekhar Tatikonda David Pollard Program in Applied Mathematics Yale University New Heaven CT USA Department of Electrical Engineering Yale University New Heaven CT USA Department of Statistics Yale University New Heaven CT USA

ISBN: (纸本)142440505X

In this paper we study convergence of the max-product (MP) algorithm on general graphs with cycles. Our analysis follows analogously to that given for the convergence of the sum-product algorithm. We do not work with Gibbs measures but instead we introduce and work with local maxifiers. The contributions of this paper include: reformulation of the MP algorithm on cyclic graphs as max-marginalization on an associated computation tree; existence of local maxifiers and proof that uniqueness of the local maxifier is sufficient for convergence of MP; a Gibbsian theory of local maxifiers and interpretation as operators; an example of non-uniqueness which does not exhibit a phase transition like its Gibbs measure counterpart; and insights into the limitations of Dobrushin-type uniqueness conditions

关键词： Algorithm design and analysis Convergence mathematics Statistical analysis Error correction codes Equations Sum product algorithm Tree graphs Phase measurement Bipartite graph

Leveraging Google’s Tensor Processing Units for tsunami-risk mitigation planning in the Pacific Northwest and beyond

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arXiv 2023年

作者： Madden, Ian Marras, Simone Suckale, Jenny Institute for Computational and Mathematical Engineering Stanford University United States Department of Mechanical Engineering Center for Applied Mathematics and Statistics New Jersey Institute of Technology United States Department of Geophysics Doerr School of Sustainability Stanford University United States

Tsunami-risk and flood-risk mitigation planning has particular importance for communities like those of the Pacific Northwest, where coastlines are extremely dynamic and a seismically-active subduction zone looms large. The challenge does not stop here for risk managers: mitigation options have multiplied since communities have realized the viability and benefits of nature-based solutions. To identify suitable mitigation options for their community, risk managers need the ability to rapidly evaluate several different options through fast and accessible tsunami models, but may lack high-performance computing infrastructure. The goal of this work is to leverage the newly developed Google’s Tensor Processing Unit (TPU), a high-performance hardware accessible via the Google Cloud framework, to enable the rapid evaluation of different tsunami-risk mitigation strategies available to all communities. We establish a starting point through a numerical solver of the nonlinear shallow-water equations that uses a fifth-order Weighted Essentially Non-Oscillatory method with the Lax-Friedrichs flux splitting, and a Total Variation Diminishing third-order Runge-Kutta method for time discretization. We verify numerical solutions through several analytical solutions and benchmarks, reproduce several findings about one particular tsunami-risk mitigation strategy, and model tsunami runup at Crescent City, California whose topography comes from a high-resolution Digital Elevation Model. The direct measurements of the simulations performance, energy usage, and ease of execution show that our code could be a first step towards a community-based, user-friendly virtual laboratory that can be run by a minimally trained user on the cloud thanks to the ease of use of the Google Cloud Platform. © 2023, CC BY.

关键词： Tensors

Coupled chemical reactions: Effects of electric field, diffusion, and boundary control

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Physical Review E 2023年第6期108卷 064413-064413页

作者： Shixin Xu Robert Eisenberg Zilong Song Huaxiong Huang Zu Chongzhi Center for Mathematics and Computational Sciences Duke Kunshan University 8 Duke Ave Kunshan Jiangsu 215316 China Department of Applied Mathematics Illinois Institute of Technology Chicago Illinois 60616 USA and Department of Physiology and Biophysics Rush University Chicago Ilinois 60612 USA Math and Statistics Department Utah State University Old Main Hill Logan Utah 84322 USA Research Center for Mathematics Advanced Institute of Natural Sciences Beijing Normal University Zhuhai Guangdong 519088 China Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science BNU-HKBU United International College Zhuhai Guangdong 519088 China Laboratory of Mathematics and Complex Systems MOE Beijing Normal University Beijing 100875 China and Department of Mathematics and Statistics York University Toronto Ontario Canada M3J 1P3

Chemical reactions involve the movement of charges, and this paper presents a mathematical model for describing chemical reactions in electrolytes. The model is developed using an energy variational method that aligns with classical thermodynamics principles. It encompasses both electrostatics and chemical reactions within consistently defined energetic and dissipative functionals. Furthermore, the energy variation method is extended to account for open systems that involve the input and output of charge and mass. Such open systems have the capability to convert one form of input energy into another form of output energy. In particular, a two-domain model is developed to study a reaction system with self-regulation and internal switching, which plays a vital role in the electron transport chain of mitochondria responsible for ATP generation—a crucial process for sustaining life. Simulations are conducted to explore the influence of electric potential on reaction rates and switching dynamics within the two-domain system. It shows that the electric potential inhibits the oxidation reaction while accelerating the reduction reaction.

关键词： Continuum mechanics Dissipative dynamics Electric field effects

Dynamics of O6+ + H electron capture in Debye plasmas and properties of resulting O5+(nl) emission spectra

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Physical Review A 2010年第2期82卷 022701-022701页

作者： Y. Q. Zhao L. Liu J. G. Wang R. K. Janev P. Xue Department of Physics and Key Laboratory of Atomic and Molecular Nanosciences Tsinghua University Beijing 100084 China Institute of Applied Physics and Computational Mathematics P.O. Box 8009 Beijing 100088 China Macedonian Academy of Sciences and Arts P.O. Box 428 1000 Skopje Macedonia

The electron capture in O6+-H collisions is studied by the two-center atomic orbital close-coupling (TC-AOCC) method when the interactions of charged particles are screened and have a Yukawa form. Atomic orbitals and eigenenergies of n ⩽ 6 states on O5+ and n ⩽ 2 states on H are calculated as a function of the interaction screening parameter and used in the AOCC dynamics scheme to calculate the electron capture cross sections in the energy range 1–200 keV/u. It is shown that the degree of interaction screening reduces the electron binding energies, the number of open electron capture channels, and the strength of exchange couplings, thus affecting the entire collision dynamics and the magnitude and energy behavior of electron capture cross sections. The state-selective electron capture cross sections in this collision system, as well as the intensities of a number of charge exchange spectral lines, for a number of representative interaction screening strengths are presented and discussed.

关键词： Quantum chemistry

Characterizing the inverses of block tridiagonal, block Toeplitz matrices

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computational Science and Discovery 2015年第1期8卷

作者： Boffi, Nicholas M. Hill, Judith C. Reuter, Matthew G. Center for Nanophase Materials Sciences Oak Ridge National Laboratory Oak RidgeTN37831 United States National Center for Computational Sciences Oak Ridge National Laboratory Oak RidgeTN37831 United States Computer Science and Mathematics Division Oak Ridge National Laboratory Oak RidgeTN37831 United States Department of Applied Mathematics and Statistics Institute for Advanced Computational Science Stony Brook University Stony BrookNY11794 United States

We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. Using matrix Möbius transformations, we first present an O(1) representation (with respect to the number of block rows and block columns) for the inverse matrix and subsequently use this representation to characterize the inverse matrix. There are four symmetry-distinct cases where the blocks of the inverse matrix (i) decay to zero on both sides of the diagonal, (ii) oscillate on both sides, (iii) decay on one side and oscillate on the other and (iv) decay on one side and grow on the other. This characterization exposes the necessary conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. Finally, we present numerical examples of these matrix types. © 2015 IOP Publishing Ltd.

关键词： Matrix algebra