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

Dynamics and energetics of emergent magnetic monopoles in chiral magnets

The formation and destruction of topologically quantized magnetic whirls, so-called skyrmions, in chiral magnets is driven by the creation and motion of singular hedgehog defects. These can be identified with emergent magnetic monopoles and antimonopoles. We investigate how the energetics of and forces between monopoles and antimonopoles influence their creation rate and dynamics. We study a single skyrmion line defect in the helical phase using both micromagnetic simulations and a Ginzburg-Landau analysis. Monopole-antimonople pairs are created in a thermally activated process, largely controlled by the (core) energy of the monopole. The force between monopoles and antimonopoles is linear in distance and described by a string tension. The sign and size of the string tension determines the stability of the phases and the velocity of the monopoles.Comment: 4 pages, 5 figure

Improving the Lattice QCD Hamiltonian

Improvement of the Hamiltonian in lattice gauge theory is considered. We give explicit expressions for classical improvement and discuss also quantum corrections.Comment: 3 pages, Latex. Presented at Lattice 97: 15th International Symposium on Lattice Field Theory, Edinburgh, Scotland, 22-26 Jul 1997, to appear in Nucl. Phys. B(Proc. Suppl.

Numerical Investigation of the Aerodynamic Properties of a Flying Wing Configuration

The numerical investigations of a generic UCAV configuration are presented. These investigations are part of the DLR internal project UCAV-2010. Compressible speed conditions are considered and presented. The DLR-F17E UCAV configuration is a flying lambda delta wing with sweep angle of 53° and varying leading edge radius. The flow field of this UCAV configuration is dominated by vortex structures and vortex-to-vortex interaction. The paper aims to give a comparison between numerical- and experimental investigations in order to gain a deeper understanding of the complex flow physics. Furthermore, it will highlight the influence of Mach- and Reynolds number change on the flow and the overall aerodynamic behavior of the configuration. The DLR TAU-Code is used to simulate the flow field, using an unstructured grid and the turbulence model of Spalart-Allmaras. Forces and moment measurements taken in the DNW-TWG, Göttingen, on the DLR-F17E configuration serve as the experimental basis to validate the numerical findings. Findings on the SACCON configuration serve as a comparison case aiming to show possible portability between different model scales but also to find analogies between low speed (M=0.15) and compressible speed (M=0.5) scenarios. This paper builds up upon the finding within the NATO/RTO AVT-161 Research Task Group on “Assessment and Control Predictions for NATO Air and Sea Vehicles” and its findings shall serve as a basis for further experimental investigations of medium to high speed wind tunnel experiments. Furthermore, this paper addresses the importance of understanding and the ability to predict controlled- and uncontrolled flow separation and the interaction of vortex systems in order to estimate the aerodynamic behavior within the entire flight envelope and to meet Stability- and Control needs

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