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 $x$. 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 $f(x)=F(Tx)$ 由定义在维数 $n>m$ 的空间上、具有可积傅里叶变换的可分离函数 $F$，以及由满秩矩阵 $T$ 给出的线性映射构成，且 $\mu\geq 0$ 为常数。例如，右端项可显式依赖于 $x$ 的分量之差 $x_i-x_j$。我们证明了该方程的解可展开为具有相同结构函数的和，并在此框架下发展了一种同样简洁且快速的迭代计算方法。该方法基于以下观察：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 $\|y\|=1$ 上偏离其均值的程度随维度 $m$ 的增大而减小，这是一种测度集中效应。维度 $m$ 越高，迭代收敛速度越快。