Pancharatnam and Berry Phases in Three-Level Photonic Systems

A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate co-linear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic 'curves'in Hilbert space. Explicit expressions for Pancharatnam, dynamical and geometrical phases are derived for the transformations produced by linear phase-converters. The problem of gauge invariance is treated along all the article

emgr - The Empirical Gramian Framework

System Gramian matrices are a well-known encoding for properties of input-output systems such as controllability, observability or minimality. These so-called system Gramians were developed in linear system theory for applications such as model order reduction of control systems. Empirical Gramian are an extension to the system Gramians for parametric and nonlinear systems as well as a data-driven method of computation. The empirical Gramian framework - emgr - implements the empirical Gramians in a uniform and configurable manner, with applications such as Gramian-based (nonlinear) model reduction, decentralized control, sensitivity analysis, parameter identification and combined state and parameter reduction

Diffusion Maps Kalman Filter for a Class of Systems with Gradient Flows

In this paper, we propose a non-parametric method for state estimation of high-dimensional nonlinear stochastic dynamical systems, which evolve according to gradient flows with isotropic diffusion. We combine diffusion maps, a manifold learning technique, with a linear Kalman filter and with concepts from Koopman operator theory. More concretely, using diffusion maps, we construct data-driven virtual state coordinates, which linearize the system model. Based on these coordinates, we devise a data-driven framework for state estimation using the Kalman filter. We demonstrate the strengths of our method with respect to both parametric and non-parametric algorithms in three tracking problems. In particular, applying the approach to actual recordings of hippocampal neural activity in rodents directly yields a representation of the position of the animals. We show that the proposed method outperforms competing non-parametric algorithms in the examined stochastic problem formulations. Additionally, we obtain results comparable to classical parametric algorithms, which, in contrast to our method, are equipped with model knowledge.Comment: 15 pages, 12 figures, submitted to IEEE TS

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.Comment: 18 pages, 5 figures, Extended version of Max Planck Institute report (MPIMD/13-21

Time-varying signal processing using multi-wavelet basis functions and a modified block least mean square algorithm

This paper introduces a novel parametric modeling and identification method for linear time-varying systems using a modified block least mean square (LMS) approach where the time-varying parameters are approximated using multi-wavelet basis functions. This approach can be used to track rapidly or even sharply varying processes and is more suitable for recursive estimation of process parameters by combining wavelet approximation theory with a modified block LMS algorithm. Numerical examples are provided to show the effectiveness of the proposed method for dealing with severely nonstatinoary processes