论文标题
参数椭圆特征值问题的分析性和应用于准蒙特卡的应用
Analyticity of Parametric Elliptic Eigenvalue Problems and Applications to Quasi-Monte Carlo Methods
论文作者
论文摘要
在本文中，我们研究了带有随机系数的线性椭圆形偏差操作员最左侧特征值的分析性，并分析了准蒙特卡利方法的收敛速率，以近似此数量的预期。假定随机系数由仿射扩展表示$ a_0（\ boldsymbol {x}）+\ sum_ {j \ in \ mathbb {n}} y_ja_j（\ boldsymbol {x}）$ \ Mathbb {n}} \在u^\ infty $中是独立的，并且在$ u：= [ - \ frac {1} {2} {2}，\ frac {1} {2} {2}] $上均匀分布。在假设下$ \ | \ sum_ { $ p \在（0,1] $中，我们表明，对于任何$ \ boldsymbol {y} \ in U^\ infty $，椭圆形的部分差分运算符具有无限数量的eigenvalues $（λ_j（λ_j（λ_j（\ boldsymbol {y}））此外，频谱差距$λ_2（\ boldsymbol {y}） - λ_1（\ boldsymbol {y}）$在$ u^\ indty $中均匀地为正，我们从中证明了$λ_1（\ boldsymbol {y Infty complect $ { and estimate mixed derivatives of $λ_1(\boldsymbol{y})$ 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 $λ_1(\boldsymbol{y})$.
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}}ρ_j|a_j|\|_{L_\infty(D)} <\infty$ with some positive sequence $(ρ_j)_{j\in \mathbb{N}}\in \ell_p(\mathbb{N})$ 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 $(λ_j(\boldsymbol{y}))_{j\in \mathbb{N}}$ which can be ordered non-decreasingly. Moreover, the spectral gap $λ_2(\boldsymbol{y})-λ_1(\boldsymbol{y})$ is uniformly positive in $U^\infty$. From this, we prove the holomorphic extension property of $λ_1(\boldsymbol{y})$ to a complex domain in $\mathbb{C}^\infty$ and estimate mixed derivatives of $λ_1(\boldsymbol{y})$ 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 $λ_1(\boldsymbol{y})$.
