9,798 research outputs found
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
A Light Signalling Approach to Node Grouping for Massive MIMO IoT Networks
Massive MIMO is a promising technology to connect very large numbers of
energy constrained nodes, as it offers both extensive spatial multiplexing and
large array gain. A challenge resides in partitioning the many nodes in groups
that can communicate simultaneously such that the mutual interference is
minimized. We here propose node partitioning strategies that do not require
full channel state information, but rather are based on nodes' respective
directional channel properties. In our considered scenarios, these typically
have a time constant that is far larger than the coherence time of the channel.
We developed both an optimal and an approximation algorithm to partition users
based on directional channel properties, and evaluated them numerically. Our
results show that both algorithms, despite using only these directional channel
properties, achieve similar performance in terms of the minimum
signal-to-interference-plus-noise ratio for any user, compared with a reference
method using full channel knowledge. In particular, we demonstrate that
grouping nodes with related directional properties is to be avoided. We hence
realise a simple partitioning method requiring minimal information to be
collected from the nodes, and where this information typically remains stable
over a long term, thus promoting their autonomy and energy efficiency
Multi-Spectrally Constrained Low-PAPR Waveform Optimization for MIMO Radar Space-Time Adaptive Processing
This paper focuses on the joint design of transmit waveforms and receive
filters for airborne multiple-input-multiple-output (MIMO) radar systems in
spectrally crowded environments. The purpose is to maximize the output
signal-to-interference-plus-noise-ratio (SINR) in the presence of
signal-dependent clutter. To improve the practicability of the radar waveforms,
both a multi-spectral constraint and a peak-to-average-power ratio (PAPR)
constraint are imposed. A cyclic method is derived to iteratively optimize the
transmit waveforms and receive filters. In particular, to tackle the
encountered non-convex constrained fractional programming in designing the
waveforms (for fixed filters), we resort to the Dinkelbach's transform,
minorization-maximization (MM), and leverage the alternating direction method
of multipliers (ADMM). We highlight that the proposed algorithm can iterate
from an infeasible initial point and the waveforms at convergence not only
satisfy the stringent constraints, but also attain superior performance
Quasi second-order methods for PDE-constrained forward and inverse problems
La conception assistée par ordinateur (CAO), les effets visuels, la robotique et de nombreux autres domaines tels que la biologie computationnelle, le génie aérospatial, etc. reposent sur la résolution de problèmes mathématiques. Dans la plupart des cas, des méthodes de calcul sont utilisées pour résoudre ces problèmes. Le choix et la construction de la méthode de calcul ont un impact important sur les résultats et l'efficacité du calcul. La structure du problème peut être utilisée pour créer des méthodes, qui sont plus rapides et produisent des résultats qualitativement meilleurs que les méthodes qui n'utilisent pas la structure. Cette thèse présente trois articles avec trois nouvelles méthodes de calcul s'attaquant à des problèmes de simulation et d'optimisation contraints par des équations aux dérivées partielles (EDP).
Dans le premier article, nous abordons le problème de la dissipation d'énergie des solveurs fluides courants dans les effets visuels. Les solveurs de fluides sont omniprésents dans la création d'effets dans les courts et longs métrages d'animation. Nous présentons un schéma d'intégration temporelle pour la dynamique des fluides incompressibles qui préserve mieux l'énergie comparé aux nombreuses méthodes précédentes. La méthode présentée présente une faible surcharge et peut être intégrée à un large éventail de méthodes existantes. L'amélioration de la conservation de l'énergie permet la création d'animations nettement plus dynamiques.
Nous abordons ensuite la conception computationelle dont le but est d'exploiter l'outils computationnel dans le but d'améliorer le processus de conception. Plus précisément, nous examinons l'analyse de sensibilité, qui calcule les sensibilités du résultat de la simulation par rapport aux paramètres de conception afin d'optimiser automatiquement la conception. Dans ce contexte, nous présentons une méthode efficace de calcul de la direction de recherche de Gauss-Newton, en tirant parti des solveurs linéaires directs épars modernes. Notre méthode réduit considérablement le coût de calcul du processus d'optimisation pour une certaine classe de problèmes de conception inverse.
Enfin, nous examinons l'optimisation de la topologie à l'aide de techniques d'apprentissage automatique. Nous posons deux questions : Pouvons-nous faire de l'optimisation topologique sans maillage et pouvons-nous apprendre un espace de solutions d'optimisation topologique. Nous appliquons des représentations neuronales implicites et obtenons des résultats structurellement sensibles pour l'optimisation topologique sans maillage en guidant le réseau neuronal pendant le processus d'optimisation et en adaptant les méthodes d'optimisation topologique par éléments finis. Notre méthode produit une représentation continue du champ de densité. De plus, nous présentons des espaces de solution appris en utilisant la représentation neuronale implicite.Computer-aided design (CAD), visual effects, robotics and many other fields such as computational biology, aerospace engineering etc. rely on the solution of mathematical problems. In most cases, computational methods are used to solve these problems. The choice and construction of the computational method has large impact on the results and the computational efficiency. The structure of the problem can be used to create methods, that are faster and produce qualitatively better results than methods that do not use the structure. This thesis presents three articles with three new computational methods tackling partial differential equation (PDE) constrained simulation and optimization problems.
In the first article, we tackle the problem of energy dissipation of common fluid solvers in visual effects. Fluid solvers are ubiquitously used to create effects in animated shorts and feature films. We present a time integration scheme for incompressible fluid dynamics which preserves energy better than many previous methods. The presented method has low overhead and can be integrated into a wide range of existing methods. The improved energy conservation leads to noticeably more dynamic animations.
We then move on to computational design whose goal is to harnesses computational techniques for the design process. Specifically, we look at sensitivity analysis, which computes the sensitivities of the simulation result with respect to the design parameters to automatically optimize the design. In this context, we present an efficient way to compute the Gauss-Newton search direction, leveraging modern sparse direct linear solvers. Our method reduces the computational cost of the optimization process greatly for a certain class of inverse design problems.
Finally, we look at topology optimization using machine learning techniques. We ask two questions: Can we do mesh-free topology optimization and can we learn a space of topology optimization solutions. We apply implicit neural representations and obtain structurally sensible results for mesh-free topology optimization by guiding the neural network during optimization process and adapting methods from finite element based topology optimization. Our method produces a continuous representation of the density field. Additionally, we present learned solution spaces using the implicit neural representation
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