arXiv subjects

André Luiz Corrêa Vianna Filho

Publications and source records attributed to André Luiz Corrêa Vianna Filho.

5 recordsLinked to original sources

arXiv2025

A graph-based approach to customer segmentation using the RFM model

The present article proposes a graph-based approach to customer segmentation, combining the RFM analysis with the classical optimization max-$k$-cut problem. We consider each customer as a vertex of a weighted graph, and the edge weights are given by the distances between the vectors corresponding to the $(R,F,M)$-scores of the customers. We design a procedure to build a reduced graph with fewer vertices and edges, and the customer segmentation is obtained by solving the max-$k$-cut for this reduced graph. We prove that the optimal objective function values of the original and the reduced problems are equal. Additionally, we show that an optimal solution to the original problem can be easily obtained from an optimal solution to the reduced problem, which provides an advantage in dealing with computational complexity in large instances. Applying our methodology to a real customer dataset allowed us to identify distinct behaviors between groups and analyze their meaning and value from a business perspective.

math.OC↗

arXiv2024

Optimal control related to weak solutions of a chemotaxis-consumption model

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $Ω\subset \mathbb{R}^3$: $$\partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v + f \,v\, 1_{Ω_c},$$ with $s \geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, being $u$ the cell density, $v$ the chemical concentration and $f$ the control acting in the $v$-equation through the bilinear term $f \,v\, 1_{Ω_c}$, in a subdomain $Ω_c \subset Ω$. We address the existence of optimal control restricted to a weak solution setting, where, in particular, uniqueness of state $(u,v)$ given a control $f$ is not clear. Then by considering weak solutions satisfying an adequate energy inequality, we prove the existence of optimal control subject to uniformly bounded controls. Finally, we discuss the relation between the considered control problem and two other related ones, where the existence of optimal solution can not be proved.

math.OC↗

arXiv2024

A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models

In the present review we focus on the chemotaxis-consumption model $\partial_t u - Δu = - \nabla \cdot (u \nabla v)$ and $\partial_t v - Δv = - u^s v$ in $(0,T) \times Ω$, for any fixed $s \geq 1$, endowed with isolated boundary conditions and nonnegative initial conditions, where $(u,v)$ model cell density and chemical signal concentration. Our objective is to present an overview of the related literature and latest results on the aforementioned model concerning the following three distinct research lines we have obtained in [12,24-26]: the mathematical analysis, the numerical analysis and the related optimal control theory with a bilinear control acting on the chemical equation.

math.AP↗

arXiv2023

Convergence of a time discrete scheme for a chemotaxis-consumption model

In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v \quad \hbox{in $(0,T)\times Ω$,}$$ endowed with isolated boundary conditions and initial conditions, where $(u,v)$ model cell density and chemical signal concentration. The proposed scheme is defined via a reformulation of the model, using the auxiliary variable $z = \sqrt{v + α^2}$ combined with a Backward Euler scheme for the $(u,z)$-problem and a upper truncation of $u$ in the nonlinear chemotaxis and consumption terms. Then, two different ways of retrieving an approximation for the function $v$ are provided. We prove the existence of solution to the time discrete scheme and establish uniform in time \emph{a priori} estimates, yielding the convergence of the scheme towards a weak solution $(u,v)$ of the chemotaxis-consumption model.

math.NA↗

arXiv2023

An optimal control problem subject to strong solutions of chemotaxis-consumption models

We consider a bilinear optimal control problem associated to the following chemotaxis-consumption model in a bounded domain $Ω\subset \mathbb{R}^3$ during a time interval $(0,T)$: $$\partial_t u - Δu = - \nabla \cdot (u \nabla v), \quad \partial_t v - Δv = - u^s v + f v 1_{Ω_c},$$ with $s \geq 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$, $u$ being the cell density, $v$ the chemical concentration and $f$ the bilinear control acting in a subdomain $Ω_c \subset Ω$. The existence of weak solutions $(u,v)$ to this model given $f \in L^q((0,T) \times Ω)$, for some $q > 5/2$, has been proved in [F. Guillén-González and A. L. Corrêa Vianna Filho, Optimal Control Related to Weak Solutions of a Chemotaxis-Consumption Model, arXiv:2211.14612, 2022]. In this paper, we study a related optimal control problem in the strong solution setting. First, imposing the regularity criterion $u ^s \in L^q((0,T) \times Ω)$ ($q > 5/2$) for a given weak solution, we prove existence and uniqueness of global-in-time strong solutions. Then, the existence of a global optimal solution can be deduced. Finally, using a Lagrange multipliers theorem, we establish first order optimality conditions for any local optimal solution, proving existence, uniqueness and regularity of the associated Lagrange multipliers.

math.OC↗