Convergence analysis of a proximal Gauss-Newton method

An extension of the Gauss-Newton algorithm is proposed to find local minimizers of penalized nonlinear least squares problems, under generalized Lipschitz assumptions. Convergence results of local type are obtained, as well as an estimate of the radius of the convergence ball. Some applications for solving constrained nonlinear equations are discussed and the numerical performance of the method is assessed on some significant test problems

A study on iterative methods for solving Richards` equation

This work concerns linearization methods for efficiently solving the Richards` equation,a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media.The discretization of Richards` equation is based on backward Euler in time and Galerkin finite el-ements in space. The most valuable linearization schemes for Richards` equation, i.e. the Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are presented and theirperformance is comparatively studied. The convergence, the computational time and the conditionnumbers for the underlying linear systems are recorded. The convergence of theLscheme is theo-retically proved and the convergence of the other methods is discussed. A new scheme is proposed,theLscheme/Newton method which is more robust and quadratically convergent. The linearizationmethods are tested on illustrative numerical examples

To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case

In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way

An almost symmetric Strang splitting scheme for nonlinear evolution equations

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described the classic Strang splitting scheme, while still a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation