NUMERICAL CALCULATION OF THE COMPLEX BERRY PHASE IN NON-HERMITIAN SYSTEMS

We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by <em>PT</em>-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional <em>PT</em>-symmetric lattice system and verify our numerical results by an analytical calculation.

A Simplified Calculation of Reduced HCT--Basis Functions in a Finite Element Context

We present simple formulas for the definition of the basis functions on a reduced Hsieh-Clough-Tocher (rHCT) element. These formulas use the P_3-biorthogonal basis in the master triangle and form the resulting basis with the help of the edge vectors of the triangle only. This allows for a simple and efficient algorithm to compute the stiffness matrices.

On the convergence of the wavelet-Galerkin method for nonlinear filtering

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.