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Mark Ainsworth

Publications and source records attributed to Mark Ainsworth.

22 records · Page 2Linked to original sources
arXiv2016

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.

math.NA↗
arXiv2015

Computing the Bézier Control Points of the Lagrangian Interpolant in Arbitrary Dimension

The Bernstein-Bézier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory and, more recently, for high order finite element methods for the solution of partial differential equations. However, if one wishes to compute the classical Lagrange interpolant relative to the Bernstein basis, then the resulting Bernstein-Vandermonde matrix is found to be highly ill-conditioned. In the univariate case of degree $n$, Marco and Martinez showed that using Neville elimination to solve the system exploits the total positivity of the Bernstein basis and results in an $\mathcal{O}(n^2)$ complexity algorithm. Remarkable as it may be, the Marco-Martinez algorithm has some drawbacks: The derivation of the algorithm is quite technical; the interplay between the ideas of total positivity and Neville elimination are not part of the standard armoury of many non-specialists; and, the algorithm is strongly associated to the univariate case. The present work addresses these issues. An alternative algorithm for the inversion of the univariate Bernstein-Vandermonde system is presented that has: The same complexity as the Marco-Martinez algorithm and whose stability does not seem to be in any way inferior; a simple derivation using only the basic theory of Lagrange interpolation (at least in the univariate case); and, a natural generalisation to the multivariate case.

math.NA↗
arXiv2015

Robust error bounds for finite element approximation of reaction-diffusion problems with non-constant reaction coefficient in arbitrary space dimension

We present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction-diffusion problems with mixed boundary conditions and piecewise constant reaction coefficient formulated in arbitrary dimension. The estimator provides a guaranteed upper bound on the energy norm of the error and it is robust for all values of the reaction coefficient, including the singularly perturbed case. The approach is based on robustly equilibrated boundary flux functions of Ainsworth and Oden (Wiley 2000) and on subsequent robust and explicit flux reconstruction. This paper simplifies and extends the applicability of the previous result of Ainsworth and Vejchodský (Numer. Math. 119 (2011) 219-243) in three aspects: (i) arbitrary dimension, (ii) mixed boundary conditions, and (iii) non-constant reaction coefficient. It is the first robust upper bound on the error with these properties. An auxiliary result that is of independent interest is the derivation of new explicit constants for two types of trace inequalities on simplices.

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