In this article, we investigate and analyze new methods for the numerical solution of three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these combine Galerkin finite element (FEMs) spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on backward Euler (BE) method Crank-Nicolson (CN) method, respectively, from which, ...

We give a highly efficient, accurate and unconditionally stable algorithm to solve the partial differential equation for simulating the action potential propagation through cardiac tissue. In the new algorithm, we discretize the space domain by combining a compact finite difference scheme with an alternating direction implicit (ADI) scheme, which has fourth-order accuracy for interior mesh poin...

Diierential cryptanalysis is a powerful attack developed by Eli Biham and Adi Shamir. It has been successfully applied to many DES-like cryptosystems. We provide a brief introduction to their paper BS91] and show how to apply diierential cryptanalysis to attack the Data Encryption Standard (DES) reduced to 5 rounds.

We consider a nonlinear fourth-order diffusion equation that arises in denoising of image densities. We propose an alternative direction implicit (ADI) splitting method for its numerical solution. To treat the high-order and mixed derivative terms in the equation we adopt an ADI method by Hundsdorfer and Verwer to the present setting. The paper is furnished with numerical results for the evolut...

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 ...

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence (CDI) FDTD method. In this paper, formulations from time-collocated to fundamental are presented ADI CDI methods. For implemented using E-E E-H update procedures, which comprise simple concise right-hand sides...

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...

We present a sparse grid high-order alternating direction implicit (ADI) scheme for option pricing in stochastic volatility models. The scheme is second-order in time and fourth-order in space. Numerical experiments confirm the computational efficiency gains achieved by the sparse grid combination technique.