高阶拟蒙特卡洛积分用于全纯参数算子方程
Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations
AI总结：
研究高阶拟蒙特卡洛积分在处理具有分布不确定参数的全纯参数算子方程中的收敛性，提出基于SPOD权重的构造方法，扩展了线性算子家族的正则性结果。
摘要：
我们分析了高阶拟蒙特卡洛（QMC）求积在解函数到可数参数、非线性算子方程中的收敛性，这些方程具有分布在可分巴纳赫空间X中的不确定参数，X具有无条件Schauder基。此类方程出现在具有随机场输入的数值不确定性量化中。无条件基使随机输入和正向问题的解成为可数参数、确定性的。我们证明这些参数解属于具有特定QMC求积权重结构的加权Bochner空间，权重结构包括问题依赖的有限维（可能较大）部分使用乘积权重，超过此维度则使用SPOD权重来描述解的正则性。本文的正则性结果扩展了[1]中对于仿射参数线性算子族的正则性结果；特别地，这些结果推导出在这些问题类中高效构造（序列）QMC求积方法的构造方法。我们提出了一种快速组件-组件构造（CBC）的混合版本，用于构造某一类型的高阶数字网。
英文摘要：
We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space $X$ admitting an unconditional Schauder basis. Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of $X$ render the random inputs and the solutions of the forward problem countably parametric, deterministic. We show that these parametric solutions belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension, product weights can be used, and beyond this dimension, weighted spaces with so-called SPOD weights recently introduced in [F.Y.~Kuo, Ch.~Schwab, I.H.~Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal., 50, 3351--3374, 2012.] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC (Petrov-)Galerkin discretization for parametric operator equations. SIAM J. Numer. Anal., 52, 2676 -- 2702, 2014.] established for affine parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component (CBC for short) construction of a certain type of higher order digital net.
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