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65
Structured Eigenvalue Methods for the Computation of Corner Singularities in . . .
, 2001
"... 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 eigenv ..."
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Cited by 39 (13 self)
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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 selfadjoint, positive definite sesquilinear forms and a skewHermitian 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.
Surface Parameterization for Meshing by Triangulation Flattening
 Proc. 9th International Meshing Roundtable
, 2000
"... 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 c ..."
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Cited by 36 (9 self)
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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 nonconvex domain boundaries. We use only the necessary and sufficient constraints for a valid twodimensional triangulation. As a result, the existence of a theoretical solution to the minimization procedure is guaranteed.
Parallel MATLAB: Doing it right
 Proceedings of the IEEE
"... MATLAB is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides highperformance computational routines and an easytouse, Clike scripting language. It started out as an interactive interface to EISPACK and LINPACK an ..."
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Cited by 33 (2 self)
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MATLAB is one of the most widely used mathematical computing environments in technical computing. It is an interactive environment that provides highperformance computational routines and an easytouse, Clike scripting language. It started out as an interactive interface to EISPACK and LINPACK and has remained a serial program. In 1995, C. Moler of Mathworks argued that there was no market at the time for a parallel MATLAB. But times have changed and we are seeing increasing interest in developing a parallel MATLAB, from both academic and commercial sectors. In a recent survey, 27 parallel MATLAB projects have been identified. In this paper, we expand upon that survey and discuss the approaches the projects have taken to parallelize MATLAB. Also, we describe innovative features in some of the parallel MATLAB projects. Then we will conclude with an idea of a “right ” parallel MATLAB. Finally we will give an example of what we think is a
The Anderson model of localization: a challenge for modern eigenvalue methods
 SIAM J. Sci. Comp
, 1999
"... 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 ..."
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Cited by 17 (6 self)
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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 shiftandinvert 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.
Efficient Solution Strategies for Building Energy System Simulation.” Energy and Buildings
 Energy and Buildings
, 2001
"... The efficiencies of methods employed in solution of building simulation models are considered and compared by means of benchmark testing. Direct comparisons between the Simulation Problem Analysis and Research Kernel (SPARK) and the HVACSIM+ programs are presented, as are results for SPARK versus co ..."
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Cited by 15 (4 self)
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The efficiencies of methods employed in solution of building simulation models are considered and compared by means of benchmark testing. Direct comparisons between the Simulation Problem Analysis and Research Kernel (SPARK) and the HVACSIM+ programs are presented, as are results for SPARK versus conventional and sparse matrix methods. An indirect comparison between SPARK and the IDA program is carried out by solving one of the benchmark test suite problems using the sparse methods employed in that program. The test suite consisted of two problems chosen to span the range of expected performance advantage. SPARK execution times versus problem size are compared to those obtained with conventional and sparse matrix implementations of these problems. Then, to see if the results of these limiting cases extend to actual problems in building simulation, a detailed control system for a heating, ventilating and air conditioning (HVAC) system is simulated with and without the use of SPARK cut set reduction. Execution times for the reduced and nonreduced SPARK models are compared with those for an HVACSIM+ model of the same system. Results show that the graphtheoretic techniques employed in SPARK offer significant speed advantages over the other methods for significantly reducible problems, and that by using sparse methods in combination with graph theoretic methods even problem portions with little reduction potential can be solved efficiently.
Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes
, 2001
"... This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding opera ..."
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Cited by 13 (1 self)
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This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two selfadjoint, positive definite sesquilinear forms and a skewHermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.
Preconditioners For Indefinite Linear Systems Arising In Surface Parameterization
, 2001
"... 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 pro ..."
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Cited by 13 (6 self)
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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.
Numerical Solution of Large Scale Structured Polynomial or Rational Eigenvalue Problems
 In Foundations of Computational Mathematics
, 2003
"... 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 eige ..."
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Cited by 11 (3 self)
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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.
Ml 3.1 smoothed aggregation user’s guide
, 2004
"... ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n £ n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial d ..."
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Cited by 9 (6 self)
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ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n £ n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial di®erential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a standalone package or to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the Aztec 2.1 and AztecOO iterative packages [19]. However, other solvers can be used by supplying a few functions. This document describes one speci¯c algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and nonsymmetric systems of equations (like elliptic equations, or compressible and incompressible °uid dynamics problems). Other methods exist within ML but are not described in this document. Examples
Multilevel Preconditioners for Solving Eigenvalue Problems Occuring in the Design of Resonant Cavities
, 2003
"... 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 ..."
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Cited by 6 (3 self)
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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.