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A wong-zakai approximation for random slow manifolds with application to parameter estimation

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arXiv 2018年

作者： He, Ziying Zhang, Xinyong Jiang, Tao Liu, Xianming Center for Mathematical Sciences Huazhong University of Sciences and Technology Wuhan430074 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Mathematical Sciences Tsinghua University Beijing100084 China Collaborative Innovation Center of China Pilot Reform Exploration and Assessment Hubei Sub-Center Hubei University of Economics Wuhan430205 China

We study a Wong-Zakai approximation for the random slow manifold of a slow-fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein-Uhlenbeck (O-U) process. Then we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this *** Codes Primary: 37L55, 35R60, Secondary: 60H15, 58J65 Copyright © 2018, The Authors. All rights reserved.

关键词： Parameter estimation

Fourth-Order Conservative Non-Splitting Semi-Lagrangian Hermite Weno Schemes for Kinetic and Fluid Simulations

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SSRN

SSRN 2022年

作者： Zheng, Nanyi Cai, Xiaofeng Qiu, Jing-Mei Qiu, Jianxian School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Research Center for Mathematics Beijing Normal University Zhuhai519087 China BNU-HKBU United International College Zhuhai519087 China Department of Mathematical Sciences University of Delaware NewarkDE19716 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass conservation naturally holds due to the weak formulation of the characteristic Galerkin method and the design of the HWENO reconstructions. We apply a positive-preserving limiter to maintain the positivity of numerical solutions when needed. Although the proposed SL framework allows us to take large time steps for improving computational efficiency, it also brings challenges to the spatial reconstruction technique;we construct two kind of novel HWENO reconstructions to fit the need for the proposed SL framework. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes. © 2022, The Authors. All rights reserved.

关键词： Computational efficiency

Fourth-order conservative non-splitting semi-Lagrangian Hermite WENO schemes for kinetic and fluid simulations

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arXiv 2022年

关键词： Computational efficiency

mathematical approaches for characterization, control, calibration and validation of a quantum computing device

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arXiv 2023年

作者： Peng, Zhichao Appelö, Daniel Petersson, N. Anders Garcia, Fortino Cho, Yujin Mathematics Department Michigan State University United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Courant Institute of Mathematical Sciences New York University United States Quantum Coherent Device Physics Lawrence Livermore National Laboratory United States

Quantum computing has received significant amounts of interest from many different research communities over the last few years. Although there are many introductory texts that focus on the algorithmic parts of quantum computing, there is a dearth of publications that describe the modeling, calibration and operation of current quantum computing devices. One aim of this report is to fill that void by providing a case study that walks through the entire procedure from the characterization and optimal control of a qudit device at Lawrence Livermore National Laboratory (LLNL) to the validation of the results. A goal of the report is to provide an introduction for students and researchers, especially computational mathematicians, who are interested in but new to quantum computing. Both experimental and mathematical aspects of this procedure are discussed. We present a description of the LLNL QuDIT testbed, the mathematical models that are used to describe it, and the numerical methods that are used to to design optimal controls. We also present experimental and computational methods that can be used to characterize a quantum device. Finally, an experimental validation of an optimized control pulse is presented, which relies on the accuracy of the characterization and the optimal control methodologies. © 2023, CC BY-NC-ND.

关键词： Numerical methods

Deterministic and Bayesian Characterization of Quantum computing Devices

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arXiv 2023年

作者： Peng, Zhichao Appelö, Daniel Petersson, N. Anders Motamed, Mohammad Garcia, Fortino Cho, Yujin Department of Mathematics Michigan State University United States Department of Mathematics Virginia Tech United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Department of Mathematics and Statistics University of New Mexico United States Courant Institute of Mathematical Sciences New York University United States Quantum Coherent Device Physics Lawrence Livermore National Laboratory United States

Motivated by the noisy and fluctuating behavior of current quantum computing devices, this paper presents a data-driven characterization approach for estimating transition frequencies and decay times in a Lindbladian dynamical model of a superconducting quantum device. The data includes parity events in the transition frequency between the first and second excited states. A simple but effective mathematical model, based upon averaging solutions of two Lindbladian models, is demonstrated to accurately capture the experimental observations. A deterministic point estimate of the device parameters is first performed to minimize the misfit between data and Lindbladian simulations. These estimates are used to make an informed choice of prior distributions for the subsequent Bayesian inference. An additive Gaussian noise model is developed for the likelihood function, which includes two hyper-parameters to capture the noise structure of the data. The outcome of the Bayesian inference are posterior probability distributions of the transition frequencies, which for example can be utilized to design risk neutral optimal control pulses. The applicability of our approach is demonstrated on experimental data from the Quantum Device and Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory, using a tantalum-based superconducting transmon *** Codes 62F15, 81P68 © 2023, CC BY-SA.

