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Efficient Numerical Computation of Time-Fractional Nonlinear Schrodinger Equations in Unbounded Domain

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Communications in computational Physics 2019年第1期25卷 218-243页

作者： Jiwei Zhang Dongfang Li Xavier Antoine Beijing Computational Science Research Center Beijing 100193P.R.China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Institut Elie Cartan de Lorraine UMR CNRS 7502Universit́e de LorraineInria Nancy-Grand EstSPHINX TeamF-54506 Vandoeuvre-l`es-Nancy CedexFrance

The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrodinger equation set in an unbounded *** first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in[47,48]and a linearization ***,the initial boundary-value problem for the fractional system with ABCs is discretized,a stability analysis is developed and the error estimate O(h^(2)+τ)is *** accel-erate the L1-scheme in time,a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional *** resulting algorithm is highly efficient for long time ***,we end the paper by reporting some numerical simulations to validate the properties(accuracy and efficiency)of the derived scheme.

关键词： Time-fractional nonlinear Schrodinger equation absorbing boundary condition sta-bility analysis convergence analysis sum-of-exponentials approximation

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

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Communications in computational Physics 2018年第6期24卷 86-103页

作者： Dongfang Li Hong-Lin Liao Weiwei Sun Jilu Wang Jiwei Zhang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Department of Mathematics City University of Hong KongKowloonHong Kong Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing211106China Department of Scientific Computing Florida StateUniversityTallahasseeFL 32306USA Beijing Computational Science Research Center Beijing 100094China.

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element *** analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type *** this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional *** terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear *** theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

关键词： Time-fractional nonlinear parabolic problems L1-Galerkin FEMs Error estimates discrete fractional Gronwall type inequality Linearized schemes

Optimal radial basis for density-based atomic representations

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arXiv 2021年

作者： Goscinski, Alexander Musil, Félix Pozdnyakov, Sergey Nigam, Jigyasa Ceriotti, Michele Laboratory of Computational Science and Modeling Institute of Materials École Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland

The input of almost every machine learning algorithm targeting the properties of matter at the atomic scale involves a transformation of the list of Cartesian atomic coordinates into a more symmetric representation. Many of the most popular representations can be seen as an expansion of the symmetrized correlations of the atom density, and differ mainly by the choice of basis. Considerable effort has been dedicated to the optimization of the basis set, typically driven by heuristic considerations on the behavior of the regression target. Here we take a different, unsupervised viewpoint, aiming to determine the basis that encodes in the most compact way possible the structural information that is relevant for the dataset at hand. For each training dataset and number of basis functions, one can determine a unique basis that is optimal in this sense, and can be computed at no additional cost with respect to the primitive basis by approximating it with splines. We demonstrate that this construction yields representations that are accurate and computationally efficient, particularly when constructing representations that correspond to high-body order correlations. We present examples that involve both molecular and condensed-phase machine-learning models. Copyright © 2021, The Authors. All rights reserved.

关键词： Atoms

Computing the absolute gibbs free energy in atomistic simulations: Applications to defects in solids

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arXiv 2017年

作者： Cheng, Bingqing Ceriotti, Michele Laboratory of Computational Science and Modeling Institute of Materials Ecole Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland

The Gibbs free energy is the fundamental thermodynamic potential underlying the relative stability of different states of matter under constant-pressure conditions. However, computing this quantity from atomic-scale simulations is far from trivial. As a consequence, all too often the potential energy of the system is used as a proxy, overlooking entropic and anharmonic effects. Here we discuss a combination of different thermodynamic integration routes to obtain the absolute Gibbs free energy of a solid system starting from a harmonic reference state. This approach enables the direct comparison between the free energy of different structures, circumventing the need to sample the transition paths between them. We showcase this thermodynamic integration scheme by computing the Gibbs free energy associated with a vacancy in BCC iron, and the intrinsic stacking fault free energy of nickel. These examples highlight the pitfalls of estimating the free energy of crystallographic defects only using the minimum potential energy, which overestimates the vacancy free energy by 60% and the stacking-fault energy by almost 300% at temperatures close to the melting point. Copyright © 2017, The Authors. All rights reserved.

关键词： Gibbs free energy

Atom-Density Representations for Machine Learning

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arXiv 2018年

作者： Willatt, Michael J. Musil, Félix Ceriotti, Michele Laboratory of Computational Science and Modeling Institute of Materials École Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland

The applications of machine learning techniques to chemistry and materials science become more numerous by the day. The main challenge is to devise representations of atomic systems that are at the same time complete and concise, so as to reduce the number of reference calculations that are needed to predict the properties of different types of materials reliably. This has led to a proliferation of alternative ways to convert an atomic structure into an input for a machine-learning model. We introduce an abstract definition of chemical environments that is based on a smoothed atomic density, using a bra-ket notation to emphasize basis set independence and to highlight the connections with some popular choices of representations for describing atomic systems. The correlations between the spatial distribution of atoms and their chemical identities are computed as inner products between these feature kets, which can be given an explicit representation in terms of the expansion of the atom density on orthogonal basis functions, that is equivalent to the smooth overlap of atomic positions (SOAP) power spectrum, but also in real space, corresponding to n-body correlations of the atom density. This formalism lays the foundations for a more systematic tuning of the behavior of the representations, by introducing operators that represent the correlations between structure, composition, and the target properties. It provides a unifying picture of recent developments in the field and indicates a way forward towards more effective and computationally affordable machine-learning schemes for molecules and materials. Copyright © 2018, The Authors. All rights reserved.

