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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

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

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

From Obstacle Problems to Neural Insights: Feed Forward Neural Network Modeling of Ice Thickness

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arXiv 2024年

作者： Chawla, Kapil Holmes, William Temam, Roger Department of Mathematics The Institute for Scientific Computing and Applied Mathematics Indiana University IN47405 United States Department of Mathematics and Cognitive Science Program Indiana University IN47405 United States

In this study, we integrate the established obstacle problem formulation from ice sheet modeling [12, 20] with cutting-edge deep learning methodologies to enhance ice thickness predictions, specifically targeting the Greenland ice sheet. By harmonizing the mathematical structure with an energy minimization framework tailored for neural network approximations, our method’s efficacy is confirmed through both 1D and 2D numerical simulations. Utilizing the NSIDC-0092 dataset for Greenland [1] and incorporating bedrock topography for model pre-training, we register notable advances in prediction accuracy. Our research underscores the potent combination of traditional mathematical models and advanced computational techniques in delivering precise ice thickness estimations. © 2024, CC BY.

关键词： Ice

On Cyclic Finite-State Approximation of Data-Driven Systems

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arXiv 2019年

作者： Vides, Fredy Scientific Computing Innovation Center Department of Applied Mathematics Universidad Nacional Autónoma de Honduras UNAH Honduras Graduate Program in Structural Engineering Universidad Tecnológica Centroamericana UNITEC Honduras

In this document, some novel theoretical and computational techniques for constrained approximation of data-driven systems, are presented. The motivation for the development of these techniques came from structure-preserving matrix approximation problems that appear in the fields of system identification and model predictive control, for data-driven systems and processes. The research reported in this document is focused on finite-state approximation of data-driven systems. Some numerical implementations of the aforementioned techniques in the simulation and model predictive control of some generic data-driven systems, that are related to electrical signal transmission models, are outlined.93B28, 47N70 (primary) and 93C57, 93B40 (secondary) Copyright © 2019, The Authors. All rights reserved.

关键词： Singular value decomposition

Multilevel Monte Carlo Methods for the Grad-Shafranov Free Boundary Problem

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SSRN

SSRN 2023年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Applied Mathematics & Statistics and Scientific Computing Program University of Maryland College Park United States Department of Mathematics The University of Arizona United States

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costsby performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductionstypically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model. © 2023, The Authors. All rights reserved.

关键词： Monte Carlo methods

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

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arXiv 2023年

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 © 2023, CC BY.

关键词： Monte Carlo methods

Accelerating GMRES with deep learning in real-time

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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

Fourier phase retrieval: Uniqueness and algorithms

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arXiv 2017年

作者： Bendory, Tamir Beinerty, Robert Eldarz, Yonina C. Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Graz8010 Austria Andrew and Erna Viterbi Faculty of Electrical Engineering Technion - Israel Institute of Technology Haifa Israel

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field. Copyright © 2017, The Authors. All rights reserved.

关键词： Fourier transforms

Preconditioners for regularized image superresolution

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Preconditioners for regularized image superresolution

International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： N. Nguyan G. Golub P. Milanfar Sci. Comput. & Comput. Math. Program Stanford Univ. CA USA Scientific Computing and Computational Mathematics Program University of Stanford Stanford CA USA Applied Electromagnetics Lab SRI International Inc. Menlo Park CA USA

Superresolution reconstruction produces a high resolution image from a set of low resolution images. Previous work on superresolution had not adequately addressed the computational issues for this problem. In this paper, we propose efficient block circulant preconditioners for solving the regularized superresolution problem by conjugate gradients. The effectiveness of the preconditioners is demonstrated with superresolution results for a simulated image sequence and a FLIR image sequence.

关键词： Image resolution Layout Spatial resolution Pixel Image reconstruction Image sequences Motion estimation Solids Equations scientific computing