关键词： Probability distributions

Universal behavior of dispersive Dirac cone in gradient-index plasmonic metamaterials

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arXiv 2017年

作者： Maier, Matthias Mattheakis, Marios Kaxiras, Efthimios Luskin, Mitchell Margetis, Dionisios School of Mathematics University of Minnesota MinneapolisMN55455 United States School of Engineering and Applied Sciences Harvard University CambridgeMA02138 United States Department of Physics Harvard University CambridgeMA02138 United States Department of Mathematics Institute for Physical Science and Technology Center for Scientific Computation and Mathematical Modeling University of Maryland College ParkMD20742 United States

We demonstrate analytically and numerically that the dispersive Dirac cone emulating an epsilon-near-zero (ENZ) behavior is a universal property within a family of plasmonic crystals consisting of two-dimensional (2D) metals. Our starting point is a periodic array of 2D metallic sheets embedded in an inhomogeneous and anisotropic dielectric host that allows for propagation of transverse-magnetic (TM) polarized waves. By invoking a systematic bifurcation argument for arbitrary dielectric profiles in one spatial dimension, we show how TM Bloch waves experience an effective dielectric function that averages out microscopic details of the host medium. The corresponding effective dispersion relation reduces to a Dirac cone when the conductivity of the metallic sheet and the period of the array satisfy a critical condition for ENZ behavior. Our analytical findings are in excellent agreement with numerical simulations. Copyright © 2017, The Authors. All rights reserved.

关键词： Plasmonics

EXPLICIT RUNGE–KUTTA METHODS THAT ALLEVIATE ORDER REDUCTION

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arXiv 2023年

作者： Biswas, Abhijit Ketcheson, David I. Roberts, Steven Seibold, Benjamin Shirokoff, David Computer Electrical and Mathematical Sciences & Engineering Division King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Mathematics Temple University PhiladelphiaPA19122 United States Department of Mathematical Sciences New Jersey Institute of Technology NewarkNJ07102 United States

Explicit Runge–Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to initial-boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high-order convergence for linear problems;we refer to these conditions as weak stage order conditions. We prove a general relationship between the method’s order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear *** Codes 65L04, 65L20, 65M12 Copyright © 2023, The Authors. All rights reserved.

关键词： Runge Kutta methods

Reduced Basis Catalogs for Gravitational Wave Templates

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Physical Review Letters 2011年第22期106卷 221102-221102页

作者： Scott E. Field Chad R. Galley Frank Herrmann Jan S. Hesthaven Evan Ochsner Manuel Tiglio Department of Physics Brown University Providence Rhode Island 02912 USA Jet Propulsion Laboratory and TAPIR California Institute of Technology Pasadena California 91125 USA Department of Physics Maryland Center for Fundamental Physics Joint Space Sciences Institute Center for Scientific Computation and Mathematical Modeling University of Maryland College Park Maryland 20742 USA Division of Applied Mathematics Brown University Providence Rhode Island 02912 USA Center for Gravitation and Cosmology University of Wisconsin-Milwaukee Milwaukee Wisconsin 53201 USA Department of Physics Maryland Center for Fundamental Physics University of Maryland College Park Maryland 20742 USA

We introduce a reduced basis approach as a new paradigm for modeling, representing and searching for gravitational waves. We construct waveform catalogs for nonspinning compact binary coalescences, and we find that for accuracies of 99% and 99.999% the method generates a factor of about 10–105 fewer templates than standard placement methods. The continuum of gravitational waves can be represented by a finite and comparatively compact basis. The method is robust under variations in the noise of detectors, implying that only a single catalog needs to be generated.

关键词： Gravity waves

CONVERGENCE OF THE PLANEWAVE APPROXIMATIONS FOR QUANTUM INCOMMENSURATE SYSTEMS

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arXiv 2022年

作者： Wang, Ting Chen, Huajie Zhou, Aihui Zhou, Yuzhi Massatt, Daniel School of Mathematical Sciences Beijing Normal University Beijing100875 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China Mathematics Department Louisiana State University LA70802 United States

Incommensurate structures come from stacking the single layers of low-dimensional materials on top of one another with misalignment such as a twist in orientation. While these structures are of significant physical interest, they pose many theoretical challenges due to the loss of periodicity. This paper studies the physical observables of continuum Schrödinger operators for incommensurate systems. We characterize the density of states and local density of states in real and reciprocal spaces, and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states and local density of states via the operator kernel characterization;and (ii) propose efficient numerical schemes based on planewave approximation and trapezoidal rule in reciprocal space. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical *** Codes 65Z05, 81-08, 47G30 Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods