arXiv subjects

Manuel Santana

Publications and source records attributed to Manuel Santana.

4 recordsLinked to original sources

arXiv2022

Alternating Minimization for Computed Tomography with Unknown Geometry Parameters

Due to the COVID-19 pandemic, there is an increasing demand for portable CT machines worldwide in order to diagnose patients in a variety of settings. This has led to a need for CT image reconstruction algorithms that can produce high quality images in the case when multiple types of geometry parameters have been perturbed. In this paper we present an alternating minimization algorithm to address this issue, where one step minimizes a regularized linear least squares problem, and the other step minimizes a bounded non-linear least squares problem. Additionally, we survey existing methods to accelerate convergence of the algorithm and discuss implementation details. Finally, numerical experiments are conducted to illustrate the effectiveness of the algorithm.

math.NA↗

arXiv2021

Cordiality of Digraphs

A $(0,1)$-labelling of a set is said to be {\em friendly} if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let $g$ be a labelling of the edge set of a graph that is induced by a labelling $f$ of the vertex set. If both $g$ and $f$ are friendly then $g$ is said to be a {\em cordial} labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called $(2,3)$-cordiality. A directed graph is $(2,3)$-cordial if there is a friendly labelling $f$ of the vertex set which induces a $(1,-1,0)$-labelling of the arc set $g$ such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are $(2,3)$-cordial, which orientations of the $n$-wheel are $(2,3)$-cordial, and which orientations of the $n -$fan are $(2,3)$-cordial.

math.CO↗

arXiv2021

(2,3) Cordial Trees and Paths

Recently L. B. Beasley introduced $(2,3)$-cordial labelings of directed graphs in [1]. He made two conjectures which we resolve in this article. He conjectured that every orientation of a path of length at least five is $(2,3)$ cordial, and that every tree of max degree $n =3$ has a cordial orientation. We show these two conjectures to be false. We also discuss the $(2,3)$ cordiality of orientations of the Petersen graph, and establish an upper bound for the number of edges a graph can have and still be $(2,3)$ cordial. An application of $(2,3)$ cordial labelings is also presented.

math.CO↗

arXiv2021

Numerical Analysis of the 1-D Parabolic Optimal Transport Problem

Numerical methods for the optimal transport problem is an active area of research. Recent work of Kitagawa and Abedin shows that the solution of a time-dependent equation converges exponentially fast as time goes to infinity to the solution of the optimal transport problem. This suggests a fast numerical algorithm for computing optimal maps; we investigate such an algorithm here in the 1-dimensional case. Specifically, we use a finite difference scheme to solve the time-dependent optimal transport problem and carry out an error analysis of the scheme. A collection of numerical examples is also presented and discussed.

math.NA↗