287 research outputs found
Approximation of the effective conductivity of ergodic media by periodization
This paper is concerned with the approximation of the effective conductivity
associated to an elliptic operator where for , , is a bounded elliptic
random symmetric matrix and takes value in an ergodic
probability space . Writing the periodization of
on the torus of dimension and side we prove that
for -almost all We extend this result to
non-symmetric operators corresponding to
diffusions in ergodic divergence free flows ( is elliptic
symmetric matrix and an ergodic skew-symmetric matrix); and to
discrete operators corresponding to random walks on with ergodic jump
rates. The core of our result is to show that the ergodic Weyl decomposition
associated to \L^2(X,\mu) can almost surely be approximated by periodic Weyl
decompositions with increasing periods, implying that semi-continuous
variational formulae associated to \L^2(X,\mu) can almost surely be
approximated by variational formulae minimizing on periodic potential and
solenoidal functions.Comment: published version. The approximation result is given for general non
linear semi-continuous variational formula
Averaging versus Chaos in Turbulent Transport?
In this paper we analyze the transport of passive tracers by deterministic
stationary incompressible flows which can be decomposed over an infinite number
of spatial scales without separation between them. It appears that a low order
dynamical system related to local Peclet numbers can be extracted from these
flows and it controls their transport properties. Its analysis shows that these
flows are strongly self-averaging and super-diffusive: the delay for
any finite number of passive tracers initially close to separate till a
distance is almost surely anomalously fast (, with
). This strong self-averaging property is such that the dissipative
power of the flow compensates its convective power at every scale. However as
the circulation increase in the eddies the transport behavior of the flow may
(discontinuously) bifurcate and become ruled by deterministic chaos: the
self-averaging property collapses and advection dominates dissipation. When the
flow is anisotropic a new formula describing turbulent conductivity is
identified.Comment: Presented at Oberwolfach (October 2002), CIRM (March 2003), Lisbonne
(XIV international congress on mathematical physics: July 2003). Submitted on
October 2002, to appear in Communications in Mathematical Physics. 45 pages,
7 figure
Bayesian Numerical Homogenization
Numerical homogenization, i.e. the finite-dimensional approximation of
solution spaces of PDEs with arbitrary rough coefficients, requires the
identification of accurate basis elements. These basis elements are oftentimes
found after a laborious process of scientific investigation and plain
guesswork. Can this identification problem be facilitated? Is there a general
recipe/decision framework for guiding the design of basis elements? We suggest
that the answer to the above questions could be positive based on the
reformulation of numerical homogenization as a Bayesian Inference problem in
which a given PDE with rough coefficients (or multi-scale operator) is excited
with noise (random right hand side/source term) and one tries to estimate the
value of the solution at a given point based on a finite number of
observations. We apply this reformulation to the identification of bases for
the numerical homogenization of arbitrary integro-differential equations and
show that these bases have optimal recovery properties. In particular we show
how Rough Polyharmonic Splines can be re-discovered as the optimal solution of
a Gaussian filtering problem.Comment: 22 pages. To appear in SIAM Multiscale Modeling and Simulatio
Homogenization of Parabolic Equations with a Continuum of Space and Time Scales
This paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available. Namely, we consider divergence form linear parabolic operators in with -coefficients. It appears that the inverse operator maps the unit ball of into a space of functions which at small (time and space) scales are close in norm to a functional space of dimension . It follows that once one has solved these equations at least times it is possible to homogenize them both in space and in time, reducing the number of operation counts necessary to obtain further solutions. In practice we show under a Cordes-type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in (instead of with Euclidean coordinates). If the medium is time-independent, then it is sufficient to solve times the associated elliptic equation in order to homogenize the parabolic equation
Metric based up-scaling
We consider divergence form elliptic operators in dimension with
coefficients. Although solutions of these operators are only
H\"{o}lder continuous, we show that they are differentiable ()
with respect to harmonic coordinates. It follows that numerical homogenization
can be extended to situations where the medium has no ergodicity at small
scales and is characterized by a continuum of scales by transferring a new
metric in addition to traditional averaged (homogenized) quantities from
subgrid scales into computational scales and error bounds can be given. This
numerical homogenization method can also be used as a compression tool for
differential operators.Comment: Final version. Accepted for publication in Communications on Pure and
Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April),
Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution
figures are available at http://www.acm.caltech.edu/~owhadi
Qualitative Robustness in Bayesian Inference
The practical implementation of Bayesian inference requires numerical
approximation when closed-form expressions are not available. What types of
accuracy (convergence) of the numerical approximations guarantee robustness and
what types do not? In particular, is the recursive application of Bayes' rule
robust when subsequent data or posteriors are approximated? When the prior is
the push forward of a distribution by the map induced by the solution of a PDE,
in which norm should that solution be approximated? Motivated by such
questions, we investigate the sensitivity of the distribution of posterior
distributions (i.e. posterior distribution-valued random variables, randomized
through the data) with respect to perturbations of the prior and data
generating distributions in the limit when the number of data points grows
towards infinity
Numerical Homogenization of the Acoustic Wave Equations with a Continuum of Scales
In this paper, we consider numerical homogenization of acoustic wave
equations with heterogeneous coefficients, namely, when the bulk modulus and
the density of the medium are only bounded. We show that under a Cordes type
condition the second order derivatives of the solution with respect to harmonic
coordinates are (instead with respect to Euclidean coordinates)
and the solution itself is in (instead of
with respect to Euclidean coordinates). Then, we
propose an implicit time stepping method to solve the resulted linear system on
coarse spatial scales, and present error estimates of the method. It follows
that by pre-computing the associated harmonic coordinates, it is possible to
numerically homogenize the wave equation without assumptions of scale
separation or ergodicity.Comment: 27 pages, 4 figures, Submitte
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