Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor product approximation between the spatial variable and the parameter. In this article, we employ this fact and reverse the multilevel quadrature method via the sparse grid construction by applying differences of quadrature rules to finite element discretizations of increasing resolution. Besides being algorithmically more efficient if the underlying quadrature rules are nested, this way of performing the sparse tensor product approximation enables the easy use of non-nested and even adaptively refined finite element meshes. Especially, we present a rigorous error and regularity analysis of the fully discrete solution, taking into account the effect of polygonal approximations to a curved physical domain and the numerical approximation of the bilinear form. Our results facilitate the construction of efficient multilevel quadrature methods based on deterministic quadrature rules. Numerical results in three spatial dimensions are provided to illustrate the approach.

中文翻译：

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曲线域多边形近似情况下椭圆参数偏微分方程的多级求积

参数算子方程的多级正交方法，例如多级（拟）蒙特卡罗方法，与空间变量和参数之间的稀疏张量积近似密切相关。在本文中，我们利用这一事实，通过将正交规则的差异应用于分辨率增加的有限元离散化，通过稀疏网格构造来反转多级正交方法。如果底层正交规则是嵌套的，那么除了在算法上更有效之外，这种执行稀疏张量积近似的方法还可以轻松使用非嵌套甚至自适应细化的有限元网格。特别是，我们对完全离散的解进行了严格的误差和规律性分析，考虑到多边形近似对弯曲物理域的影响和双线性形式的数值近似。我们的结果有助于构建基于确定性正交规则的高效多级正交方法。提供了三个空间维度的数值结果来说明该方法。