This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$, where the right-hand side $f(x)=F(Tx)$ is composed of a separable function $F$ with an integrable Fourier transform on a space of a dimension $n>m$ and a linear mapping given by a matrix $T$ of full rank and $\mu\geq 0$ is a constant. For example, the right-hand side can explicitly depend on differences $x_i-x_j$ of components of $x$. Following our publication [Numer. Math. (2020) 146:219--238], we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression $\|T^ty\|^2$ deviates on the unit sphere $\|y\|=1$ the less from its mean value the higher the dimension $m$ is, a concentration of measure effect. The higher the dimension $m$, the faster the iteration converges.

翻译：本文研究高维空间 $\mathbb{R}^m$ 上的方程 $-\Delta u+\mu u=f$，其中右端项 $f(x)=F(Tx)$ 由可分离函数 $F$（其傅里叶变换在 $n>m$ 维空间上可积）和满秩矩阵 $T$ 所给出的线性映射组成，且 $\mu\geq 0$ 为常数。例如，右端项可显式依赖于分量之差 $x_i-x_j$。基于我们已发表的论文 [Numer. Math. (2020) 146:219--238]，我们证明此类方程的解可展开为相同结构函数的和，并在此框架下发展出一种同样简单且快速的迭代计算方法。该方法基于如下观测：在几乎所有情形下，对于大规模问题类，表达式 $\|T^ty\|^2$ 在单位球面 $\|y\|=1$ 上的偏差随维度 $m$ 增大而趋近于其均值——这一现象源于测度集中效应。维度 $m$ 越高，迭代收敛速度越快。