6,487 research outputs found
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
Simulation of fermionic lattice models in two dimensions with Projected Entangled-Pair States: Next-nearest neighbor Hamiltonians
In a recent contribution [Phys. Rev. B 81, 165104 (2010)] fermionic Projected
Entangled-Pair States (PEPS) were used to approximate the ground state of free
and interacting spinless fermion models, as well as the - model. This
paper revisits these three models in the presence of an additional next-nearest
hopping amplitude in the Hamiltonian. First we explain how to account for
next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS
algorithms based on simulating time evolution. Then we present benchmark
calculations for the three models of fermions, and compare our results against
analytical, mean-field, and variational Monte Carlo results, respectively.
Consistent with previous computations restricted to nearest-neighbor
Hamiltonians, we systematically obtain more accurate (or better converged)
results for gapped phases than for gapless ones.Comment: 10 pages, 11 figures, minor change
Shell Model Monte Carlo Methods
We review quantum Monte Carlo methods for dealing with large shell model
problems. These methods reduce the imaginary-time many-body evolution operator
to a coherent superposition of one-body evolutions in fluctuating one-body
fields; the resultant path integral is evaluated stochastically. We first
discuss the motivation, formalism, and implementation of such Shell Model Monte
Carlo (SMMC) methods. There then follows a sampler of results and insights
obtained from a number of applications. These include the ground state and
thermal properties of {\it pf}-shell nuclei, the thermal and rotational
behavior of rare-earth and -soft nuclei, and the calculation of double
beta-decay matrix elements. Finally, prospects for further progress in such
calculations are discussed
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
A finite state projection algorithm for the stationary solution of the chemical master equation
The chemical master equation (CME) is frequently used in systems biology to
quantify the effects of stochastic fluctuations that arise due to biomolecular
species with low copy numbers. The CME is a system of ordinary differential
equations that describes the evolution of probability density for each
population vector in the state-space of the stochastic reaction dynamics. For
many examples of interest, this state-space is infinite, making it difficult to
obtain exact solutions of the CME. To deal with this problem, the Finite State
Projection (FSP) algorithm was developed by Munsky and Khammash (Jour. Chem.
Phys. 2006), to provide approximate solutions to the CME by truncating the
state-space. The FSP works well for finite time-periods but it cannot be used
for estimating the stationary solutions of CMEs, which are often of interest in
systems biology. The aim of this paper is to develop a version of FSP which we
refer to as the stationary FSP (sFSP) that allows one to obtain accurate
approximations of the stationary solutions of a CME by solving a finite
linear-algebraic system that yields the stationary distribution of a
continuous-time Markov chain over the truncated state-space. We derive bounds
for the approximation error incurred by sFSP and we establish that under
certain stability conditions, these errors can be made arbitrarily small by
appropriately expanding the truncated state-space. We provide several examples
to illustrate our sFSP method and demonstrate its efficiency in estimating the
stationary distributions. In particular, we show that using a quantised tensor
train (QTT) implementation of our sFSP method, problems admitting more than 100
million states can be efficiently solved.Comment: 8 figure
The Dynamics of Group Codes: Dual Abelian Group Codes and Systems
Fundamental results concerning the dynamics of abelian group codes
(behaviors) and their duals are developed. Duals of sequence spaces over
locally compact abelian groups may be defined via Pontryagin duality; dual
group codes are orthogonal subgroups of dual sequence spaces. The dual of a
complete code or system is finite, and the dual of a Laurent code or system is
(anti-)Laurent. If C and C^\perp are dual codes, then the state spaces of C act
as the character groups of the state spaces of C^\perp. The controllability
properties of C are the observability properties of C^\perp. In particular, C
is (strongly) controllable if and only if C^\perp is (strongly) observable, and
the controller memory of C is the observer memory of C^\perp. The controller
granules of C act as the character groups of the observer granules of C^\perp.
Examples of minimal observer-form encoder and syndrome-former constructions are
given. Finally, every observer granule of C is an "end-around" controller
granule of C.Comment: 30 pages, 11 figures. To appear in IEEE Trans. Inform. Theory, 200
A finite state projection algorithm for the stationary solution of the chemical master equation
The chemical master equation (CME) is frequently used in systems biology to
quantify the effects of stochastic fluctuations that arise due to biomolecular
species with low copy numbers. The CME is a system of ordinary differential
equations that describes the evolution of probability density for each
population vector in the state-space of the stochastic reaction dynamics. For
many examples of interest, this state-space is infinite, making it difficult to
obtain exact solutions of the CME. To deal with this problem, the Finite State
Projection (FSP) algorithm was developed by Munsky and Khammash (Jour. Chem.
Phys. 2006), to provide approximate solutions to the CME by truncating the
state-space. The FSP works well for finite time-periods but it cannot be used
for estimating the stationary solutions of CMEs, which are often of interest in
systems biology. The aim of this paper is to develop a version of FSP which we
refer to as the stationary FSP (sFSP) that allows one to obtain accurate
approximations of the stationary solutions of a CME by solving a finite
linear-algebraic system that yields the stationary distribution of a
continuous-time Markov chain over the truncated state-space. We derive bounds
for the approximation error incurred by sFSP and we establish that under
certain stability conditions, these errors can be made arbitrarily small by
appropriately expanding the truncated state-space. We provide several examples
to illustrate our sFSP method and demonstrate its efficiency in estimating the
stationary distributions. In particular, we show that using a quantised tensor
train (QTT) implementation of our sFSP method, problems admitting more than 100
million states can be efficiently solved.Comment: 8 figure
Sensitivity Analysis for Mirror-Stratifiable Convex Functions
This paper provides a set of sensitivity analysis and activity identification
results for a class of convex functions with a strong geometric structure, that
we coined "mirror-stratifiable". These functions are such that there is a
bijection between a primal and a dual stratification of the space into
partitioning sets, called strata. This pairing is crucial to track the strata
that are identifiable by solutions of parametrized optimization problems or by
iterates of optimization algorithms. This class of functions encompasses all
regularizers routinely used in signal and image processing, machine learning,
and statistics. We show that this "mirror-stratifiable" structure enjoys a nice
sensitivity theory, allowing us to study stability of solutions of optimization
problems to small perturbations, as well as activity identification of
first-order proximal splitting-type algorithms. Existing results in the
literature typically assume that, under a non-degeneracy condition, the active
set associated to a minimizer is stable to small perturbations and is
identified in finite time by optimization schemes. In contrast, our results do
not require any non-degeneracy assumption: in consequence, the optimal active
set is not necessarily stable anymore, but we are able to track precisely the
set of identifiable strata.We show that these results have crucial implications
when solving challenging ill-posed inverse problems via regularization, a
typical scenario where the non-degeneracy condition is not fulfilled. Our
theoretical results, illustrated by numerical simulations, allow to
characterize the instability behaviour of the regularized solutions, by
locating the set of all low-dimensional strata that can be potentially
identified by these solutions
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