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Graph embedding techniques for bounding condition numbers of incomplete factor preconditioners
 ICASE, NASA LANGLEY RESEARCH
, 1997
"... We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant Mmatrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition num ..."
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Cited by 7 (1 self)
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We extend graph embedding techniques for bounding the spectral condition number of preconditioned systems involving symmetric, irreducibly diagonally dominant Mmatrices to systems where the preconditioner is not diagonally dominant. In particular, this allows us to bound the spectral condition number when the preconditioner is based on an incomplete factorization. We provide a review of previous techniques, describe our extension, and give examples both of a bound for a model problem, and of ways in which our techniques give intuitive way of looking at incomplete factor preconditioners.
RungeKutta Research at Toronto
, 1996
"... The main purpose of this paper is to review the work on RungeKutta methods at the University of Toronto during the period 1963 to the present (1996). To provide some background, brief mention is also made of related work on the numerical solution of ordinary differential equations, but, with just a ..."
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Cited by 2 (0 self)
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The main purpose of this paper is to review the work on RungeKutta methods at the University of Toronto during the period 1963 to the present (1996). To provide some background, brief mention is also made of related work on the numerical solution of ordinary differential equations, but, with just a few exceptions, specific references are given only if the referenced material has a direct bearing on RungeKutta methods and their application to a variety of problem areas. There are several main themes. New RungeKutta formulas and new error control strategies are developed, leading for example to continuous methods and their application to areas such as delay, differentialalgebraic and boundaryvalue problems. Software design and implementation are also emphasized. And so is the importance of careful testing and comparing. Other topics, such as the notion of effectiveness, taking advantage of parallelism, and handling discontinuities, are also discussed. 1 A brief prehistory Interes...
Preconditioning and Parallel Preconditioning
, 1998
"... We review current methods for preconditioning systems of equations for their solution using iterative methods. We consider the solution of unsymmetric as well as symmetric systems and discuss techniques and implementations that exploit parallelism. We particularly study preconditioning techniques ba ..."
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Cited by 2 (1 self)
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We review current methods for preconditioning systems of equations for their solution using iterative methods. We consider the solution of unsymmetric as well as symmetric systems and discuss techniques and implementations that exploit parallelism. We particularly study preconditioning techniques based on incomplete LU factorization, sparse approximate inverses, polynomial preconditioning, and block and element by element preconditioning. In the parallel implementation, we consider the effect of reordering.
Numer. Math. (1998) 80: 397–417 Using approximate inverses in algebraic multilevel methods ⋆
, 1997
"... Summary. This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the twolevel type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatri ..."
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Summary. This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the twolevel type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the effect of using an approximate inverse instead. We derive condition number estimates that are valid for any type of approximation of the Schur complement and that do not assume the use of the hierarchical basis. They show that the twolevel methods are stable when using approximate inverses based on modified ILU techniques, or explicit inverses that meet some rowsum criterion. On the other hand, we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate. These conclusions are numerically illustrated on some examples.
A Simple Proof of Gustafsson’s Conjecture in Case of Poisson Equation on Rectangular Domains
, 2015
"... We consider the standard fivepoint finite difference method for solving the Poisson equation with the Dirichlet boundary condition. Its associated matrix is a typical illconditioned matrix whose size of the condition number is as big as ()2O h −. Among ILU, SGS, modified ILU (MILU) and other ILUt ..."
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We consider the standard fivepoint finite difference method for solving the Poisson equation with the Dirichlet boundary condition. Its associated matrix is a typical illconditioned matrix whose size of the condition number is as big as ()2O h −. Among ILU, SGS, modified ILU (MILU) and other ILUtype preconditioners, Gustafson shows that only MILU achieves an enhancement of the condition number in different order as ()1O h −. His seminal work, however, is not for the MILU but for a perturbed version of MILU and he observes that without the perurbation, it seems to reach the same result in practice. In this work, we give a simple proof of Gustafsson's conjecture on the unnecessity of perturbation in case of Poisson equation on rectangular domains. Using the CuthillMckee ordering, we simplify the recursive equation in two dimensional grid nodes into a recursive one in the level that is onedimensional. Due to the simplification, our proof is easy to follow and very short.