5,095 research outputs found
Efficient Uncertainty Quantification with the Polynomial Chaos Method for Stiff Systems
The polynomial chaos method has been widely adopted as a computationally
feasible approach for uncertainty quantification. Most studies to date
have focused on non-stiff systems. When stiff systems are considered,
implicit numerical integration requires the solution of a nonlinear
system of equations at every time step. Using the Galerkin approach, the
size of the system state increases from to , where
is the number of the polynomial chaos basis functions. Solving such systems with full
linear algebra causes the computational cost to increase from to
. The -fold increase can make the computational cost
prohibitive. This paper explores computationally efficient uncertainty
quantification techniques for stiff systems using the Galerkin, collocation and collocation least-squares formulations of polynomial chaos. In the Galerkin approach, we propose a modification in the implicit time stepping process using an approximation of the
Jacobian matrix to reduce the computational cost. The numerical results
show a run time reduction with a small impact on accuracy. In
the stochastic collocation formulation, we propose a least-squares
approach based on collocation at a low-discrepancy set of
points. Numerical experiments illustrate that the collocation
least-squares approach for uncertainty quantification has similar
accuracy with the Galerkin approach, is more efficient, and does not
require any modifications of the original code
Symptoms of complexity in a tourism system
Tourism destinations behave as dynamic evolving complex systems, encompassing
numerous factors and activities which are interdependent and whose
relationships might be highly nonlinear. Traditional research in this field has
looked after a linear approach: variables and relationships are monitored in
order to forecast future outcomes with simplified models and to derive
implications for management organisations. The limitations of this approach
have become apparent in many cases, and several authors claim for a new and
different attitude.
While complex systems ideas are amongst the most promising interdisciplinary
research themes emerged in the last few decades, very little has been done so
far in the field of tourism. This paper presents a brief overview of the
complexity framework as a means to understand structures, characteristics,
relationships, and explores the implications and contributions of the
complexity literature on tourism systems. The objective is to allow the reader
to gain a deeper appreciation of this point of view.Comment: 32 pages, 3 figures, 1 table; accepted in Tourism Analysi
Order out of Randomness : Self-Organization Processes in Astrophysics
Self-organization is a property of dissipative nonlinear processes that are
governed by an internal driver and a positive feedback mechanism, which creates
regular geometric and/or temporal patterns and decreases the entropy, in
contrast to random processes. Here we investigate for the first time a
comprehensive number of 16 self-organization processes that operate in
planetary physics, solar physics, stellar physics, galactic physics, and
cosmology. Self-organizing systems create spontaneous {\sl order out of chaos},
during the evolution from an initially disordered system to an ordered
stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical
resonances, or gyromagnetic resonances. The internal driver can be gravity,
rotation, thermal pressure, or acceleration of nonthermal particles, while the
positive feedback mechanism is often an instability, such as the
magneto-rotational instability, the Rayleigh-B\'enard convection instability,
turbulence, vortex attraction, magnetic reconnection, plasma condensation, or
loss-cone instability. Physical models of astrophysical self-organization
processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra
equation systems.Comment: 61 pages, 38 Figure
Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
There have been significant recent advances in our understanding of the
potential use and limitations of early-warning signs for predicting drastic
changes, so called critical transitions or tipping points, in dynamical
systems. A focus of mathematical modeling and analysis has been on stochastic
ordinary differential equations, where generic statistical early-warning signs
can be identified near bifurcation-induced tipping points. In this paper, we
outline some basic steps to extend this theory to stochastic partial
differential equations with a focus on analytically characterizing basic
scaling laws for linear SPDEs and comparing the results to numerical
simulations of fully nonlinear problems. In particular, we study stochastic
versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a
scaling law of the covariance operator in a regime where linearization is
expected to be a good approximation for the local fluctuations around
deterministic steady states. We compare these results to direct numerical
simulation, and study the influence of noise level, noise color, distance to
bifurcation and domain size on early-warning signs.Comment: Published in Communications in Nonlinear Science and Numerical
Simulation (2014
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