Numerical Stability in Problems of Linear Algebra

Babuvška, I.

doi:10.1137/0709008

Cited by 51 publications

(27 citation statements)

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“…The amount of computational work depends on the number of iterations performed, and an efficient PDE solver should use this principal advantage by stopping the algebraic solver whenever the algebraic error drops to the level at which it does not significantly affect the whole error (cf. [5]). In other words, the linear algebraic system is affected by the errors on the preceding stages (modeling and discretization errors) of the solution process.…”

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confidence: 99%

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

Jiránek¹,

Strakoš²,

Vohralı́k³

2010

SIAM J. Sci. Comput.

116

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Abstract. We consider the finite volume and the lowest-order mixed finite element discretizations of a second-order elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be simply bounded using the algebraic residual vector. Much better results are, however, obtained using the complementary energy of an equilibrated Raviart-Thomas-Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. The second goal of this paper is to construct efficient stopping criteria for iterative solvers such as the conjugate gradients, GMRES, or Bi-CGStab. We claim that the discretization error, implied by the given numerical method, and the algebraic one should be in balance, or, more precisely, that it is enough to solve the linear algebraic system to the accuracy which guarantees that the algebraic part of the error does not contribute significantly to the whole error. Our estimates allow a reliable and cheap comparison of the discretization and algebraic errors. One can thus use them to stop the iterative algebraic solver at the desired accuracy level, without performing an excessive number of unnecessary additional iterations. Under the assumption of the relative balance between the two errors, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. Several numerical experiments illustrate the theoretical results.Key words. Second-order elliptic partial differential equation, finite volume method, mixed finite element method, a posteriori error estimates, iterative methods for linear algebraic systems, stopping criteria.

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confidence: 99%

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

Jiránek¹,

Strakoš²,

Vohralı́k³

2010

SIAM J. Sci. Comput.

116

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“…. , n.°¡ Babuska has proved the estimate o¡R/o¡D < 9 in [2], using a backward error analysis. For the case n = 3, Miller [12] has shown numerically that o2/o2 is about 5.5 so that our estimate (2) seems to be sharp for n > 3.…”

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confidence: 99%

“…Miller's papers [12], [13] use condition numbers of the kind defined by the second equation in (10). The associated stability constants 1 + aR/atD are denoted by Í in [1], [2]. Miller [12], [13] and Miller-Spooner [14] use the notation p and co, for oR/a,D.…”

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confidence: 99%

“…. ., n. Under perturbations, an evaluation algorithm yields approximations t>, of u, that can be written in the form (2) vt -(1 + e,)F,iv), t = 0,...,n.…”

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confidence: 99%

“…Let v, = G,ie), then u, = G,(0) and, by Taylor expansion, (8) Au, = G,'i0)e + 6>,(tj2). Babuska's stability constants A in [1], [2] are defined in the form of the first equation in (10). Miller's papers [12], [13] use condition numbers of the kind defined by the second equation in (10).…”

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confidence: 99%

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Perturbation Theory for Evaluation Algorithms of Arithmetic Expressions

Stummel¹

1981

Mathematics of Computation

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Abstract. The paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations, rounding error in arithmetic floating-point operations, and approximations in 'built-in' functions. The error analysis is based on the linearization method that has been proposed by many authors in various forms. Fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors. Derivations, representations, and properties of these condition numbers are studied in detail. The condition numbers enable simple general, quantitative definitions of numerical stability, backward analysis, well-and ill-conditioning of a problem and an algorithm. The well-known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds. For many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori. The theoretical results of the paper have been applied to a series of concrete algorithms, including Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

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Rounding errors in numerical solutions of two linear equations in two unknowns

Stummel

Wendland²

1982

Math Methods in App Sciences

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Dedicated to Professor Wolfgang Haack on the occasion of his eightieth birthday Communicated by W. WendlandNew condition numbers and stability constants for the numerical behaviour of Cramer's rule and Gaussian elimination for solving two linear equations in two unknowns under data perturbations and rounding errors of floating-point arithmetic are established. By these means fundamental error estimates and stability theorems are proved. The error estimates are illustrated by a series o f numerical examples.

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Numerical Stability in Problems of Linear Algebra

Cited by 51 publications

References 1 publication

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers

Perturbation Theory for Evaluation Algorithms of Arithmetic Expressions

Rounding errors in numerical solutions of two linear equations in two unknowns

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