Consistency of Dirichlet Partitions
Braxton Osting, Todd Harry Reeb

TL;DR
This paper proves that solutions to discrete Dirichlet partition problems on graphs converge to solutions of the continuous problem as the number of sample points increases, establishing a connection between discrete and continuum partitions.
Contribution
It extends previous results to show convergence of discrete Dirichlet partitions on geometric graphs to continuum partitions, with statistical consistency when sampling uniformly.
Findings
Discrete Dirichlet energies $ ext{Gamma}$-converge to continuum energies.
Minimizers of discrete problems converge to continuum minimizers.
Dirichlet partitions on graphs converge to partitions of the sampled space.
Abstract
A Dirichlet -partition of a domain is a collection of pairwise disjoint open subsets such that the sum of their first Laplace-Dirichlet eigenvalues is minimal. A discrete version of Dirichlet partitions has been posed on graphs with applications in data analysis. Both versions admit variational formulations: solutions are characterized by minimizers of the Dirichlet energy of mappings from into a singular space . In this paper, we extend results of N.\ Garc\'ia Trillos and D.\ Slep\v{c}ev to show that there exist solutions of the continuum problem arising as limits to solutions of a sequence of discrete problems. Specifically, a sequence of points from is sampled i.i.d.\ with respect to a given probability measure on and for all , a geometric graph is…
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