On the convergence of Krylov methods with low-rank truncations

Low-rank Krylov methods are one of the few options available in the literature to address the numerical solution of large-scale general linear matrix equations. These routines amount to well-known Krylov schemes that have been equipped with a couple of low-rank truncations to maintain a feasible storage demand in the overall solution procedure. However, such truncations may affect the convergence properties of the adopted Krylov method. In this paper we show how the truncation steps have to be performed in order to maintain the convergence of the Krylov routine. Several numerical experiments validate our theoretical findings.

A spectral-in-time Newton–Krylov method for nonlinear PDE-constrained optimization

We devise a method for nonlinear time-dependent partial-differential-equation-constrained optimization problems that uses a spectral-in-time representation of the residual, combined with a Newton–Krylov method to drive the residual to zero. We also propose a preconditioner to accelerate this scheme. Numerical results indicate that this method can achieve fast and accurate solution of nonlinear problems for a range of mesh sizes and problem parameters. The numbers of outer Newton and inner Krylov iterations required to reach the attainable accuracy of a spatial discretization are robust with respect to the number of collocation points in time and also do not change substantially when other problem parameters are varied.

Jacobian Free Methods for Coupling Transport with Chemistry in Heterogenous Porous Media

Reactive transport plays an important role in various subsurface applications, including carbon dioxide sequestration, nuclear waste storage, biogeochemistry and the simulation of hydro–thermal reservoirs. The model couples a set of partial differential equations, describing the transport of chemical species, to nonlinear algebraic or differential equations, describing the chemical reactions. Solution methods for the resulting large nonlinear system can be either fully coupled or can iterate between transport and chemistry. This paper extends previous work by the authors where an approach based on the Newton–Krylov method applied to a reduced system has been developed. The main feature of the approach is to solve the nonlinear system in a fully coupled manner while keeping transport and chemistry modules separate. Here we extend the method in two directions. First, we take into account mineral precipitation and dissolution reactions by using an interior point Newton method, so as to avoid the usual combinatorial approach. Second, we study two-dimensional heterogeneous geometries. We show how the method can make use of an existing transport solver, used as a black box. We detail the methods and algorithms for the individual modules, and for the coupling step. We show the performance of the method on synthetic examples.