The ADI iteration is closely related to the rational Krylov projection methods for constructing low rank approximations to the solution of Sylvester equation. In this paper we show that the ADI and rational Krylov approximations are in fact equivalent when a special choice of shifts are employed in both methods. We will call these shifts pseudo H2-optimal shifts. These shifts are also optimal i...

We present a multilevel high order ADI method for separable generalized Helmholtz equations. The discretization method we use is a 1-D fourth order compact nite di erence applied to each directional component of the Laplace operator, resulting a discrete system e ciently solvable by ADI methods. We apply this high order di erence scheme to all levels of grids, and then starting from the coarses...

We present a parallel nite-diierence algorithm for the solution of the 3D cylindrical Poisson equation. The algorithm is based on a data-transpose technique, in which all computations are performed independently on each node, and all communications are restricted to global 3D data-transposition between nodes. The data-transpose technique aids us in implementing two sequential algorithms for the...

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting me...

The convergence of an alternating direction implicit method for Maxwell’s equations on product domains is investigated. Unlike the classical Yee scheme and most other integrators proposed in the literature, this method is both unconditionally stable and computationally cheap. We prove second-order convergence of the time-discretization in the framework of operator semigroup theory. In contrast ...

This paper deals with the numerical solution of the Heston partial differential equation (PDE) that plays an important role in financial option pricing theory, Heston (1993). A feature of this time-dependent, twodimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes f...

In this paper we consider the numerical solution of large-scale projected generalized continuous-time and discrete-time Sylvester equations with low-rank right-hand sides. First, we present the results on the sufficient conditions for the existence, uniqueness, and analytic formula of the solutions of these equations. Second, we review the low-rank alternating direction implicit method and the ...

LOW-RANK ITERATIVE METHODS FOR PROJECTED GENERALIZED LYAPUNOV EQUATIONS TATJANA STYKEL Abstract. We generalize an alternating direction implicit method and the Smith method for large-scale projected generalized Lyapunov equations. Such equations arise in model reduction of descriptor systems. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximat...