Can I read An iterative method for the solution of Laplace-like equations in high and very high space dimensions on EtoBox?

An iterative method for the solution of Laplace-like equations in high and very high space dimensions by Harry Yserentant is a Engineering article available to read on EtoBox.

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

## Abstract This paper deals with the equation $$-\varDelta u+\mu u=f$$ - Δ u + μ u = f on high-dimensional spaces $${\mathbb {R}}^m$$ R m , where the right-hand side $$f(x)=F(Tx)$$ f ( x ) = F ( T x ) is composed of a separable function __F__ with an integrable Fourier transform on a space of a dimension $$n>m$$ n > m and a linear mapping given by a matrix __T__ of full rank and $$\mu \ge 0$$ μ ≥ 0 is a constant. For example, the right-hand side can explicitly depend on differences $$x\_i-x\_j$$ x i - x j of components of __x__. Following our publication (Yserentant in Numer Math 146:219–238, 2020), 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 $$\Vert T^ty\Vert ^2$$ ‖ T t y ‖ 2 deviates on the unit sphere $$\Vert y\Vert =1$$ ‖ 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.

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

Author

Harry Yserentant

Publisher

Springer Science and Business Media LLC

Published

2024

Language

EN

Field

Engineering (Physical Sciences)