An iterative method for the solution of Laplace-like equations in high and very high space dimensions

Abstract

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 $\mu \ge 0$ 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 $\Vert T^{t}y{\Vert }^2$ deviates on the unit sphere $\Vert y\Vert =1$ 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.

arXiv commentThis version is a largely rewritten version of the paper that has been published in Numerische Mathematik. The central new result is Theorem 3.1 and, in its wake, the representation 3.18 of the probability distribution. Lemma 3.5 expresses the variance directly in terms of the vertex degrees. A new reference regarding the approximation of $1/r$ has been added

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