Consistent Dynamic Mode Decomposition

We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of data alignment penalty terms and constitutive orthogonality constraints. Our method does not make any assumptions on the structure of the data or their size, and thus it is applicable to a wide range of problems including non-linear scenarios or extremely small observation sets. In addition, our technique is robust to noise that is independent of the dynamics and it does not require input data to be sequential. Our key idea is to introduce a regularization term for the forward and backward dynamics. The obtained minimization problem is solved efficiently using the Alternating Method of Multipliers (ADMM) which requires two Sylvester equation solves per iteration. Our numerical scheme converges empirically and is similar to a provably convergent ADMM scheme. We compare our approach to various state-of-the-art methods on several benchmark dynamical systems

Binary Stars and Globular Cluster Dynamics

In this brief proceedings article I summarize the review talk I gave at the IAU 246 meeting in Capri, Italy, glossing over the well-known results from the literature, but paying particular attention to new, previously unpublished material. This new material includes a careful comparison of the apparently contradictory results of two independent methods used to simulate the evolution of binary populations in dense stellar systems (the direct N-body method of Hurley, et al. 2007 and the approximate Monte Carlo method of Ivanova, et al. 2005), that shows that the two methods may not actually yield contradictory results, and suggests future work to more directly compare the two methods.Comment: 7 pages, 1 figure, to appear in "Dynamical Evolution of Dense Stellar Systems", IAUS 246, ed. E. Vesperin

Performance analysis of continuous-time solvers for quantum impurity models

Impurity solvers play an essential role in the numerical investigation of strongly correlated electrons systems within the "dynamical mean field" approximation. Recently, a new class of continuous-time solvers has been developed, based on a diagrammatic expansion of the partition function in either the interactions or the impurity-bath hybridization. We investigate the performance of these two complementary approaches and compare them to the well-established Hirsch-Fye method. The results show that the continuous-time methods, and in particular the version which expands in the hybridization, provide substantial gains in computational efficiency

Fast spin dynamics algorithms for classical spin systems

We have proposed new algorithms for the numerical integration of the equations of motion for classical spin systems. In close analogy to symplectic integrators for Hamiltonian equations of motion used in Molecular Dynamics these algorithms are based on the Suzuki-Trotter decomposition of exponential operators and unlike more commonly used algorithms exactly conserve spin length and, in special cases, energy. Using higher order decompositions we investigate integration schemes of up to fourth order and compare them to a well established fourth order predictor-corrector method. We demonstrate that these methods can be used with much larger time steps than the predictor-corrector method and thus may lead to a substantial speedup of computer simulations of the dynamical behavior of magnetic materials.Comment: 9 pages RevTeX with 8 figure

Kinetic theory for scalar fields with nonlocal quantum coherence

We derive quantum kinetic equations for scalar fields undergoing coherent evolution either in time (coherent particle production) or in space (quantum reflection). Our central finding is that in systems with certain space-time symmetries, quantum coherence manifests itself in the form of new spectral solutions for the dynamical 2-point correlation function. This spectral structure leads to a consistent approximation for dynamical equations that describe coherent evolution in presence of decohering collisions. We illustrate the method by solving the bosonic Klein problem and the bound states for the nonrelativistic square well potential. We then compare our spectral phase space definition of particle number to other definitions in the nonequilibrium field theory. Finally we will explicitly compute the effects of interactions to coherent particle production in the case of an unstable field coupled to an oscillating background.Comment: 33 pages, 7 figures, replaced with the version published in JHE

Accessing the dynamics of large many-particle systems using Stochastic Series Expansion

The Stochastic Series Expansion method (SSE) is a Quantum Monte Carlo (QMC) technique working directly in the imaginary time continuum and thus avoiding "Trotter discretization" errors. Using a non-local "operator-loop update" it allows treating large quantum mechanical systems of many thousand sites. In this paper we first give a comprehensive review on SSE and present benchmark calculations of SSE's scaling behavior with system size and inverse temperature, and compare it to the loop algorithm, whose scaling is known to be one of the best of all QMC methods. Finally we introduce a new and efficient algorithm to measure Green's functions and thus dynamical properties within SSE.Comment: 11 RevTeX pages including 7 figures and 5 table

The Globular Cluster Luminosity Function as a Distance Indicator: Dynamical Effects

The dynamical evolution of the globular cluster systems in galaxies is predicted, based on the standard dynamical theory normalized to the example of the Milky Way. The major processes varying with the galactocentric distance are the tidal shocks and dynamical friction. Our simple model explains, on a quantitative basis, the observed differences of the inner and outer populations of globular clusters. We can thus calculate corrections for dynamical evolution for the luminosity function of globular clusters with the assumption that the initial luminosity function is identical in all galaxies (and we can test this assumption as well, in certain cases). Then we can compute the expected distribution of absolute magnitudes and compare it with the observed distribution of apparent magnitudes to estimate the distance moduli for M31 and M87. Using this new method we find dm(M31)=24.05 +- 0.23, dm(M87)=30.83 +- 0.17, as compared to current best estimates using other methods of dm(M31)=24.30 +- 0.20, dm(M87)=31.0 +- 0.1. As a check on the method we compute, and compare with observations, the differences between the inner and outer globular clusters in all three galaxies. This new method, coupled with HST observations, promises to provide an independent method of estimating distances to galaxies with recession velocities < 10,000 km/s, or D < 100 Mpc.Comment: 12 pages, 2 figures; submitted to ApJ Letter

The Potential-Density Phase Shift Method for Determining the Corotation Radii in Spiral and Barred Galaxies

We have developed a new method for determining the corotation radii of density waves in disk galaxies, which makes use of the radial distribution of an azimuthal phase shift between the potential and density wave patterns. The approach originated from improved theoretical understandings of the relation between the morphology and kinematics of galaxies, and on the dynamical interaction between density waves and the basic-state disk stars which results in the secular evolution of disk galaxies. In this paper, we present the rationales behind the method, and the first application of it to several representative barred and grand-design spiral galaxies, using near-infrared images to trace the mass distributions, as well as to calculate the potential distributions used in the phase shift calculations. We compare our results with those from other existing methods for locating the corotations, and show that the new method both confirms the previously-established trends of bar-length dependence on galaxy morphological types, as well as provides new insights into the possible extent of bars in disk galaxies. Application of the method to a larger sample and the preliminary analysis of which show that the phase shift method is likely to be a generally-applicable, accurate, and essentially model-independent method for determining the pattern speeds and corotation radii of single or nested density wave patterns in galaxies. Other implications of this work are: most of the nearby bright disk galaxies appear to possess quasi-stationary spiral modes; that these density wave modes and the associated basic state of the galactic disk slowly transform over time; and that self-consistent N-particle systems contain physics not revealed by the passive orbit analysis approaches.Comment: 48 pages, 16 figures. Accepted for publication in the Astronomical Journa