1,989 research outputs found
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
We find the weak rate of convergence of the spatially semidiscrete finite
element approximation of the nonlinear stochastic heat equation. Both
multiplicative and additive noise is considered under different assumptions.
This extends an earlier result of Debussche in which time discretization is
considered for the stochastic heat equation perturbed by white noise. It is
known that this equation has a solution only in one space dimension. In order
to obtain results for higher dimensions, colored noise is considered here,
besides white noise in one dimension. Integration by parts in the Malliavin
sense is used in the proof. The rate of weak convergence is, as expected,
essentially twice the rate of strong convergence.Comment: 19 page
An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations
We establish an optimal, linear rate of convergence for the stochastic
homogenization of discrete linear elliptic equations. We consider the model
problem of independent and identically distributed coefficients on a
discretized unit torus. We show that the difference between the solution to the
random problem on the discretized torus and the first two terms of the
two-scale asymptotic expansion has the same scaling as in the periodic case. In
particular the -norm in probability of the \mbox{-norm} in space of
this error scales like , where is the discretization
parameter of the unit torus. The proof makes extensive use of previous results
by the authors, and of recent annealed estimates on the Green's function by
Marahrens and the third author
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