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Algebraic multigrid methods based on element preconditioning

INTERNATIONAL JOURNAL OF COMPUTER mathematics 2001年第4期78卷 575-598页

作者： Haase, G Langer, U Reitzinger, S Schöberl, J Institute for Analysis and Computational Mathematics Department of Computational Mathematics and Optimization Special Research Program (SFB) F013 Numerical and Symbolic Scientific Computing

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stuben's method. Ruge/Stuben's algorithm is robust for M-matrices, but unfortunately the "region of robustness" between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy. For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem. The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.

关键词： finite element equations algebraic multigrid element preconditioning technique preconditioned conjugate gradient method

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Numerical analysis of dynamical systems

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Acta Numerica 1994年 3卷 467-572页

作者： Stuart, Andrew M. Program in Scientific Computing and Computational Mathematics Division of Applied Mechanics Stanford University California CA 94305-4040 United States

This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration. Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors. Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method. © 1994, Cambridge University Press. All rights reserved.

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A class of nonsymmetric preconditioners for saddle point problems

A class of nonsymmetric preconditioners for saddle point pro...

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作者： Botchev, Mike A. Golub, Gene H. Department of Applied Mathematics University of Twente P.O. Box 217 7500 AE Enschede Netherlands Scientific Computing and Computational Mathematics Program Stanford University Stanford CA 94305-9025

For the iterative solution of saddle point problems, a nonsymmetric preconditioner is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate the efficiency of the new preconditioner, especially when the upper-left block is far from symmetric. © 2006 Society for Industrial and Applied mathematics.

关键词： Problem solving

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Numerical solution of saddle point problems

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Acta Numerica 2005年 14卷 1-137页

作者： Benzi, Michele Golubt, Gene H. Liesen, Jörg Department of Mathematics and Computer Science Emory University Atlanta GA 30322 United States Scientific Computing and Computational Mathematics Program Stanford University Stanford CA 94305-9025 United States Institut für Mathematik Technische Universität Berlin D-10623 Berlin Germany

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems. © Cambridge University Press, 2005.

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Convergence properties of preconditioned hermitian and skew-hermitian splitting methods for non-hermitian positive semidefinite matrices

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mathematics of Computation 2007年第257期76卷 287-298页

作者： Bai, Zhong-Zhi Golub, Gene H. Li, Chi-Kwong Department of Mathematics Fudan University Shanghai 200433 China State Key Laboratory of Scientific Acad. of Math. and Syst. Sci. Chinese Acad. of Sci. Beijing 100080 P.O. BOX 2719 China Scientific Computing and Computational Mathematics Program Department of Computer Science Stanford University Stanford CA 94305-9025 United States Department of Mathematics College of William and Mary Williamsburg VA 23187-8795 P.O. Box 8795 United States

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditiorrs for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. © 2006 American Mathematical Society.

关键词： Convergence Hermitian and skew-Hermitian splitting Non-Hermitian matrix Positive semidefinite matrix Splitting iteration method

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An efficient wavelet-based algorithm for image superresolution

An efficient wavelet-based algorithm for image superresoluti...

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IEEE International Conference on Image Processing

作者： N. Nguyen P. Milanfar Scientific Computing and Computational Mathematics Program Stanford CA USA Department of Electrical Engineering University of California Santa Cruz Santa Cruz CA USA

Superresolution produces high quality, high resolution images from a set of degraded, low resolution frames. We present a new and efficient wavelet-based algorithm for image superresolution. The algorithm is a combination of interpolation and restoration processes. Unlike previous work, our method exploits the interlaced sampling structure in the low resolution data. Numerical experiments and analysis demonstrate the effectiveness of our approach and illustrate why the computational complexity only doubles for 2-D superresolution versus 1-D case.

关键词： Image resolution Interpolation Image sampling Degradation Spatial resolution scientific computing mathematics Image restoration Pixel Additive noise

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Modeling subgrid-scale force and divergence of heat flux of compressible isotropic turbulence by artificial neural network

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Physical Review Fluids 2019年第10期4卷 104605-104605页

作者： Chenyue Xie Ke Li Chao Ma Jianchun Wang Shenzhen Key Laboratory of Complex Aerospace Flows Center for Complex Flows and Soft Matter Research Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China Institute of Computational Mathematics and Scientific Engineering Computing Chinese Academy of Sciences Beijing 100190 People's Republic of China The Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In this paper, the subgrid-scale (SGS) force and the divergence of SGS heat flux of compressible isotropic turbulence are modeled directly by an artificial neural network (ANN), which serves as a data-driven SGS modeling tool for large-eddy simulations (LESs). The unclosed SGS force and divergence of SGS heat flux are modeled based on the local stencil geometry with Galilean invariance. The input features include the first-order and second-order derivatives of filtered velocity and temperature, filtered density, and its first-order derivative. It is shown that the proposed ANN-F7 model shows an advantage over the gradient model in the a priori test. Specifically, the ANN-F7 model gives larger correlation coefficients and smaller relative errors than the gradient model. In an a posteriori analysis, the ANN-F7 model performs better than the dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM) in the prediction of the statistical properties of flow fields at the Taylor microscale Reynolds number Reλ ranging from 180 to 250. The DSM and DMM models lead to the typical tilted spectral distribution of velocity, where low wave numbers are too energy rich, while those near the cutoff are damped too strongly. In contrast, it is shown that the velocity spectrum predicted by the ANN-F7 model almost overlaps with the filtered direct numerical simulation data. Besides, the ANN-F7 model reconstructs the probability density functions of SGS force and divergence of SGS heat flux much better than the DSM and DMM models. An artificial neural network with reasonable physical input features can deepen our understanding of turbulence modeling.

关键词： Compressible flows Turbulence Artificial neural networks

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OnsagerNet: Learning stable and interpretable dynamics using a generalized Onsager principle

arXiv

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arXiv 2020年

作者： Yu, Haijun Tian, Xinyuan Weinan, E. Li, Qianxiao NCMIS 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 Department of Mathematics The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics National University of Singapore Singapore119077 Singapore Institute of High Performance Computing A*STAR Singapore138632 Singapore

We propose a systematic method for learning stable and physically interpretable dynamical models using sampled trajectory data from physical processes based on a generalized Onsager principle. The learned dynamics are autonomous ordinary differential equations parameterized by neural networks that retain clear physical structure information, such as free energy, diffusion, conservative motion and external forces. For high dimensional problems with a low dimensional slow manifold, an autoencoder with metric preserving regularization is introduced to find the low dimensional generalized coordinates on which we learn the generalized Onsager dynamics. Our method exhibits clear advantages over existing methods on benchmark problems for learning ordinary differential equations. We further apply this method to study Rayleigh-Bénard convection and learn Lorenz-like low dimensional autonomous reduced order models that capture both qualitative and quantitative properties of the underlying dynamics. This forms a general approach to building reduced order models for forced dissipative *** Codes 76E30, 34D20, 68T05/07, 82C35 Copyright © 2020, The Authors. All rights reserved.

关键词： Ordinary differential equations

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Accelerating GMRES with deep learning in real-time

arXiv

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arXiv 2021年

作者： Luna, Kevin Klymko, Katherine Blaschke, Johannes P. Program in Applied Mathematics The University of Arizona United States Computational Research Division Lawrence Berkeley National Laboratory United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory United States

GMRES is a powerful numerical solver used to find solutions to extremely large systems of linear equations. These systems of equations appear in many applications in science and engineering. Here we demonstrate a real-time machine learning algorithm that can be used to accelerate the time-to-solution for GMRES. Our framework is novel in that is integrates the deep learning algorithm in an in situ fashion: the AI-accelerator gradually learns how to optimizes the time to solution without requiring user input (such as a pre-trained data set). We describe how our algorithm collects data and optimizes GMRES. We demonstrate our algorithm by implementing an accelerated (MLGMRES) solver in Python. We then use MLGMRES to accelerate a solver for the Poisson equation - a class of linear problems that appears in may applications. Informed by the properties of formal solutions to the Poisson equation, we test the performance of different neural networks. Our key takeaway is that networks which are capable of learning non-local relationships perform well, without needing to be scaled with the input problem size, making them good candidates for the extremely large problems encountered in high-performance computing. For the inputs studied, our method provides a roughly 2× acceleration. Copyright © 2021, The Authors. All rights reserved.

关键词： Poisson equation

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General black-scholes models accounting for increased market volatility from hedging strategies

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International Journal of Phytoremediation 1998年第1期21卷 45-82页

作者： Ronnie Sircar, K. Papanicolaou, George Scientific Computing and Computational Mathematics Program Stanford University Stanford CA 94305-9025 Gates Building 2B United States Department of Mathematics Stanford University Stanford CA 94305-2125 United States

Increases in market volatility of asset prices have been observed and analysed in recent years and their cause has generally been attributed to the popularity of portfolio insurance strategies for derivative securities. The basis of derivative pricing is the Black-Scholes model and its use is so extensive that it is likely to influence the market itself. In particular it has been suggested that this is a factor in the rise in volatilities. A class of pricing models is presented that accounts for the feedback effect from the Black-Scholes dynamic hedging strategies on the price of the asset, and from there back onto the price of the derivative. These models do predict increased implied volatilities with minimal assumptions beyond those of the Black-Scholes theory. They are characterized by a nonlinear partial differential equation that reduces to the Black-Scholes equation when the feedback is removed. We begin with a model economy consisting of two distinct groups of traders: reference traders who are the majority investing in the asset expecting gain, and programme traders who trade the asset following a Black-Scholes type dynamic hedging strategy, which is not known a priori, in order to insure against the risk of a derivative security. The interaction of these groups leads to a stochastic process for the price of the asset which depends on the hedging strategy of the programme traders. Then following a Black-Scholes argument, we derive nonlinear partial differential equations for the derivative price and the hedging strategy. Consistency with the traditional Black-Scholes model characterizes the class of feedback models that we analyse in detail. We study the nonlinear partial differential equation for the price of the derivative by perturbation methods when the programme traders are a small fraction of the economy, by numerical methods, which are easy to use and can be implemented efficiently, and by analytical methods. The results clearly support the observed i

关键词： Black-scholes model Dynamic hedging Feedback effects Option pricing Volatility

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