Analyticity of Parametric and Stochastic Elliptic Eigenvalue Problems with an Application to Quasi-Monte Carlo Methods
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In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operator with random coefficient and analyze the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a_0(x)+∑_j∈ℕy_ja_j(x), where elements of the parameter vector y=(y_j)_j∈ℕ∈ U^∞ are independent and identically uniformly distributed on U:=[-1/2,1/2]. Under the assumption ∑_j∈ℕρ_j|a_j|_L_∞(D) <∞ with some positive sequence (ρ_j)_j∈ℕ∈ℓ_p(ℕ) for p∈ (0,1] we show that for any y∈ U^∞, the elliptic partial differential operator has a countably infinite number of eigenvalues (λ_j(y))_j∈ℕ which can be ordered non-decreasingly. Moreover, the spectral gap λ_2(y)-λ_1(y) is uniformly positive in U^∞. From this, we prove the holomorphic extension property of λ_1(y) to a complex domain in ℂ^∞ and estimate mixed derivatives of λ_1(y) with respect to the parameters y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ_1(y). In this case, the computational cost of fast component-by-component algorithm for generating quasi-Monte Carlo N-points scales linearly in terms of integration dimension.
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