Mario Arioli

A stopping criterion for the conjugate gradient algorithm in a finite element method framework

SummaryThe Conjugate Gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self-adjoint elliptic partial differential equations. Taking into account recent results [13, 19, 20, 22] which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced in [3]. Moreover, we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion.

Stopping criteria for iterations in finite element methods

Summary.This work extends the results of Arioli [1], [2] on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive-definite problems. We show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context. We then use Krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria.

Null Space Algorithm and Spanning Trees in Solving Darcy's Equation

A Null Space algorithm is considered to solve the augmented system produced by the mixed finite element approximation of Darcys Law. The method is based on the combination of a LU factorization technique for sparse matrices with an iterative Krylov solver. The computational efficiency of the method relies on the use of spanning trees to compute the LU factorization without fill-in and on a suitable stopping criterion for the iterative solver. We experimentally investigate its performance on a realistic set of selected application problems.

Componentwise error analysis for FFTs with applications to fast Helmholtz solvers

We analyze the stability of the Cooley-Tukey algorithm for the Fast Fourier Transform of ordern=2k and of its inverse by using componentwise error analysis.We prove that the components of the roundoff errors are linearly related to the result in exact arithmetic. We describe the structure of the error matrix and we give optimal bounds for the total error in infinity norm and inL2 norm.The theoretical upper bounds are based on a “worst case” analysis where all the rounding errors work in the same direction. We show by means of a statistical error analysis that in realistic cases the max-norm error grows asymptotically like the logarithm of the sequence length by machine precision.Finally, we use the previous results for introducing tight upper bounds on the algorithmic error for some of the classical fast Helmholtz equation solvers based on the Faster Fourier Transform and for some algorithms used in the study of turbulence.

Linear regression models, least-squares problems, normal equations, and stopping criteria for the conjugate gradient method ☆

Abstract Minimum-variance unbiased estimates for linear regression models can be obtained by solving least-squares problems. The conjugate gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from these least-squares problems. Taking into account the results of Golub and Meurant (1997, 2009) [10] , [11] , Hestenes and Stiefel (1952) [17] , and Strakos and Tichý (2002) [16] , which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli (2005) [18] to the normal equations taking into account the statistical properties of the underpinning linear regression problem. Moreover, we show how the energy norm of the error is linked to the χ 2 -distribution and to the Fisher–Snedecor distribution. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criteria.

Chebyshev acceleration of iterative refinement

It is well known that the FGMRES algorithm can be used as an alternative to iterative refinement and, in some instances, is successful in computing a backward stable solution when iterative refinement fails to converge. In this study, we analyse how variants of the Chebyshev algorithm can also be used to accelerate iterative refinement without loss of numerical stability and at a computational cost at each iteration that is less than that of FGMRES and only marginally greater than that of iterative refinement. A component-wise error analysis of the procedure is presented and numerical tests on selected sparse test problems are used to corroborate the theory.