We study average case tractability of non-homogeneous tensor product problems with the absolute error criterion. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. For general non-homogeneous tensor product problems, we obtain the matching necessary and sufficient conditions for strong polynomial tractability in terms of the one-dimensional eigenvalues. We give some examples to show that strong polynomial tractability is not equivalent to polynomial tractability, and polynomial tractability is not equivalent to quasi-polynomial tractability. But for non-homogeneous tensor product problems with decreasing eigenvalues, we prove that strong polynomial tractability is always equivalent to polynomial tractability, and strong polynomial tractability is even equivalent to quasi-polynomial tractability when the one-dimensional largest eigenvalues are less than one. In particular, we find an example that quasi-polynomial tractability with the absolute error criterion is not equivalent to that with the normalized error criterion even if all the one-dimensional largest eigenvalues are one. Finally we consider a special class of non-homogeneous tensor product problems with improved monotonicity condition of the eigenvalues.

我们使用绝对误差准则研究非齐次张量积问题的平均情况易处理性。我们考虑使用任意线性泛函的有限多次评估的算法。对于一般的非齐次张量积问题，我们根据一维特征值得到强多项式易处理性的匹配充要条件。我们给出一些例子来说明强多项式的易处理性不等同于多项式的易处理性，多项式的易处理性也不等同于拟多项式的易处理性。但是对于特征值递减的非齐次张量积问题，我们证明强多项式易处理性总是等价于多项式易处理性，当一维最大特征值小于一时，强多项式易处理性甚至等同于拟多项式易处理性。特别是，我们发现了一个例子，即即使所有一维最大特征值都是一个，具有绝对误差准则的准多项式易处理性也不等同于具有归一化误差准则的拟多项式易处理性。最后，我们考虑一类特殊的非齐次张量积问题，其具有改进的特征值单调性条件。