Steven F. Ashby

A parallel multigrid preconditioned conjugate gradient algorithm for groundwater flow simulations

The numerical simulation of groundwater flow through heterogeneous porous media is discussed. The focus is on the performance of a parallel multigrid preconditioner for accelerating convergence of conjugate gradients, which is used to compute the pressure head. The numerical investigation considers the effects of boundary conditions, coarse grid solver strategy, increasing the grid resolution, enlarging the domain, and varying the geostatistical parameters used to define the subsurface realization. Scalability is also examined. The results were obtained using the PARFLOW groundwater flow simulator on the CRAY T3D massively parallel computer.

A taxonomy for conjugate gradient methods

The conjugate gradient method of Hestenes and Stiefel is an effective method for solving large, sparse Hermitian positive definite (hpd) systems of linear equations,

Analysis of subsurface contaminant migration and remediation using high performance computing

Ax = b

Adaptive polynomial preconditioning for hermitian indefinite linear systems

. Generalizations to non-hpd matrices have long been sought. The recent theory of Faber and Manteuffel gives necessary and sufficient conditions for the existence of a CG method. This paper uses these conditions to develop and organize such methods. It is shown that any CG method for

The Role of the Inner Product in Stopping Criteria for Conjugate Gradient Iterations

Ax = b

A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems

is characterized by an hpd inner product matrix B and a left preconditioning matrix C. At each step the method minimizes the B-norm of the error over a Krylov subspace. This characterization is then used to classify known and new methods. Finally, it is shown how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos. Such estimates allow implementation of a stopping criterion based more nearly on the true error.

A Numerical Simulation of Groundwater Flow and Contaminant Transport on the CRAY T3D and C90 Supercomputers

Abstract Highly resolved simulations of groundwater flow, chemical migration and contaminant recovery processes are used to test the applicability of stochastic models of flow and transport in a typical field setting. A simulation domain encompassing a portion of the upper saturated aquifer materials beneath the Lawrence Livermore National Laboratory was developed to hierarchically represent known hydrostratigraphic units and more detailed stochastic representations of geologic heterogeneity within them. Within each unit, Gaussian random field models were used to represent hydraulic conductivity variation, as parameterized from well test data and geologic interpretation of spatial variability. Groundwater flow, transport and remedial extraction of two hypothetical contaminants were made in six different statistical realizations of the system. The effective flow and transport behavior observed in the simulations compared reasonably with the predictions of stochastic theories based upon the Gaussian models, even though more exacting comparisons were prevented by inherent nonidealities of the geologic model and flow system. More importantly, however, biases and limitations in the hydraulic data appear to have reduced the applicability of the Gaussian representations and clouded the utility of the simulations and effective behavior based upon them. This suggests a need for better and unbiased methods for delineating the spatial distribution and structure of geologic materials and hydraulic properties in field systems. High performance computing can be of critical importance in these endeavors, especially with respect to resolving transport processes within highly variable media.©1998 Elsevier Science Limited. All rights reserved

Modeling groundwater flow on MPPs

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(λ), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.

A numerical investigation of the conjugate gradient method as applied to three‐dimensional groundwater flow problems in randomly heterogeneous porous media

Two natural and efficient stopping criteria are derived for conjugate gradient (CG) methods, based on iteration parameters. The derivation makes use of the inner product matrix B-defining the CG method. In particular, the relationship between the eigenvalues and B-norm of a matrix is investigated, and it is shown that the ratio of largest to smallest eigenvalues defines the B-condition number of the matrix. Upper and lower bounds on various measures of the error are also given. The compound stopping criterion presented here is an obvious “default” in software packages because it does not require any additional norm computations.

ParFlow User's Manual

This paper explores the use of adaptive polynomial preconditioning for Hermitian positive definite linear systems,