arXiv subjects

Samuel Mercer

Publications and source records attributed to Samuel Mercer.

2 recordsLinked to original sources

arXiv2026

Uniform Sobolev inequalities on geometric graphs

There is significant interest in the study of calculus on graphs, especially regarding the use of gradient-based methods for applications in data driven problems such as classification, clustering and regularisation for inverse problems. Geometric graphs, whose vertices are take from from a Euclidean domain and whose edge structure is determined by the distance between the nodes in the domain, have been central in theoretical studies. Typical approaches for analysis, such as studying consistency and the existence of continuum limits, rely on $\Gamma$-convergence. This technique has some limitations, as it requires the typical length scale which determines the connectivity structure of the graph to be much larger than the scales frequently used for applications. Moreover, it may fail to provide quantitative results. This paper provides necessary and sufficient conditions on the asymptotic behaviour of this length scale for the existence of a uniform collection of Sobolev inequalities on a sequence of geometric graphs. Furthermore, these inequalities hold when the length scales are much smaller than what is typically assumed for $\Gamma$-convergence results and within the range of what is used for data-driven problems. The Sobolev inequalities provide a quantitative estimate on the $L^q$-regularisation effect of discrete gradients.

math.AP↗

arXiv2025

An extension to Banach stackings of the Brezis--Pazy semigroup-convergence theorem, with applications to $\lambda$-convex gradient flows

A 1972 theorem by Brezis and Pazy establishes the uniform convergence of nonlinear semigroups generated by $\omega$-accretive operators on a Banach space. Our goal is to expand the setting of this theorem to include nonlinear semigroups that are acting on different Banach spaces. This is useful, for example, to prove discrete-to-continuum convergence for graph-based gradient flows. We name the general setting in which our theorem holds a Banach stacking. We give three main applications of the extended theorem that are of independent interest. The first establishes uniform convergence of semigroups in a Banach stacking if the generators of the semigroups converge pointwise. The second is a proof of uniform convergence for gradient flows of $\Gamma$-converging $\lambda$-convex functions on a Banach stacking of Hilbert spaces; the third a proof of uniform convergence for gradient flows of $\Gamma$-converging functions that satisfy a convexity condition that was formulated by B\'enilan and Crandall in a 1991 publication (and which we term `$P_0$-convexity') on a Banach stacking of $L^p$ spaces, corresponding to the $TL^p$ space introduced by Garc\'ia Trillos and Slep\v{c}ev in a 2016 paper.

math.AP↗