We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for , , , and we are given and we seek to approximate with error measured in the -norm. Information is standard, that is, function values of . Our results extend previous work of Heinrich and Sindambiwe (1999) for and Wiegand (2006) for . Wiegand's analysis was carried out under the assumption that is continuously embedded in (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. 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.

中文翻译：

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参数积分的随机复杂性和自适应的作用 II。索博列夫空间

我们研究了依赖于参数的积分随机计算的复杂性，以及来自索博列夫空间的被积函数。也就是说，对于 、 、 、 ，我们已知并且我们寻求用 -范数中测量的误差来近似。信息是标准的，即 的函数值。我们的结果扩展了 Heinrich 和 Sindambiwe (1999) 以及 Wiegand (2006) 之前的工作。韦根的分析是在连续嵌入（嵌入条件）的假设下进行的。我们还研究了嵌入条件不成立的情况。为此，开发了一种新成分——随机离散技术。在第一部分中，解决了基于信息的复杂性的基本问题，即随机设置中线性问题的适应能力。这里解决了这个问题的另一个方面。