Differentiation on spaces of triangulations and optimized triangulations

TL;DR

This paper introduces a smooth Fr"olicher space structure on CW complexes and triangulation spaces, enabling differential methods for optimization, exemplified by improved triangulations for solving PDEs.

Contribution

It presents a novel Fr"olicher space framework on triangulation spaces, facilitating differential techniques for optimization in geometric and PDE contexts.

Findings

Fr"olicher space structure on CW complexes and triangulations

Application to optimized triangulations for PDE solutions

Enabling differential methods in geometric optimization

Abstract

We describe a smooth structure, called Fr\"olicher space, on CW complexes and spaces of triangulations. This structure enables differential methods for e.g. minimization of functionnals. As an application, we exhibit how an optimized triangulation can be obtained in order to solve a standard PDE.