关键词： Orthogonal functions

Computing the Tolman length for solid-liquid interfaces

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arXiv 2018年

作者： Cheng, Bingqing Ceriotti, Michele Laboratory of Computational Science and Modeling Institute of Materials Ecole Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland

The curvature dependence of interfacial free energy, which is crucial in quantitatively predicting nucleation kinetics and the stability of bubbles and droplets, can be described in terms of the Tolman length d. For solid-liquid interfaces, however, d has never been computed directly due to various theoretical and practical challenges. Here we present a general method that enables the direct evaluation of the Tolman length from atomistic simulations of a solid-liquid planar interface in out-of-equilibrium conditions. This method works by first measuring the surface tension from the amplitude of thermal capillary fluctuations of a localized version of Gibbs dividing surface, and by then computing the free energy difference between the surface of tension and the equimolar dividing surface. For benchmark purposes, we computed d for a model potential, and compared the results to less rigorous indirect approaches. Copyright © 2018, The Authors. All rights reserved.

关键词： Free energy

Roughness-induced transient growth

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Roughness-induced transient growth

35th AIAA Fluid Dynamics Conference and Exhibit

作者： Choudhari, Meelan Fischer, Paul Computational Modeling and Simulation Branch NASA Langley Research Center M.S. 128 Hampton VA 23681 United States Argonne National Laboratory Mathematics and Computer Science Division 9700 S. Cass Avenue Argonne IL 60439 United States

ISBN: (纸本)9781624100598

Numerical simulations are used to model the boundary layer disturbance field due to a spanwise periodic array of circular disks at the surface. Our earlier computations (Fischer and Choudhari 2004) reproduced some of the major trends associated with roughness induced transient growth as measured by White and Ergin (2003), confirming the accuracy of their data reduction technique. The present paper explores the effects of roughness height, shape, and size on the transient growth along the wake of the roughness array. Results indicate the range of roughness heights over which the energy levels of the dominant stationary disturbance are consistent with the empirically determined Rek2 scaling of White and Ergin. Computations further reveal the intricate effects of disturbance nonlinearity on transient growth characteristics. At sufficiently large roughness heights, simulations indicate spontaneous vortex shedding behind the roughness array, which has been identified in previous experiments as a precursor to premature transition (i.e., tripping) of the laminar boundary layer. The effect of stationary, suboptimal transient growth disturbances on the amplification of Tollmien-Schlichting instability waves in the boundary layer is also assessed in the light of the recent findings concerning the stabilizing influence of optimal growth streaks (Cossu and Brandt 2002).

关键词： Laminar boundary layer

Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems

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Physical Review Letters 2018年第3期120卷 036002-036002页

作者： Andrea Grisafi David M. Wilkins Gábor Csányi Michele Ceriotti Laboratory of Computational Science and Modeling IMX École Polytechnique Fédérale de Lausanne 1015 Lausanne Switzerland Engineering Laboratory University of Cambridge Trumpington Street Cambridge CB21PZ United Kingdom

Statistical learning methods show great promise in providing an accurate prediction of materials and molecular properties, while minimizing the need for computationally demanding electronic structure calculations. The accuracy and transferability of these models are increased significantly by encoding into the learning procedure the fundamental symmetries of rotational and permutational invariance of scalar properties. However, the prediction of tensorial properties requires that the model respects the appropriate geometric transformations, rather than invariance, when the reference frame is rotated. We introduce a formalism that extends existing schemes and makes it possible to perform machine learning of tensorial properties of arbitrary rank, and for general molecular geometries. To demonstrate it, we derive a tensor kernel adapted to rotational symmetry, which is the natural generalization of the smooth overlap of atomic positions kernel commonly used for the prediction of scalar properties at the atomic scale. The performance and generality of the approach is demonstrated by learning the instantaneous response to an external electric field of water oligomers of increasing complexity, from the isolated molecule to the condensed phase.

关键词： Chemical Physics & Physical Chemistry Dielectric properties Molecular spectra Nonlinear optics Density functional theory Machine learning Quantum chemistry methods

Learning the electronic density of states in condensed matter

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Physical Review B 2020年第23期102卷 235130-235130页

作者： Chiheb Ben Mahmoud Andrea Anelli Gábor Csányi Michele Ceriotti Laboratory of Computational Science and Modeling IMX École Polytechnique Fédérale de Lausanne 1015 Lausanne Switzerland Engineering Laboratory University of Cambridge Trumpington Street Cambridge CB21PZ United Kingdom

The electronic density of states (DOS) quantifies the distribution of the energy levels that can be occupied by electrons in a quasiparticle picture and is central to modern electronic structure theory. It also underpins the computation and interpretation of experimentally observable material properties such as optical absorption and electrical conductivity. We discuss the challenges inherent in the construction of a machine-learning (ML) framework aimed at predicting the DOS as a combination of local contributions that depend in turn on the geometric configuration of neighbors around each atom, using quasiparticle energy levels from density functional theory as training data. We present a challenging case study that includes configurations of silicon spanning a broad set of thermodynamic conditions, ranging from bulk structures to clusters and from semiconducting to metallic behavior. We compare different approaches to represent the DOS, and the accuracy of predicting quantities such as the Fermi level, the electron density at the Fermi level, or the band energy, either directly or as a side product of the evaluation of the DOS. We find that the performance of the model depends crucially on the resolution chosen to smooth the DOS and that there is a tradeoff to be made between the systematic error associated with the smoothing and the error in the ML model for a specific structure. We find however that the errors are not strongly correlated among similar structures, and so the average DOS over an ensemble of configurations is in very good agreement with the reference electronic structure calculations, despite the large nominal error on individual configurations. We demonstrate the usefulness of this approach by computing the density of states of a large amorphous silicon sample, for which it would be prohibitively expensive to compute the DOS by direct electronic structure calculations and show how the atom-centered decomposition of the DOS that is obtained through

关键词： Amorphous materials Density functional theory Electronic structure Machine learning Molecular dynamics