In this book, which focuses on the use of iterative methods for solving large sparse systems of linear equations, templates are introduced to meet the needs of both the traditional user and the high-performance specialist Templates, a description of a general algorithm rather than the executable object or source code more commonly found in a conventional software library, offer whatever degree of customization the user may desire

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Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods

Basic error estimates for elliptic problems

In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

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Conjugate Gradient Methods for Toeplitz Systems

The problem of detection of orientation in finite dimensional Euclidean spaces is solved in the least squares sense. The theory is developed for the case when such orientation computations are necessary at all local neighborhoods of the n-dimensional Euclidean space. Detection of orientation is shown to correspond to fitting an axis or a plane to the Fourier transform of an n-dimensional structure. The solution of this problem is related to the solution of a well-known matrix eigenvalue problem. The computations can be performed in the spatial domain without actually doing a Fourier transformation. Along with the orientation estimate, a certainty measure, based on the error of the fit, is proposed. Two applications in image analysis are considered: texture segmentation and optical flow. The theory is verified by experiments which confirm accurate orientation estimates and reliable certainty measures in the presence of noise. The comparative results indicate that the theory produces algorithms computing robust texture features as well as optical flow. >

Multidimensional orientation estimation with applications to texture analysis and optical flow

Presents a new method for determining the minimal nonrigid deformation between two 3D surfaces, such as those which describe anatomical structures in 3D medical images. Although the authors match surfaces, they represent the deformation as a volumetric transformation. Their method performs a least squares minimization of the distance between the two surfaces of interest. To quickly and accurately compute distances between points on the two surfaces, the authors use a precomputed distance map represented using an octree spline whose resolution increases near the surface. To quickly and robustly compute the deformation, the authors use a second octree-spline to model the deformation function. The coarsest level of the deformation encodes the global (e.g., affine) transformation between the two surfaces, while finer levels encode smooth local displacements which bring the two surfaces into closer registration. The authors present experimental results on both synthetic and real 3D surfaces. >

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Matching 3-D anatomical surfaces with non-rigid deformations using octree-splines

Finite Element Solution of Boundary Value Problems: Theory and Computation provides a thorough, balanced introduction to both the theoretical and the computational aspects of the finite element method for solving boundary value problems for partial differential equations. Although significant advances have been made in the finite element method since this book first appeared in 1984, the basics have remained the same, and this classic, well-written text explains these basics and prepares the reader for more advanced study. Useful as both a reference and a textbook, complete with examples and exercies, it remains as relevant today as it was when originally published. This book is written for advanced undergraduate and graduate students in the areas of numerical analysis, mathematics, and computer science, as well as for theoretically inclined practitioners in engineering and the physical science.

Finite Element Solution of Boundary Value Problems: Theory and Computation

This chapter discusses the conjugate gradient method to solve positive definite system of equations, preconditioning by symmetric successive overrelaxation (SSOR), and incomplete factorization considering the case in which the matrix of the system arises from the finite element treatment of a boundary value problem. It discusses the spectral condition number κ ( K ) associated with SSOR preconditioning, the quantity that determines the rate of convergence of the iterations. The chapter discusses how the computation can be extended to the case of higher-degree basis functions without loss of rate of convergence. It discusses the residual or gradient vector in the context of computational labor and numerical stability. The chapter presents a comparison of the performance of iterative and direct methods for solving finite element systems of equations and presents an account of multigrid methods for solving finite element problems.

Iterative Solution of Finite Element Equations

The application of the finite element method to a boundary value problem leads to a system of equations K α = G, where the stiffness matrix K is often large, sparse, and positive definite. This chapter reviews the solution of such systems by Gaussian elimination and the closely related Cholesky method. These so-called direct, that is, non-iterative, methods are the most widely used methods for finite element computation today. Apart from being important in their own right, the direct methods provide the basis for the incomplete factorization technique used to precondition the conjugate gradient method. The chapter discusses node-ordering strategies aimed at making K a matrix with a small band or envelope and presents storage schemes appropriate for such matrices. The mathematics of Gaussian elimination and the Cholesky method is basic regardless of how the nodes are ordered. It then examines some special techniques for solving the stiffness equations. Each essentially corresponds to a node-ordering strategy that differs from the band-minimizing type and the implementation of Gaussian elimination differs accordingly. The chapter also explains the main types of errors that arise in constructing and solving finite element equations.

Direct Methods for Solving Finite Element Equations

One of the most important problems of mathematical physics is the boundary value problem in which one seeks a function that satisfies some differential equation in a region Ω and satisfies specified conditions on the boundary of Ω. Many problems of this type have the property that the solution minimizes a certain functional f defined on some set of functions V or, more generally, is a stationary point of such a functional. Thus, the task of solving a boundary value problem is equivalent to that of finding a function in V that makes f stationary and this latter problem is called the variational formulation of the boundary value problem. This chapter discusses the functional of the above type and their associated boundary value problems. It presents some fundamental definitions and theorems for functionals in one-space dimension and a number of examples. It discusses quadratic functionals as they are the source of linear boundary value problems. The chapter discusses the important distinction between essential boundary conditions, which must be imposed on the functions in V , and natural boundary conditions, which need not be imposed. It also presents a number of physical problems that give rise to the mathematical problem of minimizing a quadratic functional.

Variational Formulation of Boundary Value Problems: Part II

This chapter describes the Ritz–Galerkin method for finding the approximate solution of a boundary value problem. Every method of obtaining an approximate solution of a boundary value problem requires some kind of discretization. In the case of the Ritz method, considered in the setting of the Lax–Milgram lemma, this is done by selecting an N -dimensional subspace V N ⊂ V. The chapter discusses how the minimal property of the Ritz method leads to important error estimates.

V.A. Barker

Papers

Finite Element Solution of Boundary Value Problems: Theory and Computation

Iterative Solution of Finite Element Equations

Direct Methods for Solving Finite Element Equations

Variational Formulation of Boundary Value Problems: Part II

The Ritz–Galerkin Method