Extending TRANSIMS Technology to an Integrated Multilevel Representation

The TRANSIMS system developed at Los Alamos in the USA over the past decade is a world leader in providing an integrated land-use transportation dynamical model for large areas with a million or more inhabitants. TRANSIMS uses standard survey data to create synthetic micropopulations, including family structure, to simulate trip making and emergent traffic dynamics. We propose to extend TRANSIMS by adapting it to a new multi-level representation, allowing dynamics to be algebraically integrated at the micro-, meso- and macro-levels. The new representation builds a lattice hierarchy in a way that integrates non-partitional hierarchies of links and routes based on the usual hierarchy of geographical zones, e.g. neighbourhoods, districts, cities, counties and countries. Applying the representation to a big city starts by defining sets of zones at different levels. At the first level, N, is the street. This can be subdivided to building plots at level N-1, buildings at level N-2, and even rooms at level N-3. At level N+1 are the neighbourhoods, at level N+2 is the set of district zones (each of them containing the different neighbourhoods in the previous level), and at the top level N+3 (in this case), is just one zone, the city itself. If a larger study area is to be considered, we would have a whole set of N+3 zones defining N+4-level areas, and so on, extending to the level of counties, countries or even continents. This paper will explain the fundamentals of TRANSIMS technology and compare it to other systems. We will show how TRANSIMS and the new multi-level representation can be brought together to give new insights into the macro-dynamics of very large road systems such as London, England and even the whole of Europe

Multidimensional approximation of nonlinear dynamical systems

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems

Real-time modelling and interpolation of spatio-temporal marine pollution

Due to the complexity of the interactions involved in various dynamic systems, known physical, biological or chemical laws cannot adequately describe the dynamics behind these processes. The study of these systems thus depends on measurements often taken at various discrete spatial locations through time by noisy sensors. For this reason, scientists often necessitate interpolative, visualisation and analytical tools to deal with the large volumes of data common to these systems. The starting point of this study is the seminal research by C. Shannon on sampling and reconstruction theory and its various extensions. Based on recent work on the reconstruction of stochastic processes, this paper develops a novel real-time estimation method for non- stationary stochastic spatio-temporal behaviour based on the Integro-Di erence Equation (IDE). This meth- odology is applied to collected marine pollution data from a Norwegian fjord. Comparison of the results obtained by the proposed method with interpolators from state-of-the-art Geographical Information System (GIS) packages will show, that signifi cantly superior results are obtained by including the temporal evolution in the spatial interpolations.peer-reviewe

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.Comment: 26 pages, double columns; 15 figure

Multiconfiguration time-dependent Hartree impurity solver for nonequilibrium dynamical mean-field theory

Nonequilibrium dynamical mean-field theory (DMFT) solves correlated lattice models by obtaining their local correlation functions from an effective model consisting of a single impurity in a self-consistently determined bath. The recently developed mapping of this impurity problem from the Keldysh time contour onto a time-dependent single-impurity Anderson model (SIAM) [C. Gramsch et al., Phys. Rev. B 88, 235106 (2013)] allows one to use wave function-based methods in the context of nonequilibrium DMFT. Within this mapping, long times in the DMFT simulation become accessible by an increasing number of bath orbitals, which requires efficient representations of the time-dependent SIAM wave function. These can be achieved by the multiconfiguration time-dependent Hartree (MCTDH) method and its multi-layer extensions. We find that MCTDH outperforms exact diagonalization for large baths in which the latter approach is still within reach and allows for the calculation of SIAMs beyond the system size accessible by exact diagonalization. Moreover, we illustrate the computation of the self-consistent two-time impurity Green's function within the MCTDH second quantization representation.Comment: 12 pages, 8 figure