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Thermal disorder and phonon softening in the ferroelectric phase transition of lead titanate

Physical Review B 2025年第9期111卷 094113-094113页

作者： Pinchen Xie Yixiao Chen Weinan E Roberto Car Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA AI for Science Institute Beijing China and Center for Machine Learning Research and School of Mathematical Sciences Peking University Beijing China

We report a molecular dynamics study of ab initio quality of the ferroelectric phase transition in crystalline PbTiO3. We model anharmonicity accurately in terms of potential energy and polarization surfaces trained on density functional theory data with modern machine learning techniques. Our simulations demonstrate that the transition has a strong order-disorder character, in agreement with diffraction experiments, and provide fresh insight into the approach to equilibrium across the phase transition. We find that the emergence and disappearance of the macroscopic polarization is driven by dipolar switching at the nanometer scale. We also computed the infrared optical absorption spectra in both the ferroelectric and the paraelectric phases, finding good agreement with the experimental Raman frequencies. Often, the almost ideal displacive character of the soft mode detected by Raman scattering in the paraelectric phase has been contrasted with the order-disorder character of the transition suggested by diffraction experiments. We settle this issue by showing that the soft mode coexists with a strong Debye relaxation associated with thermal disordering of the dipoles. The Debye relaxation feature is centered at zero frequency and appears near the transition temperature in both the ferroelectric and the paraelectric phases.

关键词： Ferroelectricity

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Unbiased Parameter Estimation for Bayesian Inverse Problems

arXiv

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arXiv 2025年

作者： Chada, Neil K. Jasra, Ajay Maama, Mohamed Tempone, Raul Department of Mathematics City University of Hong Kong China School of Data Science The Chinese University of Hong Kong Shenzhen CN Shenzhen China 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

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximizer of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE. Copyright © 2025, The Authors. All rights reserved.

关键词： Markov processes

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Asymptotic Analysis of Quantum Dynamics in Crystals: the Bloch-Wigner Transform, Bloch Dynamics and Berry Phase

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Acta Mathematicae Applicatae Sinica 2013年第3期29卷 465-476页

作者： Weinan E Jian-feng LU Xu YANG Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University School of Mathematical Sciences Peking University

We study the semi-classical limit of the Schro¨dinger equation in a crystal in the presence of an external potential and magnetic field. We first introduce the Bloch-Wigner transform and derive the asymptotic equations governing this transform in the semi-classical setting. For the second part, we focus on the appearance of the Berry curvature terms in the asymptotic equations. These terms play a crucial role in many important physical phenomena such as the quantum Hall effect. We give a simple derivation of these terms in different settings using asymptotic analysis.

关键词： semiclassical limit Bloch-Wigner transform Bloch dynamics Berry phase asymptotic analysis

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The Illinois resource information system: Early innovations in geographic information system design

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Cartography and Geographic Information Science 2006年第2期33卷 97-114页

作者： Armstrong, Marc P. Department of Geography Program in Applied Mathematical and Computational Sciences The University of Iowa 316 Jessup Hall Iowa City IA 52242 United States

The Illinois Resource Information System (IRIS) was designed and implemented in the early 1970s by computer scientists at the University of Illinois. These researchers did not continue to work in the emerging field of Geographic Information Systems (GIS), and as a consequence, IRIS innovations were not reported in the GIS literature. IRIS operated across the then-new ARPANet, making it the first networked GIS. The system also had an interactive query language with a rich syntax. Retrievals in IRIS were optimized to promote high levels of interactivity. The development of IRIS had elements in common with the first supercomputer (ILLIAC IV), which was designed at the University of Illinois during that time period. A student who worked on the IRIS project also produced the first doctoral dissertation with "Geographic Information Systems" in its title.

关键词： Geographic information systems

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Domains of mobility for a planar body moving among obstacles

Domains of mobility for a planar body moving among obstacles

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ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference, DETC-CIE 1996

作者： Haug, Edward J. Adkins, Frederick A. Coroian, Dan I. Department of Mechanical Engineering Program in Applied Mathematical and Computational Sciences The University of Iowa United States

ISBN: (纸本)9780791897591

A formulation is presented for defining domains of mobility for a planar convex body moving with three degrees-of-freedom among convex planar obstacles. Applications included are determination of areas of a factory floor or material storage facility in which objects can be manipulated without impacting fixed obstacles. Mobility of the moving body is defined to encompass (1) dextrous mobility of the body;i.e., points that can be reached by a reference point on the body and at which the body can be rotated through its full range of admissible orientations without penetrating any stationary obstacle, and (2) limited mobility of the body;i.e., points that can be reached by the reference point and at which the body does not penetrate any stationary obstacle, for some admissible orientation. Analytical criteria for points on boundaries of domains of mobility are derived and numerical methods suitable for mapping these boundaries are summarized. An elementary example involving a moving and a stationary ellipse, with and without orientation restrictions, is solved analytically to illustrate the method. A more general application with one moving body and three stationary obstacles is solved numerically. © 1996 American Society of Mechanical Engineers (ASME). All rights reserved.

