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159
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 180 (30 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Finite Element Methods for Active Contour Models and Balloons for 2D and 3D Images
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1991
"... The use of energyminimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Terzopoulos [23]. A balloon model was introduced in [12] as a way to generalize and solve some of the problems encountered with the original method. We present a 3 ..."
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Cited by 161 (22 self)
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The use of energyminimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Terzopoulos [23]. A balloon model was introduced in [12] as a way to generalize and solve some of the problems encountered with the original method. We present a 3D generalization of the balloon model as a 3D deformable surface, which evolves in 3D images. It is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential. We also show properties of energyminimizing surfaces concerning their relationship with 3D edge points. To solve the minimization problem for a surface, two simplified approaches are shown first, defining a 3D surface as a series of 2D planar curves. Then, after comparing Finite Element Method and Finite Difference Method in the 2D problem, we solve the 3D model using the Finite Element Method yielding greater stability and faster convergence. We have a...
The Minpack2 Test Problem Collection
, 1991
"... The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory are collaborating on the development of the software package MINPACK2. As part of the MINPACK2 project we are developing a collection ..."
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Cited by 47 (5 self)
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The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory are collaborating on the development of the software package MINPACK2. As part of the MINPACK2 project we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes the problems in the preliminary version of this collection. 1 Introduction The Army High Performance Computing Research Center at the University of Minnesota and the Mathematics and Computer Science Division at Argonne National Laboratory have initiated a collaboration for the development of the software package MINPACK2. As part of the MINPACK2 project, we are developing a collection of significant optimization problems to serve as test problems for the package. This report describes some of the problems in the preliminary version of this collection. Optimization software has often bee...
Parallel NewtonKrylovSchwarz Algorithms For The Transonic Full Potential Equation
, 1998
"... We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finite ..."
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Cited by 42 (27 self)
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We study parallel twolevel overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, NewtonKrylovSchwarz (NKS), employs an inexact finitedifference Newton method and a Krylov space iterative method, with a twolevel overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and economical for this class of mixed elliptichyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fillin in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributedmemory parallel computer.
A stable finite element for the Stokes equations
 Calcolo
, 1984
"... ABSTRACT. We present in this paper a new velocitypressure finite element for the computation of Stokes flow. We discretize the velocity field with continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies ..."
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Cited by 40 (4 self)
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ABSTRACT. We present in this paper a new velocitypressure finite element for the computation of Stokes flow. We discretize the velocity field with continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual infsup condition and converges with first order for both velocities and pressure. Finally we relate this element to families of higer order elements and to the popuIar TaylorHood element. 1. Introduction. We consider approximations of the Stckes problem for a viscous incompressible
On the computation of crystalline microstructure
 Acta Numerica
, 1996
"... Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the num ..."
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Cited by 39 (16 self)
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Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.
A preconditioner for generalized saddle point problems
 SIAM J. Matrix Anal. Appl
, 2004
"... Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matri ..."
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Cited by 34 (26 self)
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Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skewsymmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical
Multigrid And Krylov Subspace Methods For The Discrete Stokes Equations
 INT. J. NUMER. METH. FLUIDS
, 1994
"... Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the perfo ..."
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Cited by 32 (3 self)
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Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations, a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper, we compare the performance of four such methods: variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual, and multigrid methods, for solving several twodimensional model problems. The results indicate that where it is applicable, multigrid with smoothing based on incomplete factorizaton is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantages of being both independent of iteration parameters and widely applicable.
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 32 (9 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
Nonlinear Variational Method for Optical Flow Computation
, 1993
"... We present a new method for optical flow computation based on the minimization of a nonquadratic functional. The solution of the obtained nonlinear di#erential equations is done with a time dependent approach leading to the successive solutions of linear systems. This new method allows to compute o ..."
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Cited by 29 (1 self)
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We present a new method for optical flow computation based on the minimization of a nonquadratic functional. The solution of the obtained nonlinear di#erential equations is done with a time dependent approach leading to the successive solutions of linear systems. This new method allows to compute optical flow fields while insuring a unique solution and preserving the flow discontinuities. This method seems to be more appropriate since it does not enforce the optical flow to be smooth in the boundaries of moving objects and reconstruct the optical flow discontinuities without any specific processing of these points.