Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations

Abstract

Abstract. Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data is far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase.In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized partial differential equation with a Lipschitz continuous, but non-differentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient.The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches require problem-specific components in an algorithm. The method in this paper addresses the nonsmoothness directly.Key words. Pseudo-transient Continuation, Nonlinear equations, Semismooth functions, Clarke differential AMS subject classifications. 65H10, 65H20, 65L05, 1. Introduction. In this paper we show how pseudo-transient continuation (Ψtc ) can be used to solve a class of nonsmooth nonlinear equations. Ψtc is a predictor-corrector method for efficient integration of a time-dependent differential equation to steady state. The objective of the method is not temporal accuracy, but rather to resolve the transient behavior of the solution until the iteration is close to steady state, and then to increase the "time step" and transition to a fast Newton-like method.In this paper we extend the theoretical convergence results of [7,18] to problems with certain nonsmooth nonlinearities and, thereby, partially explain the results reported in [8,10]. We also show how generalized derivatives can be approximated by finite differences, and how those approximate derivatives can be used effectively both in locally convergent iterations, such as those which arise in temporal integration, and in the context of Ψtc . This aspect of the work is motivated by several papers on simulation of unsaturated flow, [10,14,15,24,30,31], in which Lipschitz continuous spline approximations to the non-Lipschitz continuous van Geneuchten and Mualem [25,33] constitutive laws are used. These nonsmooth functions are then differentiated with finite differences as if they were smooth. The results in this paper explain the success reported in those papers. Another aspect of the paper is an extension of the local results in [9,21,27,28].Ψtc methods are particularly appropriate for the types of nonsmooth nonlinearities which we discuss in this paper. Traditional methods for globalizing iterative methods for nonlinear equa-

Projected Pseudotransient Continuation

“…Even in the unconstrained case (where P(u) = u for all u ∈ R N ) Theorem 2.1 extends the results from [9,13,24] by replacing the semismoothness and the inexact Newton condition with a general condition on the convergence of the local iteration. If the stability condition (1.1) holds, then Ψtc with SER is a convergent iteration for unconstrained optimization, even if one does not use exact Hessians.…”

“…The complete proof is based on the same ideas but requires more bookkeeping. The outline of the proof follows those in [9,13,24].…”

“…If F is semismooth, which is the case considered in [13], and H(u n ) ∈ ∂F (u n ), then the local iteration (2.7) is superlinearly convergent, if all matrices in ∂F (u * ) are nonsingular. Hence, we may recover the results from [13] from Theorem 2.1, the extension to DAE dynamics being the same as that in [13] and not relevant to this paper.…”

“…We refer the reader to [9,12,13,15,24] for the existing convergence results and references to applications.…”

“…In section 2 we will describe an algorithm for constrained Ψtc and prove a convergence theorem which not only allows for constraints but has a weaker stability assumption than was used in [9,13,24] and includes more general assumptions on the iteration itself. We close section 2 with some remarks on applications of the theory.…”