关键词： Numerical methods

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NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

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Journal of computational Mathematics 2019年第4期37卷 506-523页

作者： Weimin Han Ziping Huang Cheng Wang Wei Xu Program in Applied Mathematical and Computational Sciences (A MCS)& Department of Mathematics University of Iowa Iowa City IA 522/2 USA School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College J taxing 314051 China School of Mathematical Sciences Tojigji University Shanghai 200092 China School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College Jiaxing 314051 China

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

关键词： Hemivariational inequality Interior semipermeability Boundary semipermeability Finite element method Error estimate

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Effective Maxwell Equations from Time-dependent Density Functional Theory

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Acta Mathematica Sinica,English Series 2011年第2期27卷 339-368页

作者： Weinan E Jianfeng LU Xu YANG Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Beijing International Center for Mathematical Research and School of Mathematical Sciences Peking University Beijing 100871 P. R. China Department of Mathematics Courant Institute of Mathematical Sciences New York University New York NY 10012 USA

The behavior of interacting electrons in a perfect crystal under macroscopic external electric and magnetic fields is studied. Effective Maxwell equations for the macroscopic electric and magnetic fields are derived starting from time-dependent density functional theory. Effective permittivity and permeability coefficients are obtained.

关键词： Time-dependent density functional theory Maxwell equations effective permittivity electromagnetic fields

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Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

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Communications in computational Physics 2011年第3期9卷 668-687页

作者： Shi Jin Hao Wu Xu Yang Department of Mathematics University of WisconsinMadisonWI 53706USA Department of Mathematical Sciences Tsinghua UniversityBeijing 10084China Program in Applied and Computational Mathematics Princeton UniversityNJ 08544USA

A novel Eulerian Gaussian beam method was developed in[8]to compute the Schrödinger equation efficiently in the semiclassical *** this paper,we introduce an efficient semi-Eulerian implementation of this *** new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing *** difference lies in that,we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these *** yields effectively a local level set implementation,and the beam summation can be carried out on the initial physical space instead of the phase *** a consequence,it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam ***,the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods,which is the topic of the second part of this *** numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.

关键词： Schrödinger equation semi-Eulerian Gaussian beam method high order methods

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Mean field limit of a dynamical model of polymer systems

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Science China Mathematics 2013年第12期56卷 2591-2598页

作者： E Weinan SHEN Hao School of Mathematical Sciences and BICMR Peking University Department of Mathematics and PACM Princeton UniversityPrinceton NJ 08544 USA Program in Applied and Computational Mathematics Princeton UniversityPrinceton NJ 08544 USA

This paper provides a mathematically rigorous foundation for self-consistent mean field theory of the polymeric physics. We study a new model for dynamics of mono-polymer systems. Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces. Every two points on the same string or on two different strings also interact under a pairwise potential V. The dynamics of the system is described by a system of N coupled stochastic partial differential equations （SPDEs）. We show that the mean field limit as N -＋ c~ of the system is a self-consistent McKean-Vlasov type equation, under suitable assumptions on the initial and boundary conditions and regularity of V. We also prove that both the SPDE system of the polymers and the mean field limit equation are well-posed.

关键词： polymers mean field limit stochastic PDEs McKean-Vlasov equation

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English-Chinese Translation using Neural Machine Models

English-Chinese Translation using Neural Machine Models

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2022 IEEE Conference on Telecommunications, Optics and Computer Science, TOCS 2022

作者： Liu, Qi Ou, Pengfei University of Washington Applied and Computational Mathematical Sciences Program Seattle United States University of Illinois at Urbana Champaign Department of Mathematics Urbana United States

ISBN: (纸本)9781665470537

During the last few years, neural machine translation (NMT) as a notable branch of machine translation has been increasing its popularity both in research and in practice. In particular, neural machine translation between English and Chinese, which are two of the most widely used languages all over the world, is receiving more and more attention. In this review, we discuss current mainstream models for neural machine translation, including recurrent neural network (RNN), convolution neural network (CNN), and self-attention network (SAN) or transformer. The mechanisms of these models are illustrated. Moreover, examples of studies and applications of them are analyzed. In addition, comparisons on the performance of different models are implemented. © 2022 IEEE.

关键词： Neural machine translation

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