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The connected Grundy coloring problem: Formulations and a local-search enhanced biased random-key genetic algorithm arxiv.org/abs/2411.14533

The connected Grundy coloring problem: Formulations and a local-search enhanced biased random-key genetic algorithm

Given a graph G=(V,E), a connected Grundy coloring is a proper vertex coloring that can be obtained by a first-fit heuristic on a connected vertex sequence. A first-fit coloring heuristic is one that attributes to each vertex in a sequence the lowest-index color not used for its preceding neighbors. A connected vertex sequence is one in which each element, except for the first one, is connected to at least one element preceding it. The connected Grundy coloring problem consists of obtaining a connected Grundy coloring maximizing the number of colors. In this paper, we propose two integer programming (IP) formulations and a local-search enhanced biased random-key genetic algorithm (BRKGA) for the connected Grundy coloring problem. The first formulation follows the standard way of partitioning the vertices into color classes while the second one relies on the idea of representatives in an attempt to break symmetries. The BRKGA encompasses a local search procedure using a newly proposed neighborhood. A theoretical neighborhood analysis is also presented. Extensive computational experiments indicate that the problem is computationally demanding for the proposed IP formulations. Nonetheless, the formulation by representatives outperforms the standard one for the considered benchmark instances. Additionally, our BRKGA can find high-quality solutions in low computational times for considerably large instances, showing improved performance when enhanced with local search and a reset mechanism. Moreover we show that our BRKGA can be easily extended to successfully tackle the Grundy coloring problem, i.e., the one without the connectivity requirements.

arXiv.org

Deep operator network models for predicting post-burn contraction arxiv.org/abs/2411.14555

Deep operator network models for predicting post-burn contraction

Burn injuries present a significant global health challenge. Among the most severe long-term consequences are contractures, which can lead to functional impairments and disfigurement. Understanding and predicting the evolution of post-burn wounds is essential for developing effective treatment strategies. Traditional mathematical models, while accurate, are often computationally expensive and time-consuming, limiting their practical application. Recent advancements in machine learning, particularly in deep learning, offer promising alternatives for accelerating these predictions. This study explores the use of a deep operator network (DeepONet), a type of neural operator, as a surrogate model for finite element simulations, aimed at predicting post-burn contraction across multiple wound shapes. A DeepONet was trained on three distinct initial wound shapes, with enhancement made to the architecture by incorporating initial wound shape information and applying sine augmentation to enforce boundary conditions. The performance of the trained DeepONet was evaluated on a test set including finite element simulations based on convex combinations of the three basic wound shapes. The model achieved an $R^2$ score of $0.99$, indicating strong predictive accuracy and generalization. Moreover, the model provided reliable predictions over an extended period of up to one year, with speedups of up to 128-fold on CPU and 235-fold on GPU, compared to the numerical model. These findings suggest that DeepONets can effectively serve as a surrogate for traditional finite element methods in simulating post-burn wound evolution, with potential applications in medical treatment planning.

arXiv.org

Unconditionally stable symplectic integrators for the Navier-Stokes equations and other dissipative systems arxiv.org/abs/2411.13569

Unconditionally stable symplectic integrators for the Navier-Stokes equations and other dissipative systems

Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{dissipative} systems is inherently difficult due to dissipative systems' lack of symplectic structure. Leveraging the intrinsic variational structure of higher-order dynamics, this paper presents a general technique for applying existing symplectic integration schemes to dissipative systems, with particular emphasis on viscous fluids modeled by the Navier-Stokes equations. Two very simple such schemes are developed here. Not only are these schemes unconditionally stable for dissipative systems, they also outperform traditional methods with a similar degree of complexity in terms of accuracy for a given time step. For example, in the case of viscous flow between two infinite, flat plates, one of the schemes developed here is found to outperform both the implicit Euler method and the explicit fourth-order Runge-Kutta method in predicting the velocity profile. To the authors' knowledge, this is the very first time that a symplectic integration scheme has been applied successfully to the Navier-Stokes equations. We interpret the present success as direct empirical validation of the canonical Hamiltonian formulation of the Navier-Stokes problem recently published by Sanders~\emph{et al.} More sophisticated symplectic integration schemes are expected to exhibit even greater performance. It is hoped that these results will lead to improved numerical methods in computational fluid dynamics.

arXiv.org

Higher-Order Spectral Element Methods for Electromagnetic Modeling of Complex Anisotropic Waveguides arxiv.org/abs/2411.13573

Higher-Order Spectral Element Methods for Electromagnetic Modeling of Complex Anisotropic Waveguides

This research thesis presents a novel higher-order spectral element method (SEM) formulated in cylindrical coordinates for analyzing electromagnetic fields in waveguides filled with complex anisotropic media. In this study, we consider a large class of cylindrical waveguides: radially-bounded and radially-unbounded domains; homogeneous and inhomogeneous waveguides; concentric and non-concentric geometries; Hermitian and non-Hermitian anisotropic media tensors. This work explores different wave equation formulations for one-layer eccentric and multilayer cylindrical waveguides. For the first case, we can define a new normalized scalar Helmholtz equation for decoupling TM and TE modes, and for the second, a vectorial Helmholtz equation for hybrid modes in multilayered anisotropic structures. Additionally, we formulate a transformation optics (TO) framework to include non-symmetric and non-Hermitian media tensors for non-concentric multilayer waveguides. Lastly, we model excitation sources for logging sensors applied in geophysical problems using the fields obtained by SEM. We validate the proposed approach against analytical solutions, perturbation-based and mode-matching-based methods, finite-elements, and finite-integration numerical methods. Our technique obtains accurate results with fewer elements and degrees of freedom (DoF) than Cartesian-based SEM and ordinary finite-element approaches. To this end, we use higher-order two-dimensional basis functions associated with the zeros of the completed Lobatto polynomial to model the fields in each reference element. The convergence analysis demonstrates the absence of the Runge effect as the expansion order increases. Numerical results show that our formulation is efficient and accurate for modeling cylindrical waveguided geometries filled with complex media.

arXiv.org
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