Applied Math And Analysis Seminar
Monday, March 16, 2009, 4:30pm, 119 Physics
Guillaume Bal (Columbia University, Dept. of Applied Physics & Applied Mathematics)
Some convergence results in equations with random coefficients
Abstract:- The theory of homogenization for equations with random
coefficients is now quite well-developed. What is less studied is the theory
for the correctors to homogenization, which asymptotically characterize the
randomness in the solution of the equation and as such are important to
quantify in many areas of applied sciences. I will present recent results in
the theory of correctors for elliptic and parabolic problems and briefly
mention how such correctors may be used to improve reconstructions in
inverse problems. Homogenized (deterministic effective medium) solutions are
not the only possible limits for solutions of equations with highly
oscillatory random coefficients as the correlation length in the medium
converges to zero. When fluctuations are sufficiently large, the limit may
take the form of a stochastic equation and stochastic partial differential
equations (SPDE) are routinely used to model small scale random forcing. In
the very specific setting of a parabolic equation with large, Gaussian,
random potential, I will show the following result: in low spatial
dimensions, the solution to the parabolic equation indeed converges to the
solution of a SPDE, which however needs to be written in a (somewhat
unconventional) Stratonovich form; in high spatial dimension, the solution
to the parabolic equation converges to a homogenized (hence deterministic)
equation and randomness appears as a central limit-type corrector. One of
the possible corollaries for this result is that SPDE models may indeed be
appropriate in low spatial dimensions but not necessarily in higher spatial
dimensions. [video]
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