Comparison of topologies

Summary

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used. Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in τ2: :\tau_1 \subseteq \tau_2. That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1. If additionally :\tau_1 \neq \tau_2 we say τ1 is strict

Official source

https://en.wikipedia.org/wiki/Comparison_of_topologies

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related publications

Related people

Related units

Related concepts (20)

Related concepts

Related courses (5)

Related courses

Related lectures (7)

Related lectures

A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible inviscid aerodynamics

Pénélope Leyland, Fabio Nobile, Michele Pisaroni

In this work we apply the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed in Collier et al. (2014) to efficiently propagate operating and geometric uncertainties in inviscid compressible aerodynamics numerical simulations. The key idea of MLMC is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids (levels). The expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy. By this way, most of the computational effort is transported from the finest level (as in a standard Monte Carlo approach) to the coarsest one. The continuation algorithm (C-MLMC) is a robust and self-tuning version that estimates on the fly the optimal number of level and realizations per level. In this work we describe in detail how C-MLMC can be adapted to perform uncertainty quantification analysis in compressible aerodynamics and we apply it to the relevant test cases of a quasi 1D convergent–divergent Laval nozzle and the 2D transonic RAE-2822 airfoil.

Elsevier2017

, ,

In this work we apply the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed by [Collier et al, BIT 2014] to efficiently propagate operational and geometrical uncertainties in compressible aerodynamics numerical simulations. The key idea of MLMC is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids (levels). The expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy. By this way, most of the computational effort is transported from the finest level (as in a standard Monte Carlo approach) to the coarsest one. In the continuation algorithm (C-MLMC) the parameters that control the number of levels and realizations per level are computed on the y to further reduce the overall computational cost. The C-MLMC is applied to the quasi 1D convergent-divergent Laval nozzle and the 2D transonic RAE-2822 airfoil.

MATHICSE2016