355 research outputs found
Edgeworth expansions for slow-fast systems with finite time scale separation
We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation
Estimating eddy diffusivities from noisy Lagrangian observations
The problem of estimating the eddy diffusivity from Lagrangian observations
in the presence of measurement error is studied in this paper. We consider a
class of incompressible velocity fields for which is can be rigorously proved
that the small scale dynamics can be parameterised in terms of an eddy
diffusivity tensor. We show, by means of analysis and numerical experiments,
that subsampling of the data is necessary for the accurate estimation of the
eddy diffusivity. The optimal sampling rate depends on the detailed properties
of the velocity field. Furthermore, we show that averaging over the data only
marginally reduces the bias of the estimator due to the multiscale structure of
the problem, but that it does significantly reduce the effect of observation
error
Multiscale Modeling, Homogenization and Nonlocal Effects: Mathematical and Computational Issues
In this work, we review the connection between the subjects of homogenization
and nonlocal modeling and discuss the relevant computational issues. By further
exploring this connection, we hope to promote the cross fertilization of ideas
from the different research fronts. We illustrate how homogenization may help
characterizing the nature and the form of nonlocal interactions hypothesized in
nonlocal models. We also offer some perspective on how studies of nonlocality
may help the development of more effective numerical methods for
homogenization
OBMeshfree: An optimization-based meshfree solver for nonlocal diffusion and peridynamics models
We present OBMeshfree, an Optimization-Based Meshfree solver for compactly
supported nonlocal integro-differential equations (IDEs) that can describe
material heterogeneity and brittle fractures. OBMeshfree is developed based on
a quadrature rule calculated via an equality constrained least square problem
to reproduce exact integrals for polynomials. As such, a meshfree
discretization method is obtained, whose solution possesses the asymptotically
compatible convergence to the corresponding local solution. Moreover, when
fracture occurs, this meshfree formulation automatically provides a sharp
representation of the fracture surface by breaking bonds, avoiding the loss of
mass. As numerical examples, we consider the problem of modeling both
homogeneous and heterogeneous materials with nonlocal diffusion and
peridynamics models. Convergences to the analytical nonlocal solution and to
the local theory are demonstrated. Finally, we verify the applicability of the
approach to realistic problems by reproducing high-velocity impact results from
the Kalthoff-Winkler experiments. Discussions on possible immediate extensions
of the code to other nonlocal diffusion and peridynamics problems are provided.
OBMeshfree is freely available on GitHub.Comment: For associated code, see
https://github.com/youhq34/meshfree_quadrature_nonloca
An Algorithm for Pattern Discovery in Time Series
We present a new algorithm for discovering patterns in time series and other
sequential data. We exhibit a reliable procedure for building the minimal set
of hidden, Markovian states that is statistically capable of producing the
behavior exhibited in the data -- the underlying process's causal states.
Unlike conventional methods for fitting hidden Markov models (HMMs) to data,
our algorithm makes no assumptions about the process's causal architecture (the
number of hidden states and their transition structure), but rather infers it
from the data. It starts with assumptions of minimal structure and introduces
complexity only when the data demand it. Moreover, the causal states it infers
have important predictive optimality properties that conventional HMM states
lack. We introduce the algorithm, review the theory behind it, prove its
asymptotic reliability, use large deviation theory to estimate its rate of
convergence, and compare it to other algorithms which also construct HMMs from
data. We also illustrate its behavior on an example process, and report
selected numerical results from an implementation.Comment: 26 pages, 5 figures; 5 tables;
http://www.santafe.edu/projects/CompMech Added discussion of algorithm
parameters; improved treatment of convergence and time complexity; added
comparison to older method
Reactive Flow and Transport Through Complex Systems
The meeting focused on mathematical aspects of reactive flow, diffusion and transport through complex systems. The research interest of the participants varied from physical modeling using PDEs, mathematical modeling using upscaling and homogenization, numerical analysis of PDEs describing reactive transport, PDEs from fluid mechanics, computational methods for random media and computational multiscale methods
Applications of Asymptotic Analysis
This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and on other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods
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