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Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere

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arXiv 2019年

作者： Hamon, François P. Schreiber, Martin Minion, Michael L. Center for Computational Sciences and Engineering Lawrence Berkeley National Laboratory Berkeley United States Chair of Computer Architecture and Parallel Systems Technical University of Munich Germany Department of Applied Mathematics Lawrence Berkeley National Laboratory Berkeley United States

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field. We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space. The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes. Copyright © 2019, The Authors. All rights reserved.

关键词： Harmonic analysis

QCQP-Net: Reliably Learning Feasible Alternating Current Optimal Power Flow Solutions Under Constraints

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arXiv 2024年

作者： Zeng, Sihan Kim, Youngdae Ren, Yuxuan Kim, Kibaek J.P. Morgan AI Research United States ExxonMobil Technology and Engineering Company – Research United States Department of Computational Applied Mathematics & Operations Research Rice University United States Argonne National Laboratory United States

At the heart of power system operations, alternating current optimal power flow (ACOPF) studies the generation of electric power in the most economical way under network-wide load requirement, and can be formulated as a highly structured non-convex quadratically constrained quadratic program (QCQP). Optimization-based solutions to ACOPF (such as ADMM or interior-point method), as the classic approach, require large amount of computation and cannot meet the need to repeatedly solve the problem as load requirement frequently changes. On the other hand, learning-based methods that directly predict the ACOPF solution given the load input incur little computational cost but often generates infeasible solutions (i.e. violate the constraints of ACOPF). In this work, we combine the best of both worlds – we propose an innovated framework for learning ACOPF, where the input load is mapped to the ACOPF solution through a neural network in a computationally efficient and reliable manner. Key to our innovation is a specific-purpose "activation function" defined implicitly by a QCQP and a novel loss, which enforce constraint satisfaction. We show through numerical simulations that our proposed method achieves superior feasibility rate and generation cost in situations where the existing learning-based approaches fail. Copyright © 2024, The Authors. All rights reserved.

关键词： Numerical methods

Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics

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arXiv 2022年

作者： Rueda-Ramírez, Andrés M. Hindenlang, Florian J. Chan, Jesse Gassner, Gregor J. Department of Mathematics and Computer Science University of Cologne Weyertal 86-90 Cologne50931 Germany Max Planck Institute for Plasma Physics Boltzmannstraße 2 Garching85748 Germany Department of Computational and Applied Mathematics Rice University 6100 Main St HoustonTX77005 United States Center for Data and Simulation Science University of Cologne Cologne50931 Germany

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms. Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. [1, "Efficient Entropy Stable Gauss Collocation Methods". SIAM (2019)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system. We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart. Copyright © 2022, The Authors. All rights reserved.

关键词： Galerkin methods

$H(\mathrm {div})$ preconditioning for a mixed finite element formulation of the diffusion problem with random data

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mathematics of Computation 2010年第270期79卷 733-760页

作者： Elman, Howard Furnival, Darran Powell, Catherine Department of Computer Science and Institute for Advanced Computer Studies University of Maryland College Park Maryland 20742 Applied Mathematics and Scientific Computing Program University of Maryland College Park Maryland 20742 National Oceanographic Centre Southampton S014 32H School of Mathematics University of Manchester Oxford Road Manchester M13 9PL United Kingdom

Abstract: We study $H(\mathrm {div})$ preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for an $H(\mathrm {div})$ operator with random weight function acting on a certain tensor product space of random fields with finite variance. We build on the Arnold-Falk-Winther multigrid algorithm presented in 1997 by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed. We extend the deterministic analysis to accommodate the modified $H(\mathrm {div})$ operator and establish spectral equivalence bounds with a new multigrid V-cycle operator that are independent of the spatial and stochastic discretization parameters. We implement multigrid within a block-diagonal preconditioner for the full saddle-point problem, derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned minres.

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Boundary Integral Formulations for Flexural Wave Scattering in Thin Plates

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arXiv 2024年

作者： Nekrasov, Peter Su, Tim Askham, Travis Hoskins, Jeremy G. Committee on Computational and Applied Mathematics University of Chicago 5747 S. Ellis Avenue 60637 ChicagoIL United States Department of Mathematical Sciences New Jersey Institute of Technology Cullimore Hall NewarkNJ07102 United States

In this paper we develop second kind integral formulations for flexural wave scattering problems involving the clamped, free, and supported plate boundary conditions. While the clamped plate problem can be solved with layer potentials previously developed for the biharmonic equation [1], the free plate problem is more difficult due to the complex nature of the boundary conditions. In this paper we describe a representation for the free plate problem that uses the Hilbert transform to cancel singularities of certain layer potentials, ultimately leading to a Fredholm integral equation of the second kind. Additionally, for the supported plate problem, we improve on an existing representation to obtain a second kind integral equation. With these representations, it is possible to solve flexural wave scattering problems with high-order-accurate methods, examine the far-field patterns of scattering objects, and solve large problems involving multiple scatterers. © 2024, CC BY.

