We propose a new method to compute planar triangulations of triangulated surfaces for surface parameterization. Our method computes a projection that minimizes the distortion of the surface metric structures (lengths, angles, etc.). It can handle any manifold surface, including surfaces with large curvature gradients and non-convex domain boundaries. We use only the necessary and sufficient constraints for a valid two-dimensional triangulation. As a result, the existence of a theoretical solution to the minimization procedure is guaranteed.

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitian form. This eigenvalue problem is discretized by the finite element method. The resulting quadratic matrix eigenvalue problem is then solved with the Skew Hamiltonian Implicitly Restarted Arnoldi method (SHIRA) which is specifically adapted to the structure of this problem. Some numerical examples are given that show the performance of this approach.

Abstract. We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.

In [19] we introduced a new algorithm for computing planar triangulations of faceted surfaces for surface parameterization. Our algorithm computes a mapping that minimizes the distortion of the surface metric structures (lengths, angles, etc.). Compared with alternative approaches, the algorithm provides a significant improvement in robustness and applicability; it can handle more complicated surfaces and it does not require a convex or predefined planar domain boundary. However, our algorithm involves the solution of a constrained minimization problem. The potential high cost in solving the optimization problem has given rise to concerns about the applicability of the method, especially for very large problems. This paper is concerned with the ecient solution of the symmetric indefinite linear systems that arise when Newton's method is applied to the constrained minimization problem. In small to moderate size models the linear systems can be solved efficiently with a sparse direct method. We give examples from computations with the SuperLU package [6]. For larger models we have to use preconditioned iterative methods. We develop a new preconditioner that takes into account the structure of our linear systems. Some preliminary experimental results are shown that indicate the effectiveness of this approach.

This paper deals with the numerical solution of large scale polynomial or rational eigenvalue problems with Hamiltonian or symplectic symmetry in the spectrum. Applications where such problems arise are introduced briey. It is shown how these problems may be formulated as linear generalized eigenvalue problems that have either symmetric/skew symmetric, skew Hamiltonian/Hamiltonian or symplectic pencils. The presented numerical methods are designed to preserve these structures.

In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field. Keywords: linear systems, dense matrices, sparse matrices, tridiagonal systems, parallelism, direct methods, iterative methods, Krylov methods, preconditioning. AMS(MOS) subject classifications: 65F05, 65F50. 1 Introduction Solution methods for systems of linear equations Ax = b; (1) where A is a coefficient matrix of order n and x and b are n-vectors, are usually grouped into two distinct classes: direct methods and iterative methods. However, CCLRC - Rutherford Appleton Laboratory, Oxfordshire, England and CERFACS, Toulouse,...

We study block-diagonal preconditioners and fast iterative solvers for inde nite two-by-two block linear systems with zero (2,2) block. Our preconditioners are derived from a splitting of the (1,1) block, A = D E, where the matrix D can be eciently inverted. Dierent splittings lead to dierent preconditioners, and we analyze properties of the preconditioned matrices, in particular their eigenvalue distributions. From the preconditioned linear system we derive a xed point iteration as well as its so-called related system for solving the original block two-by-two problem. We study the convergence of the xed point iteration and the eigenvalue distribution of the related system matrix. Using our analytical results we show that solving the original system by applying GMRES [20] to the related system is typically more ecient than solving it by applying GMRES to the preconditioned system. In addition, in the case of constrained problems the use of the xed point iteration or its related system leads to approximations that satisfy the constraints exactly after one iteration. Moreover, we show how scaling the original block two-by-two system can improve convergence dramatically. Our theoretical results are con rmed by numerical experiments on a constrained optimization application. Our approach is very general, as we make almost no assumptions on the given block two-by-two system. In particular, the system matrix might be nonsymmetric, and the (1,1) block A might be inde nite, even singular. This is the rst paper in a two-part sequence. In the second paper we will study the use of our preconditioners in a broad variety of applications.

We review the influence of the advent of high performance computing on the solution of linear equations. We will concentrate on direct methods of solution and consider both the case when the coefficient matrix is dense and when it is sparse. We will examine the current performance of software in this area and speculate on what advances we might expect in the early years of the next century. Keywords: sparse matrices, direct methods, parallelism, matrix factorization, multifrontal methods. AMS(MOS) subject classifications: 65F05, 65F50. 1 Current reports available at http://www.cerfacs.fr/algor/algo reports.html. Also appeared as Technical Report RAL-TR-1999-072 from Rutherford Appleton Laboratory, Oxfordshire. 2 duff@cerfacs.fr. Also at Atlas Centre, RAL, Oxon OX11 0QX, England. Rutherford Appleton Laboratory. Contents 1 Introduction 1 2 Building blocks 1 3 Factorization of dense matrices 2 4 Factorization of sparse matrices 4 5 Parallel computation 8 6 Current situation 12 7 F...

We investigate eigensolvers for computing a few of the smallest eigenvalues of a generalized eigenvalue problem resulting from the nite element discretization of the time independent Maxwell equation. Various multilevel preconditioners are employed to improve the convergence and memory consumption of the JacobiDavidson algorithm and of the locally optimal block preconditioned conjugate gradient (LOBPCG) method. We present numerical results of very large eigenvalue problems originating from the design of resonant cavities of particle accelerators.

Abstract. Static symbolic factorization coupled with supernode partitioning and asynchronous computation scheduling can achieve high gigaflop rates for parallel sparse LU factorization with partial pivoting. This paper studies properties of elimination forests and uses them to optimize supernode partitioning/amalgamation and execution scheduling. It also proposes supernodal matrix multiplication to speed up kernel computation by retaining the BLAS-3 level efficiency and avoiding unnecessary arithmetic operations. The experiments show that our new design with proper space optimization, called S +, improves our previous solution substantially and can achieve up to 10 GFLOPS on 128 Cray T3E 450MHz nodes. Key words. Gaussian elimination with partial pivoting, LU factorization, sparse matrices, elimination forests, supernode amalgamation and partitioning, asynchronous computation scheduling AMS subject classifications. 65F50, 65F05 PII. S0895479898337385