2,524 research outputs found
Solving the stationary Liouville equation via a boundary element method
Intensity distributions of linear wave fields are, in the high frequency
limit, often approximated in terms of flow or transport equations in phase
space. Common techniques for solving the flow equations for both time dependent
and stationary problems are ray tracing or level set methods. In the context of
predicting the vibro-acoustic response of complex engineering structures,
reduced ray tracing methods such as Statistical Energy Analysis or variants
thereof have found widespread applications. Starting directly from the
stationary Liouville equation, we develop a boundary element method for solving
the transport equations for complex multi-component structures. The method,
which is an improved version of the Dynamical Energy Analysis technique
introduced recently by the authors, interpolates between standard statistical
energy analysis and full ray tracing, containing both of these methods as
limiting cases. We demonstrate that the method can be used to efficiently deal
with complex large scale problems giving good approximations of the energy
distribution when compared to exact solutions of the underlying wave equation
Data-driven model reduction and transfer operator approximation
In this review paper, we will present different data-driven dimension
reduction techniques for dynamical systems that are based on transfer operator
theory as well as methods to approximate transfer operators and their
eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out
similarities and differences between methods developed independently by the
dynamical systems, fluid dynamics, and molecular dynamics communities such as
time-lagged independent component analysis (TICA), dynamic mode decomposition
(DMD), and their respective generalizations. As a result, extensions and best
practices developed for one particular method can be carried over to other
related methods
Cluster-based reduced-order modelling of a mixing layer
We propose a novel cluster-based reduced-order modelling (CROM) strategy of
unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger's
group (Burkardt et al. 2006) and and transition matrix models introduced in
fluid dynamics in Eckhardt's group (Schneider et al. 2007). CROM constitutes a
potential alternative to POD models and generalises the Ulam-Galerkin method
classically used in dynamical systems to determine a finite-rank approximation
of the Perron-Frobenius operator. The proposed strategy processes a
time-resolved sequence of flow snapshots in two steps. First, the snapshot data
are clustered into a small number of representative states, called centroids,
in the state space. These centroids partition the state space in complementary
non-overlapping regions (centroidal Voronoi cells). Departing from the standard
algorithm, the probabilities of the clusters are determined, and the states are
sorted by analysis of the transition matrix. Secondly, the transitions between
the states are dynamically modelled using a Markov process. Physical mechanisms
are then distilled by a refined analysis of the Markov process, e.g. using
finite-time Lyapunov exponent and entropic methods. This CROM framework is
applied to the Lorenz attractor (as illustrative example), to velocity fields
of the spatially evolving incompressible mixing layer and the three-dimensional
turbulent wake of a bluff body. For these examples, CROM is shown to identify
non-trivial quasi-attractors and transition processes in an unsupervised
manner. CROM has numerous potential applications for the systematic
identification of physical mechanisms of complex dynamics, for comparison of
flow evolution models, for the identification of precursors to desirable and
undesirable events, and for flow control applications exploiting nonlinear
actuation dynamics.Comment: 48 pages, 30 figures. Revised version with additional material.
Accepted for publication in Journal of Fluid Mechanic
Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits
A new expansion scheme to evaluate the eigenvalues of the generalized
evolution operator (Frobenius-Perron operator) relevant to the
fluctuation spectrum and poles of the order- power spectrum is proposed. The
``partition function'' is computed in terms of unstable periodic orbits and
then used in a finite pole approximation of the continued fraction expansion
for the evolution operator. A solvable example is presented and the approximate
and exact results are compared; good agreement is found.Comment: CYCLER Paper 93mar00
Resonances of the cusp family
We study a family of chaotic maps with limit cases the tent map and the cusp
map (the cusp family). We discuss the spectral properties of the corresponding
Frobenius--Perron operator in different function spaces including spaces of
analytic functions. A numerical study of the eigenvalues and eigenfunctions is
performed.Comment: 14 pages, 3 figures. Submitted to J.Phys.
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
Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
In this paper, we provide a new algorithm for the finite dimensional
approximation of the linear transfer Koopman and Perron-Frobenius operator from
time series data. We argue that existing approach for the finite dimensional
approximation of these transfer operators such as Dynamic Mode Decomposition
(DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two
important properties of these operators, namely positivity and Markov property.
The algorithm we propose in this paper preserve these two properties. We call
the proposed algorithm as naturally structured DMD since it retains the
inherent properties of these operators. Naturally structured DMD algorithm
leads to a better approximation of the steady-state dynamics of the system
regarding computing Koopman and Perron- Frobenius operator eigenfunctions and
eigenvalues. However preserving positivity properties is critical for capturing
the real transient dynamics of the system. This positivity of the transfer
operators and it's finite dimensional approximation also has an important
implication on the application of the transfer operator methods for controller
and estimator design for nonlinear systems from time series data
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