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

Harry Yserentant

Abstract

This paper deals with the equation $-Δu+μu=f$ 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 $μ\geq 0$ 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.

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

Abstract

This paper deals with the equation on high-dimensional spaces , where the right-hand side is composed of a separable function with an integrable Fourier transform on a space of a dimension and a linear mapping given by a matrix of full rank and is a constant. For example, the right-hand side can explicitly depend on differences of components of . 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 deviates on the unit sphere the less from its mean value the higher the dimension is, a concentration of measure effect. The higher the dimension , the faster the iteration converges.
Paper Structure (4 sections, 15 theorems, 157 equations, 4 figures, 1 table)

This paper contains 4 sections, 15 theorems, 157 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Solutions as traces of higher-dimensional functions
  3. The angular distribution
  4. The iterative procedure

Key Result

lemma 1

Let $U:\mathbb{R}^n\to\mathbb{R}$ be a function in $\mathcal{B}_0(\mathbb{R}^n)$ and let the functions be integrable. Then the trace function (eq2.3) possesses the partial derivative that, like $u$, is itself uniformly continuous and vanishes at infinity.

Figures (4)

  • Figure 1: The probability distributions assigned to the densities (\ref{['eq3.13']}) for $m=2^k$, $k=1,\ldots,16$, and $n=2m$
  • Figure 2: The frequency distribution of the values $\|T^t\eta\|^2$ for the matrix $T$ associated with the $\mathrm{C}_{60}$-molecule
  • Figure 3: Comparison of the approximation by a rescaled beta distribution and a normal distribution
  • Figure 4: The rescaled function $s\to\phi(s\ln 10)$ approximating $1$ for $h=1$, $k_1=-2$, and $k_2=50$

Theorems & Definitions (31)

  • lemma 1
  • proof
  • theorem 1
  • proof
  • theorem 2
  • theorem 3
  • lemma 2
  • proof
  • theorem 1
  • proof
  • ...and 21 more