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.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

/pdf/active-contours-without-edges-2tt3qz9lq0.pdf

Active contours without edges

Co-Planar Stereotaxic Atlas of the Human Brain—3-Dimensional Proportional System: An Approach to Cerebral Imaging, J. Talairach, P. Tournoux. Georg Thieme Verlag, New York (1988), 122 pp., 130 figs. DM 268

Data Mining - Concepts and Techniques.

https://hal.archives-ouvertes.fr/hal-00417283/document

Independent component analysis, a new concept?

Domain decomposition (DD) has been widely used to design parallel efficient algorithms for solving elliptic problems. In this thesis, we focus on improving the efficiency of DD methods and applying them to more general problems. Specifically, we propose efficient variants of the vertex space DD method and minimize the complexity of general DD methods. In addition, we apply DD algorithms to coupled elliptic systems, singular Neumann boundary problems and linear algebraic systems.
We successfully improve the vertex space DD method of Smith by replacing the exact edge, vertex dense matrices by approximate sparse matrices. It is extremely expensive to calculate, invert and store the exact vertex and edge Schur complement dense sub-matrices in the vertex space DD algorithm. We propose several approximations for these dense matrices, by using Fourier approximation and an algebraic probing technique. Our numerical and theoretical results show that these variants retain the fast convergence rate and greatly reduce the computational cost.
We develop a simple way to reduce the overall complexity of domain decomposition methods through choosing the coarse grid size. For sub-domain solvers with different complexities, we derive the optimal coarse grid size $H\sb{opt},$ which asymptotically minimizes the total computational cost of DD methods under the sequential and parallel environments. The overall complexity of DD methods is significantly reduced by using this optimal coarse grid size.
We apply the additive and multiplicative Schwarz algorithms to solving coupled elliptic systems. Using the Dryja-Widlund framework, we prove that their convergence rates are independent of both the mesh and the coupling parameters. We also construct several approximate interface sparse matrices by using Sobolev inequalities, Fourier analysis and probe technique.
We further discuss the application of DD to the singular Neumann boundary value problems. We extend the general framework to these problems and show how to deal with the null space in practice. Numerical and theoretical results show that these modified DD methods still have optimal convergence rate.
By using the DD methodology, we propose algebraic additive and multiplicative Schwarz methods to solve general sparse linear algebraic systems. We analyze the eigenvalue distribution of the iterative matrix of each algebraic DD method to study the convergence behavior.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0962492900002427

Domain decomposition algorithms

Rank revealing QR factorizations

Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\|C - A\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.

An Optimal Circulant Preconditioner for Toeplitz Systems

Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filter-the digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational (or L/sup 1/) optimization problem, which differs from most statistical filters. Applications are given in the denoising of one dimensional (1-D) signals, two-dimensional (2-D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.

The digital TV filter and nonlinear denoising

In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. The model is a combination between more classical active contour models using mean curvature motion techniques, and the Mumford-Shah model for segmentation. We minimize an energy which can be seen as a particular case of the so-called minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable.

/pdf/an-active-contour-model-without-edges-13d2ub15dt.pdf

Tony F. Chan

Papers

Domain decomposition algorithms

Rank revealing QR factorizations

An Optimal Circulant Preconditioner for Toeplitz Systems

The digital TV filter and nonlinear denoising

An Active Contour Model without Edges