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...

Asynchronous events sequences are widely distributed in the natural world and human activities, such as earthquakes records, users’ activities social media, so on. How to distill information from these seemingly disorganized data is a persistent topic that researchers focus One of most useful models point process model, on basis, obtain many noticeable results. Moreover, recent years, foundatio...

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...

In implicit solvation models, the electrostatic contribution to the solvation energy can be estimated by solving a system of elliptic partial differential equations modeling the reaction potential. The domain of definition of such elliptic equations is the union of the van der Waals cavities corresponding to the atoms of the solute molecule. Therefore, the computations can naturally be performe...

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction-diffusion equations. The applications of reaction-diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is...

An algorithm is presented for constructing an approximate numerical solution to a large scale Lyapunov equation in low rank factored form. The algorithm is based upon a synthesis of an approximate power method and an alternating direction implicit (ADI) method. The former is parameter free and tends to be efficient in practice, but there is little theoretical understanding of its convergence pr...

A novel formulation of approximate truncated balanced realization (TBR) is introduced to unify three approaches: two iterative methods for solving the underlying Lyapunov equations – the alternating directions implicit (ADI) iteration and the rational Krylov subspace method (RKSM) – and a two-step procedure that performs a Krylov-based projection and subsequently direct TBR. The framework allow...

In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed...

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has ...