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We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with iterative refinement with a factorization in half precision. We analyze the method as an inexact Newton method. This analysis shows that, except for very poorly conditioned Jacobians, the number of nonlinear iterations needed is the same that one would get if one stored and factored the Jacobian in double precision. In many ill-conditioned cases one can use the low precision factorization as a preconditioner for a GMRES iteration. That approach can recover fast convergence of the nonlinear iteration. We present an example to illustrate the results.Dedication: To Masao Fukushima on his 75th birthday, with thanks for his many contributions to optimization.

Anderson(m) is a method for acceleration of fixed point iteration which stores m + 1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson(m) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. We prove q-linear convergence of the residuals for linear problems for Anderson(1) without the assumption of boundedness of the coefficients. We observe that the optimization problem for the coefficients can be formulated and solved in non-standard ways and report on numerical experiments which illustrate the ideas.

Abstract. Pseudo-transient continuation (Ψtc) is a well-known and physically motivated technique for computation of steady state solutions of time-dependent partial differential equations. Standard globalization strategies such as line search or trust region methods often stagnate at local minima. Ψtc succeeds in many of these cases by taking advantage of the underlying PDE structure of the problem. Though widely employed, the convergence of Ψtc is rarely discussed. In this paper we prove convergence for a generic form of Ψtc and illustrate it with two practical strategies.

Abstract. Capillary pressure-saturation-relative permeability relations described using the van Genuchten [1980] and Mualem [1976] models for nonuniform porous media lead to numerical convergence difficulties when used with Richards' equation for certain auxiliary conditions. These difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure. Convergence difficulties are illustrated using standard numerical approaches to simulate such problems. We investigate constitutive relations, interblock permeability, nonlinear algebraic system approximation methods, and two time integration approaches. An integral permeability approach approximated by Hermite polynomials is recommended and shown to be robust and economical for a set of test problems, which correspond to sand, loam, and clay loam media. An example of such a case was for infiltration from a ponded surface boundary condition into a system originally drained to static equilibrium.These experiences motivated this work, which had several objectives: (1) to document a common class of variably saturated flow problems that lack robustness when solved using standard solution approaches, (2) to determine the reason why traditional approaches lack robustness for this class of problems, (3) to investigate a variety of alternative approaches, and (4) to compare a set of alternative approaches for a range of porous media conditions to test robustness and efficiency. BackgroundFour aspects of the literature on unsaturated flow warrant at least a brief consideration: (1) constitutive relations used to describe pressure-saturation-conductivity relations and typical parameter values for natural, unconsolidated media, (2) approaches typically used to approximate RE, (3) methods for approximating relative permeabilities for a discrete approximation of RE, and (4) strategies used to estimate the relatively complex constitutive relations that are a part of the formulations of concern. Pressure-Saturation-Conductivity RelationsA well-posed formulation of RE requires that constitutive relations be specified to describe the interdependence among fluid pressures, saturations, and relative permeabilities, which will be referred to as p-s-k relations.

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