Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator

The global behavior of dynamical systems can be studied by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with the system. Two important operators which are frequently used to gain insight into the system's behavior are the Perron-Frobenius operator and the Koopman operator. Due to the curse of dimensionality, computing the eigenfunctions of high-dimensional systems is in general infeasible. We will propose a tensor-based reformulation of two numerical methods for computing finite-dimensional approximations of the aforementioned infinite-dimensional operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD). The aim of the tensor formulation is to approximate the eigenfunctions by low-rank tensors, potentially resulting in a significant reduction of the time and memory required to solve the resulting eigenvalue problems, provided that such a low-rank tensor decomposition exists. Typically, not all variables of a high-dimensional dynamical system contribute equally to the system's behavior, often the dynamics can be decomposed into slow and fast processes, which is also reflected in the eigenfunctions. Thus, the weak coupling between different variables might be approximated by low-rank tensor cores. We will illustrate the efficiency of the tensor-based formulation of Ulam's method and EDMD using simple stochastic differential equations

combined rainfed farming systems in Ishkamesh, Afghanistan

This paper deals with the multiple insecurities affecting combined rainfed farming systems in the Ishkamesh district of Takhar Province in Northern Afghanistan. It looks at how local natural resource management practices work out under conditions of recurrent and severe drought and how pasture access regimes and rainfed farming practices structure intergroup relations in an area that was affected by heavy fighting during Soviet occupation and civil war in Afghanistan. Based on findings from three periods of fieldwork in the area, Ishkamesh can be used to provide a better understanding of practices of rainfed agriculture and the construction of rural livelihoods at the Afghan periphery, which is influenced by high risk environmental conditions. Affected by scarcity of land, deficiency of water, restricted income opportunities and restricted access to education and health facilities, the threats to human security of local populations are identified and security strategies examined

The Kunduz Oasis and Military Globalisation

International audienceL'intervention Occidentale en Afghanistan est un indicateur des grands processus en cours, dans le cadre très humanitarisé, caractérisé par l'enchevêtrement d'acteurs militaires, ONGs et pouvoir politique, qui tentent d'établir une " paix libérale " dans ce pays. Dans ce contexte, la mondialisation militaire qui a pris pied en Afghanistan, influence les configurations sociales locales et les interconnexions qui touchent spécialement l'économie sociale des oasis comme pôle économique et agricole en Afghanistan. Au regard de l'exemple spécifique de l'oasis de Kunduz au nord du pays, on tentera d'analyser les changements locaux affectés par l'émergence d'acteurs militaires internationaux. L'analyse sera faite au reflet du contexte historique dans lequel se sont opérés ces changements dans l'oasis de Kunduz. Ces résultats sont issus de multiples travaux de terrain, en particulier sur le district de Chahar Dara, sur la marge orientale de l'oasis. L'étude s'intéresse à la structure du système de production agricole et les mutations en cours dans les pâturages itinérants, régulés par les changements dans les usages coutumiers du sol sur les marges arides de l'oasis. Ces processus sont accompagnés de nouvelles configurations sociales dans l'oasis de Kunduz qui s'opèrent avec l'éveil de nouvelles stratégies de contre-offensive mises en place par les militaires étrangers

Tensor-based dynamic mode decomposition

Dynamic mode decomposition (DMD) is a recently developed tool for the analysis of the behavior of complex dynamical systems. In this paper, we will propose an extension of DMD that exploits low-rank tensor decompositions of potentially high-dimensional data sets to compute the corresponding DMD modes and eigenvalues. The goal is to reduce the computational complexity and also the amount of memory required to store the data in order to mitigate the curse of dimensionality. The efficiency of these tensor-based methods will be illustrated with the aid of several different fluid dynamics problems such as the von K\'arm\'an vortex street and the simulation of two merging vortices

Nearest-Neighbor Interaction Systems in the Tensor-Train Format

Low-rank tensor approximation approaches have become an important tool in the scientific computing community. The aim is to enable the simulation and analysis of high-dimensional problems which cannot be solved using conventional methods anymore due to the so-called curse of dimensionality. This requires techniques to handle linear operators defined on extremely large state spaces and to solve the resulting systems of linear equations or eigenvalue problems. In this paper, we present a systematic tensor-train decomposition for nearest-neighbor interaction systems which is applicable to a host of different problems. With the aid of this decomposition, it is possible to reduce the memory consumption as well as the computational costs significantly. Furthermore, it can be shown that in some cases the rank of the tensor decomposition does not depend on the network size. The format is thus feasible even for high-dimensional systems. We will illustrate the results with several guiding examples such as the Ising model, a system of coupled oscillators, and a CO oxidation model

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

Stadtentwicklung in Kabul im Kontext von Staatsaufbau und militärisch-humanitärer Intervention

Stadtentwicklung in der afghanischen Hauptstadt Kabul nach der westlichen Intervention ist gekennzeichnet von der Verschränkung einer durch Staatsaufbau nach westlichem Vor bild gesteuerten Logik und der sich in diesem Rahmen ausbreitenden Aneignung des städtischen Raumes durch ganz verschiedene Akteure. Vor diesem Hintergrund wird das Ziel verfolgt, stadträumliche Entwicklungsprozesse in Kabul nach 2001 genauer in den Blick zu nehmen und mit jüngeren Debatten um urbane Informalität im globalen Süden zu verknüpfen. Dabei werden verschiedene, aufeinander bezogene Regime der Stadtplanung und ihre Raumproduktionen unter der Linse urbaner Informalität genauer betrachtet. Es soll herausgearbeitet werden, wie eine Herstellung und Zementierung städtischer Ungleichheit forciert wird, wie bestimmte soziale Dispositionen und Praktiken im Kontext von Informalität charakterisiert werden und wie urbane Informalität gezielt als Ressource zur Macht- und Wohlstandsaneignung eingesetzt werden kann

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics