We study $\epsilon$-approximation of the solution of the $d$-variateVolterra problem of the second kind, the Volterra operator having a convolution kernel. The Volterra operator is defined on an arbitrary normed function space $F_d$ that is continuously embedded in the space of square integrable functions defined on the unit $d$-cube.

Admissible information is given by continuous linear functionals on $F_d$. These functionals might be arbitrarily chosen; alternatively, we may allow only function values. (In the latter case, we can only consider spaces $F_d$ for which function values are well-defined continuous linear functionals.) The error and cost are measured in the worst case or randomized setting in the $L_2$-norm.

Our first result is that lower and upper bounds on the information complexity of the Volterra problem and of $d$-variate $L_2$-approximation differ by at most a constant factor. Using this result, we then show that necessary and sufficient conditions characterizing a given kind of tractability for multivariate approximation also characterize that same kind of tractability for the Volterra problem. We consider different kinds of algebraic tractabilities (in which we compare the information complexity to $d$ and ${\epsilon }^{-1}$) and exponential tractability (in which we compare the information complexity to $d$ and $1+ln{\epsilon }^{-1}$).

However, our comparison of Volterra to approximation falls short in one respect. Multivariate approximation is linear; for many spaces (e.g., Hilbert spaces), the combinatory cost of multivariate approximation is roughly the same as its information complexity. But since the Volterra problem is nonlinear, it is unclear what the combinatory cost will be for the Volterra problem. This means that we do not know the extent to which the combinatory cost will exceed the information complexity. We partially address this issue by seeing whether the Picard iteration can give us an approximation without too great a penalty when we include the combinatory cost; this penalty is measured by the normalized combinatory cost, defined as the ratio of the combinatory cost to the cost of one admissible operation.

In particular, suppose that we agree to compute an $\epsilon$-approximation to the Volterra problem by (randomized) Monte Carlo. Suppose further that the convolution kernels are uniformly bounded in the $L_{\infty }$-norm. We then obtain an upper bound on the normalized combinatory cost of the Monte Carlo algorithm in the randomized setting. This upper bound is larger than the information complexity by roughly a factor of ${\epsilon }^{-2}$. This factor does not change positive results for algebraic tractability, but it does affect some of the positive results for exponential tractability.

We also describe a deterministic algorithm that implements the Picard iteration. Assume that the approximation problem can be solved by linear algorithms that lie in a space of dimension proportional to the information complexity. We then find that the combinatory cost of the Picard iteration in the worst case setting is at most a power of the information complexity of $d$-variate approximation. This power is a constant if $d^{-1}ln{\epsilon }^{-1}$ is uniformly bounded. However, if $d^{-1}ln{\epsilon }^{-1}$ goes to infinity, then this power is of order $(d^{-1}ln{\epsilon }^{-1})∕ln(d^{-1}ln{\epsilon }^{-1})$. Hence if $d$ is large relative to $ln{\epsilon }^{-1}$, this result is quite positive. On the other hand, for general $d$ and $\epsilon$, only some kinds of algebraic and exponential tractabilities hold when the normalized combinatory cost is included.

我们学习 $\epsilon$-解的近似 $d$-variateVolterra第二类问题，Volterra运算符具有卷积核。Volterra运算符在任意赋范函数空间上定义$F_d$ 连续嵌入在单元上定义的方形可积函数的空间中 $d$-立方体。

允许的信息由连续线性函数给出 $F_d$。这些功能可以任意选择。或者，我们可能只允许函数值。（在后一种情况下，我们只能考虑空格 $F_d$ 误差和成本是在最坏的情况下或随机设定的情况下测得的。 ${\mathrm{大号}}_{\mathrm{2个}}$-规范。

我们的第一个结果是Volterra问题和 $d$-变量 ${\mathrm{大号}}_{\mathrm{2个}}$-近似值最多相差一个常数。使用该结果，我们然后证明了表征给定类型的多元逼近性的必要条件和充分条件也表征了针对Volterra问题的相同类型的可算性。我们考虑了不同种类的代数可扩展性（在其中我们将信息复杂度与$d$ 和 ${\epsilon }^{-\mathrm{1个}}$）和指数可扩展性（我们将信息复杂度与 $d$ 和 $\mathrm{1个}+ln{\epsilon }^{-\mathrm{1个}}$）。

但是，我们在Volterra和近似值之间的比较在一个方面不足。多元近似是线性的；对于许多空间（例如希尔伯特空间），多元逼近的组合成本与其信息复杂度大致相同。但是由于Volterra问题是非线性的，因此尚不清楚Volterra问题的组合成本是多少。这意味着我们不知道组合成本将超出信息复杂性的程度。我们通过考虑Picard迭代在包含组合成本时是否可以给我们一个近似值而不会带来太大的损失来部分解决这个问题。该损失由归一化组合成本衡量，该组合成本定义为组合成本与一项可允许操作的成本之比。

特别是，假设我们同意计算 $\epsilon$-（随机）蒙特卡洛对Volterra问题的逼近。进一步假设卷积核均匀地约束在${\mathrm{大号}}_{\infty }$-规范。然后，我们在随机设置中获得蒙特卡洛算法的标准化组合成本的上限。这个上限比信息复杂度大大约一个因素${\epsilon }^{-\mathrm{2个}}$。此因子不会改变代数可延展性的阳性结果，但会影响一些指数可延展性的阳性结果。

我们还描述了一种实现Picard迭代的确定性算法。假设可以通过线性算法解决近似问题，该线性算法位于与信息复杂度成比例的维空间中。然后，我们发现在最坏的情况下，Picard迭代的组合成本最多是信息复杂度的幂。$d$-变量近似。如果$d^{-\mathrm{1个}}ln{\epsilon }^{-\mathrm{1个}}$是一致有界的。但是，如果$d^{-\mathrm{1个}}ln{\epsilon }^{-\mathrm{1个}}$ 达到无穷大，那么这个能力是有序的 $（d^{-\mathrm{1个}}ln{\epsilon }^{-\mathrm{1个}}）∕ln（d^{-\mathrm{1个}}ln{\epsilon }^{-\mathrm{1个}}）$。因此，如果$d$ 相对于 $ln{\epsilon }^{-\mathrm{1个}}$，这个结果是相当积极的。另一方面，对于一般$d$ 和 $\epsilon$，当包括归一化的组合成本时，只有某些代数和指数的可延性成立。