Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: acc-phys