Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse 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) + Sigma(j is an element of N) y(j)a(j)(x), where elements of the parameter vector y = (y(j))(j is an element of N) is an element of U-infinity are independent and identically uniformly distributed on U := [- 1/2, 1/2]. Under the assumption ||Sigma(j is an element of N.) rho(j)|a(j)| ||L-infinity(D) < infinity with some positive sequence (rho(j))(j is an element of N) is an element of l(p)(N) for p is an element of (0, 1] we show that for any y is an element of U-infinity, the elliptic partial differential operator has a countably infinite number of eigenvalues (lambda(j)(y)) (j is an element of N) which can be ordered non-decreasingly. Moreover, the spectral gap lambda(2)(y) - lambda(1)(y) is uniformly positive in U-infinity. From this, we prove the holomorphic extension property of lambda(1)(y) to a complex domain in C-infinity and estimate partial derivatives of lambda(1)(y) with respect to the parameter 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 lambda(1)(y).