We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for $r,d_1,d_2\in{\mathbb N}$, $1\le p,q\le \infty$, $D_1= [0,1]^{d_1}$, and $D_2= [0,1]^{d_2}$ we are given $f\in W_p^r(D_1\times D_2)$ and we seek to approximate $$ Sf=\int_{D_2}f(s,t)dt\quad (s\in D_1), $$ with error measured in the $L_q(D_1)$-norm. Our results extend previous work of Heinrich and Sindambiwe (J.\ Complexity, 15 (1999), 317--341) for $p=q=\infty$ and Wiegand (Shaker Verlag, 2006) for $1\le p=q<\infty$. Wiegand's analysis was carried out under the assumption that $W_p^r(D_1\times D_2)$ is continuously embedded in $C(D_1\times D_2)$ (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed -- a stochastic discretization technique. The paper is based on Part I, where vector valued mean computation -- the finite-dimensional counterpart of parametric integration -- was studied. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.

翻译：我们研究了在Sobolev空间中，依赖参数的积分的随机化计算复杂度。具体而言，对于 $r,d_1,d_2\in{\mathbb N}$，$1\le p,q\le \infty$，$D_1= [0,1]^{d_1}$ 和 $D_2= [0,1]^{d_2}$，给定 $f\in W_p^r(D_1\times D_2)$，我们寻求逼近 $$ Sf=\int_{D_2}f(s,t)dt\quad (s\in D_1), $$ 并以 $L_q(D_1)$-范数度量误差。我们的结果扩展了 Heinrich 和 Sindambiwe (J.\ Complexity, 15 (1999), 317--341) 关于 $p=q=\infty$ 的情形以及 Wiegand (Shaker Verlag, 2006) 关于 $1\le p=q<\infty$ 的情形的工作。Wiegand 的分析基于 $W_p^r(D_1\times D_2)$ 连续嵌入到 $C(D_1\times D_2)$（嵌入条件）的假设。我们还研究了嵌入条件不成立的情况。为此，我们发展了一个新的工具——随机离散化技术。本文基于第一部分的研究，其中研究了向量值均值计算——参数化积分的有限维对应。在第一部分中，解决了信息复杂度中关于线性问题在随机化设置下自适应能力的一个基本问题。本文则解决了该问题的另一个方面。