关键词： Choquet integral

Casimir preserving numerical method for global multilayer geostrophic turbulence

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arXiv 2024年

作者： Franken, Arnout D. Luesink, Erwin Ephrati, Sagy R. Geurts, Bernard J. Mathematics of Multiscale Modelling and Simulation Department of Applied Mathematics Faculty EEMCS University of Twente PO Box 217 Enschede7500 AE Netherlands Korteweg-de-Vries Institute University of Amsterdam PO Box 94248 Science Park 107 Amsterdam1090 GE Netherlands Department of Mathematical Sciences Chalmers University of Technology Gothenburg412 96 Sweden Multiscale Physics Center for Computational Energy Research Department of Applied Physics Eindhoven University of Technology Eindhoven Netherlands

Accurate long-term predictions of large-scale flow features on planets are crucial for understanding global atmospheric and oceanic systems, necessitating the development of numerical methods that can preserve essential physical structures over extended simulation periods without excessive computational costs. Recent advancements in the study of global single-layer barotropic models have led to novel numerical methods based on Lie-Poisson discretization that preserve energy, enstrophy and higher-order moments of potential vorticity. This work extends this approach to more complex stratified quasi-geostrophic (QG) systems on the sphere. In this work, we present a formulation of the multi-layer QG equations on the full globe. This allows for extending the Lie-Poisson discretization to multi-layer QG models, ensuring consistency with the underlying structure and enabling long-term simulations without additional regularization. The numerical method is benchmarked through simulations of forced geostrophic turbulence and the long-term behaviour of unforced multi-layered systems. These results demonstrate the structure-preserving properties and robustness of the proposed numerical method, paving the way for a better understanding of the role of high-order conserved quantities in large-scale geophysical flow dynamics. Copyright © 2024, The Authors. All rights reserved.

关键词： Atmospheric movements

Ab initio spin dynamics applied to nanoparticles: Canted magnetism of a finite Co chain along a Pt(111) surface step edge

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Physical Review B 2004年第10期70卷 100404(R)-100404(R)页

作者： B. Újfalussy B. Lazarovits L. Szunyogh G. M. Stocks P. Weinberger Metals and Ceramics Division Oak Ridge National Laboratory Oak Ridge Tennessee 37831 USA Center for Computational Materials Science Technical University Vienna A-1060 Gumpendorferstr. 1.a. Vienna Austria Department of Theoretical Physics and Center for Applied Mathematics and Computational Physics Budapest University of Technology and Economics Budafoki út 8 H-1521 Budapest Hungary

In order to search for the magnetic ground state of surface nanostructures we extended first principles adiabatic spin dynamics to the case of fully relativistic electron scattering. Our method relies on a constrained density functional theory whereby the evolution of the orientations of the spin-moments results from a semi-classical Landau-Lifshitz equation. This approach is applied to a study of the ground state of a finite Co chain placed along a step edge of a Pt(111) surface. As far as the ground state spin orientation is concerned we obtain excellent agreement with the experiment. Furthermore we observe noncollinearity of the atom-resolved spin and orbital moments. In terms of magnetic force theorem calculations we also demonstrate how a reduction of symmetry leads to the existence of canted magnetic states.

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Sentiment Analysis Technology of English Newspapers Quotes Based on Neural Network as Public Opinion Influences Identification Tool

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Sentiment Analysis Technology of English Newspapers Quotes B...

International Computer Sciences and Information Technologies (CSIT)

作者： Serhii Voloshyn Victoria Vysotska Oksana Markiv Ivan Dyyak Ihor Budz Vadim Schuchmann Information Systems and Networks Department Lviv Polytechnic National University Lviv Ukraine Lviv Polytechnic National University Lviv Ukraine Osnabrück University Osnabrück Germany Applied Mathematics Department Ivan Franko National University of Lviv Lviv Ukraine Computational Mathematics and Programming Department Lviv Polytechnic National University Lviv Ukraine Software Engineering Department West Ukrainian National University Ternopil Ukraine

ISBN: (纸本)9798350334326

This article dwells upon research concerning human language processing techniques, namely emotion analysis, conducted to identify emotions of author quotations in English newspaper articles. The publication describes general information about human language processing, brief description of analogues based on machine learning and neural network to analyse emotions in text. The main goal is to develop intelligent system for sentiment analysis of English language quotations and keywords identification that influences public opinion. The article describes sequence of program actions and tools that have been used. Then, statistics of tasks performed by neural networks with various functions of activators have been described. In the publication, product demo version has been described, and primary usage scenario has been depicted. In the end, research results have been summarised.

关键词： Training Sentiment analysis Vocabulary Dictionaries Mood Neural networks Text categorization

Wavelet based data reduction and compression scheme for high volume scientific data

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Tongxin Xuebao/Journal on Communication 2008年第SUPPL.期29卷 91-95页

作者： Wu, Guo-Qing Chen, Hong Postgraduate Department China Academic of Engineering Physics Beijing 100088 China Institute of Applied Physics and Computational Mathematics Beijing 100088 China

To handle high volume numerical data from simulations, a data reduction and compression scheme based on discrete wavelet transform was presented. By discrete wavelet transform, most energy of scientific data is concentrated into a smaller region. Wavelet coefficients can be selected by thresholded and still retain high percentage of energy of the original data. To optimize the threshold, objection function is constructed to balance model accuracy and data reduction. Reduced coefficients are then quantized and finally entropy encoded. Signal noise ratio and relative energy error are used to evaluate precision loss of data in the whole process. To show applicability of scheme, results obtained with plasma simulation data is presented. Extension of the work leads to efficient analysis and understanding of the GB or TB volume scientific data.

关键词： Discrete wavelet transforms