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First order second moment analysis for stochastic interface problems based on low-rank approximation

Helmut Harbrecht, Jingzhi Li (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper, we propose a numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface...

FKN Theorem on the biased cube

Piotr Nayar (2014)

Colloquium Mathematicae

We consider Boolean functions defined on the discrete cube - γ , γ - 1 equipped with a product probability measure μ n , where μ = β δ - γ + α δ γ - 1 and γ = √(α/β). This normalization ensures that the coordinate functions ( x i ) i = 1 , . . . , n are orthonormal in L ( - γ , γ - 1 , μ n ) . We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover, in the symmetric...

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