Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

We describe the University of Florida Sparse Matrix Collection, a large and actively growing set of sparse matrices that arise in real applications The Collection is widely used by the numerical linear algebra community for the development and performance evaluation of sparse matrix algorithms It allows for robust and repeatable experiments: robust because performance results with artificially generated matrices can be misleading, and repeatable because matrices are curated and made publicly available in many formats Its matrices cover a wide spectrum of domains, include those arising from problems with underlying 2D or 3D geometry (as structural engineering, computational fluid dynamics, model reduction, electromagnetics, semiconductor devices, thermodynamics, materials, acoustics, computer graphics/vision, robotics/kinematics, and other discretizations) and those that typically do not have such geometry (optimization, circuit simulation, economic and financial modeling, theoretical and quantum chemistry, chemical process simulation, mathematics and statistics, power networks, and other networks and graphs) We provide software for accessing and managing the Collection, from MATLAB™, Mathematica™, Fortran, and C, as well as an online search capability Graph visualization of the matrices is provided, and a new multilevel coarsening scheme is proposed to facilitate this task

The university of Florida sparse matrix collection

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Much has been written about theory and practice in the law, and the tension between practitioners and theorists. Judges do not cite theoretical articles often; they rarely "apply" theories to particular cases. These arguments are not revisited. Instead the Essay explores the working and interaction of theory and practice, practitioners and theorists. The Essay starts with a story about solving a legal issue using our intellectual tools - theory, practice, and their progenies: experience and "gut." Next the Essay elaborates on the nature of theory, practice, experience and "gut." The third part of the Essay discusses theories that are helpful to practitioners and those that are less helpful. The Essay concludes that practitioners theorize, and theorists practice. They use these intellectual tools differently because the goals and orientations of theorists and practitioners, and the constraints under which they act, differ. Theory, practice, experience and "gut" help us think, remember, decide and create. They complement each other like the two sides of the same coin: distinct but inseparable.

Of Theory and Practice

This article describes the governing equations, computational algorithms, and other components entering into the Community Multiscale Air Quality (CMAQ) modeling system. This system has been designed to approach air quality as a whole by including state-ofthe-science capabilities for modeling multiple air quality issues, including tropospheric ozone, fine particles, acid deposition, and visibility degradation. CMAQ was also designed to have multiscale capabilities so that separate models were not needed for urban and regional scale air quality modeling. By making CMAQ a modeling system that addresses multiple pollutants and different spatial scales, it has a “one-atmosphere” perspective that combines the efforts of the scientific community. To implement multiscale capabilities in CMAQ, several issues (such as scalable atmospheric dynamics and generalized coordinates), which depend on the desired model resolution, are addressed. A set of governing equations for compressible nonhydrostatic atmospheres is available to better resolve atmospheric dynamics at smaller scales. Because CMAQ is designed to handle scale-dependent meteorological formulations and a large amount of flexibility, its governing equations are expressed in a generalized coordinate system. This approach ensures consistency between CMAQ and the meteorological modeling system. The generalized coordinate system determines the necessary grid and coordinate transformations, and it can accommodate various vertical coordinates and map projections. The CMAQ modeling system simulates various chemical and physical processes that are thought to be important for understanding atmospheric trace gas transformations and distributions. The modeling system contains three types of modeling components (Models-3): a meteorological modeling system for the description of atmospheric states and motions, emission models for man-made and natural emissions that are injected into the atmosphere, and a chemistry-transport modeling system for simulation of the chemical transformation and fate. The chemical transport model includes the following process modules: horizontal advection, vertical advection, mass conservation adjustments for advection processes, horizontal diffusion, vertical diffusion, gas-phase chemical reactions and solvers, photolytic rate computation, aqueous-phase reactions and cloud mixing, aerosol dynamics, size distributions and chemistry, plume chemistry effects, and gas and aerosol deposition velocity estimation. This paper describes the Models-3 CMAQ system, its governing equations, important science algorithms, and a few application examples. This review article cites 114 references. DOI: 10.1115/1.2128636

Review of the Governing Equations, Computational Algorithms, and Other Components of the Models-3 Community Multiscale Air Quality (CMAQ) Modeling System

1. The Air Pollution Problem. 2. Mathematical Models for Studying the Long-Range Transport of Air Pollutants. 3. Numerical Treatment of Large Air Pollution Models. 4. Testing the Reliability of the Numerical Algorithms. 5. Need for Efficient Algorithms. 6. Computations on High-Speed Computers. 7. Running Air Pollution Models on Vector Machines. 8. Running Models on Parallel Machines with Shared Memory. 9. Running Models on Massively Parallel Computers. 10. Numerical Experiments with the Danish Eulerian Model. References. Author Index. Subject Index.

Computer Treatment of Large Air Pollution Models

Operational air pollution forecasts from European to local scale

The mathematical models of many practical problems lead to systems of linear algebraic equations where the coefficient matrix is large and sparse. Typical examples are the solutions of partial differential equations by finite difference or finite element methods but many other applications could be mentioned. When there is a large proportion of zeros in the coefficient matrix then it is fairly obvious that we do not want to store all those zeros in the computer, but it might not be quite so obvious how to get around it. We first describe storage techniques which are convenient to use with direct solution methods, and we then show how a very efficient computational scheme can be based on Gaussian elimination and iterative refinement. A serious problem in the storage and handling of sparse matrices is the appearance of fill-ins, i.e. new elements which are created in the process of generating zeros below the diagonal. Many of these new elements tend to be smaller than the original matrix elements, and if they are smaller than a quantity which we shall call the drop tolerance we simply ignore them. In this way we may preserve the sparsity quite well but we probably introduce rather large errors in the LU decomposition to the effect that the solution becomes unacceptable. In order to retrieve the accuracy we use iterative refinement and we show theoreticaly and with practical experiments that it is ideal for the purpose. Altogether, the combination of Gaussian elimination, a large drop tolerance, and iterative refinement gives a very efficient and competitive computational scheme for sparse problems. For dense matrices iterative refinement will always require more storage and computation time, and the extra accuracy it yields may not be enough to justify it. For sparse problems, however, iterative refinement combined with a large drop tolerance will in most cases give very accurate results and reliable error estimates with less storage and computation time.

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Direct Methods for Sparse Matrices

1. Exploiting Sparsity.- 2. Storage Schemes.- 3. General Scheme for Linear Algebraic Problems.- 4. Pivotal Strategies for Gaussian Elimination.- 5. Use of Iterative Refinement in the GE Process.- 6. Implementation of the Algorithms.- 7. Solving Least Squares Problems by Augmentation.- 8. Sparse Matrix Technique for Ordinary Differential Equations.- 9. Condition Number Estimators in a Sparse Matrix Software.- 10. Parallel Direct Solvers.- 11 Parallel Orthomin for General Sparse Matrices.- 12. Orthogonalization Methods.- 13. Two Storage Schemes for Givens Plane Rotations.- 14. Pivotal Strategies for Givens Plane Rotations.- 15. Iterative Refinement after the Plane Rotations.- 16. Preconditioned Conjugate Gradients for Givens Plane Rotations.- References.- Author Index.

Zahari Zlatev

Papers

Computer Treatment of Large Air Pollution Models

Operational air pollution forecasts from European to local scale

Direct Methods for Sparse Matrices

Computational Methods for General Sparse Matrices

Comparison of numerical techniques for use in air pollution models with non-linear chemical reactions