1999

H.F. Walker.
**“An Adaptation of Krylov Subspace Methods to Path Following Problems,”** In *SIAM journal on scientific computing*, Vol. 21, No. 3, pp. 1191--1198. 1999.

DOI: 10.1137/S1064827597315376

A procedure is outlined for adapting Krylov subspace methods to solving approximately the underdetermined linear systems that arise in path following (continuation, homotopy) methods. This procedure, in addition to preserving the usual desirable features of Krylov subspace methods, has the advantages of satisfying orthogonality constraints exactly and of not introducing ill-conditioning through poor scaling.

1998

M. Sosonkina, L.T. Watson, R.K. Kapania, H.F. Walker.
**“A new adaptive GMRES algorithm for achieving high accuracy,”** In *Numerical Linear Algebra and Applications*, Vol. 5, No. 4, pp. 275--297. 1998.

DOI: 10.1002/(SICI)1099-1506(199807/08)5:43.0.CO;2-B

GMRES(*k*) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for *k* close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(*k*) which tunes the restart value *k* based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(*k*) procedure outperforms standard GMRES(*k*), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(*k*) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(*k*) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications.

M.D. Tocci, C.T. Kelley, C.T. Miller, C.E. Kees.
**“Inexact Newton Methods and the Method of Lines for Solving Richards' Equation in Two Space Dimensions,”** In *Computational Geosciences*, Vol. 2, No. 4, pp. 291--309. 1998.

DOI: 10.1023/A:1011562522244

Richards' equation (RE) is often used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which are present in more complex models of multiphase flow. The numerical solution of RE is difficult not only because of these physical effects but also because of the mathematical problems that arise in dealing with the nonlinearities. The method of lines has been shown to be very effective for solving RE in one space dimension. When solving RE in two space dimensions, direct methods for solving the linearized problem for the Newton step are impractical. In this work, we show how the method of lines and Newton-iterative methods, which solve linear equations with iterative methods, can be applied to RE in two space dimensions. We present theoretical results on convergence and use that theory to design an adaptive method for computation of the linear tolerance. Numerical results show the method to be effective and robust compared with an existing approach.

S. Vyazovkin, C.A. Wight.
**“Isothermal and non-isothermal kinetics of thermally stimulated reactions of solids,”** In *International Reviews in Physical Chemistry*, Vol. 17, No. 3, pp. 407--433. 1998.

DOI: 10.1080/014423598230108

This review covers both the history and present state of the kinetics of thermally stimulated reactions in solids. The traditional methodology of kinetic analysis, which is based on fitting data to reaction models, dates back to the very first isothermal studies. The model fitting approach suffers from an inability to determine the reaction model uniquely,and this does not allow reliable mechanistic conclusions to be drawn even from isothermal data. In non-isothermal kinetics, the use of the traditional methodology results in highly uncertain values of Arrhenius parameters that cannot be compared meaningfully with isothermal values. An alternative model-free methodology is based on the isoconversional method. The use of this model-free approach in both isothermal and non-isothermal kinetics helps to avoid the problems that originate from the ambiguous evaluation of the reaction model. The model-free methodology allows the dependence of the activation energy on the extent of conversion to be determined. This, in turn, permits reliable reaction rate predictions to be made and mechanistic conclusions to be drawn.