Analyticity of Parametric and Stochastic Elliptic Eigenvalue Problems with an Application to Quasi-Monte Carlo Methods

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 $a_0(\boldsymbol{x})+\sum_{j\in \mathbb{N}}y_ja_j(\boldsymbol{x})$, where elements of the parameter vector $\boldsymbol{y}=(y_j)_{j\in \mathbb{N}}\in U^\infty$ are independent and identically uniformly distributed on $U:=[-\frac{1}{2},\frac{1}{2}]$. Under the assumption $ \|\sum_{j\in \mathbb{N}}\rho_j|a_j|\|_{L_\infty(D)} <\infty$ with some positive sequence $(\rho_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N})$ for $p\in (0,1]$ we show that for any $\boldsymbol{y}\in U^\infty$, the elliptic partial differential operator has a countably infinite number of eigenvalues $(\lambda_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be ordered non-decreasingly. Moreover, the spectral gap $\lambda_2(\boldsymbol{y})-\lambda_1(\boldsymbol{y})$ is uniformly positive in $U^\infty$. From this, we prove the holomorphic extension property of $\lambda_1(\boldsymbol{y})$ to a complex domain in $\mathbb{C}^\infty$ and estimate mixed derivatives of $\lambda_1(\boldsymbol{y})$ with respect to the parameters $\boldsymbol{y}$ 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 $\lambda_1(\boldsymbol{y})$. 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.