For more information about this meeting, contact Leonid Berlyand.
|Title:||A quantitative theory of stochastic homogenization|
|Seminar:||Marker Lecture Series|
|Speaker:||Dr. Felix Otto, Max-Planck-Institute for Mathematics|
|If one is interested in the conductivity of a composite, say,
one has to solve an elliptic equation with coefficients that vary on a length scale
much smaller than the characteristic scale of the domain.
We are interested in a situation where the coefficients are characterized
in statistical terms: Their statistics are assumed to
be translation invariant and to decorrelate over large distances.
As is known by qualitative theory, the solution operator
behaves -- on large scales -- like the solution operator of
an elliptic problem with homogeneous, deterministic coefficients,
a huge reduction in complexity!
Theory provides a formula for the homogenized coefficients,
based on the construction of a "corrector", which defines
harmonic coordinates. In practise, this formula has to be
approximated by a "Representative Volume Element", leading
to a random and a systematic error.
We present optimal estimates of both.
We are also interested in other quantitative aspects:
How close is the actual solution to the homogenized one
--- we give an optimal answer, and point out the connections with
elliptic regularity theory (input from Nash's theory,
a new outlook on De Giorgi's theory).
We are also interested in the quantitative ergodicity properties
for the process usually called "the environment as seen from
the random walker". We give an optimal estimate that relies on a link
with (the Spectral Gap for) another stochastic process on the coefficient
fields, namely heat-bath Glauber dynamics. This connection between
statistical mechanics and stochastic homogenization has previously
been used in opposite direction (i. e. with qualitative stochastic homogenization
as an input).
This is joint work with A. Gloria, S. Neukamm, and D. Marahrens.|
Room Reservation Information
|Date:||10 / 22 / 2013|
|Time:||03:35pm - 04:35pm|