1,214 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On the Complexity of Equilibrium Computation in First-Price Auctions

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    Duality, Derivative-Based Training Methods and Hyperparameter Optimization for Support Vector Machines

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    In this thesis we consider the application of Fenchel's duality theory and gradient-based methods for the training and hyperparameter optimization of Support Vector Machines. We show that the dualization of convex training problems is possible theoretically in a rather general formulation. For training problems following a special structure (for instance, standard training problems) we find that the resulting optimality conditions can be interpreted concretely. This approach immediately leads to the well-known notion of support vectors and a formulation of the Representer Theorem. The proposed theory is applied to several examples such that dual formulations of training problems and associated optimality conditions can be derived straightforwardly. Furthermore, we consider different formulations of the primal training problem which are equivalent under certain conditions. We also argue that the relation of the corresponding solutions to the solution of the dual training problem is not always intuitive. Based on the previous findings, we consider the application of customized optimization methods to the primal and dual training problems. A particular realization of Newton's method is derived which could be used to solve the primal training problem accurately. Moreover, we introduce a general convergence framework covering different types of decomposition methods for the solution of the dual training problem. In doing so, we are able to generalize well-known convergence results for the SMO method. Additionally, a discussion of the complexity of the SMO method and a motivation for a shrinking strategy reducing the computational effort is provided. In a last theoretical part, we consider the problem of hyperparameter optimization. We argue that this problem can be handled efficiently by means of gradient-based methods if the training problems are formulated appropriately. Finally, we evaluate the theoretical results concerning the training and hyperparameter optimization approaches practically by means of several example training problems

    High-Order Mixed Finite Element Variable Eddington Factor Methods

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    We apply high-order mixed finite element discretization techniques and their associated preconditioned iterative solvers to the Variable Eddington Factor (VEF) equations in two spatial dimensions. The mixed finite element VEF discretizations are coupled to a high-order Discontinuous Galerkin (DG) discretization of the Discrete Ordinates transport equation to form effective linear transport algorithms that are compatible with high-order (curved) meshes. This combination of VEF and transport discretizations is motivated by the use of high-order mixed finite element methods in hydrodynamics calculations at the Lawrence Livermore National Laboratory. Due to the mathematical structure of the VEF equations, the standard Raviart Thomas (RT) mixed finite elements cannot be used to approximate the vector variable in the VEF equations. Instead, we investigate three alternatives based on the use of continuous finite elements for each vector component, a non-conforming RT approach where DG-like techniques are used, and a hybridized RT method. We present numerical results that demonstrate high-order accuracy, compatibility with curved meshes, and robust and efficient convergence in iteratively solving the coupled transport-VEF system and in the preconditioned linear solvers used to invert the discretized VEF equations

    Microstructure modeling and crystal plasticity parameter identification for predicting the cyclic mechanical behavior of polycrystalline metals

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    Computational homogenization permits to capture the influence of the microstructure on the cyclic mechanical behavior of polycrystalline metals. In this work we investigate methods to compute Laguerre tessellations as computational cells of polycrystalline microstructures, propose a new method to assign crystallographic orientations to the Laguerre cells and use Bayesian optimization to find suitable parameters for the underlying micromechanical model from macroscopic experiments

    Aspects Topologiques des Représentations en Analyse Calculable

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    Computable analysis provides a formalization of algorithmic computations over infinite mathematical objects. The central notion of this theory is the symbolic representation of objects, which determines the computation power of the machine, and has a direct impact on the difficulty to solve any given problem. The friction between the discrete nature of computations and the continuous nature of mathematical objects is captured by topology, which expresses the idea of finite approximations of infinite objects.We thoroughly study the multiple interactions between computations and topology, analysing the information that can be algorithmically extracted from a representation. In particular, we focus on the comparison between two representations of a single family of objects, on the precise relationship between algorithmic and topological complexity of problems, and on the relationship between finite and infinite representations.L’analyse calculable permet de formaliser le traitement algorithmique d’objets mathĂ©matiques infinis. La thĂ©orie repose sur une reprĂ©sentation symbolique des objets, dont le choix dĂ©termine les capacitĂ©s de calcul de la machine, notamment sa difficultĂ© Ă  rĂ©soudre chaque problĂšme donnĂ©. La friction entre le caractĂšre discret du calcul et la nature continue des objets est capturĂ©e par la topologie, qui exprime l’idĂ©e d’approximation finie d’objets infinis.Nous Ă©tudions en profondeur les multiples interactions entre calcul et topologie, cherchant Ă  analyser l’information qui peut ĂȘtre extraite algorithmiquement d’une reprĂ©sentation. Je me penche plus particuliĂšrement sur la comparaison entre deux reprĂ©sentations d’une mĂȘme famille d’objets, sur les liens dĂ©taillĂ©s entre complexitĂ© algorithmique et topologique des problĂšmes, ainsi que sur les relations entre reprĂ©sentations finies et infinies

    Exactly soluble models in many-body physics

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    Almost all phenomena in the universe are described, at the fundamental level, by quantum manybody models. In general, however, a complete understanding of large systems with many degrees of freedom is impossible. While in general many-body quantum systems are intractable, there are special cases for which there are techniques that allow for an exact solution. Exactly soluble models are interesting because they are soluble; beyond this, they can be used to gain intuition for further reaching many-body systems, including when they can be leveraged to help with numerical approximations for general models. The work presented in this thesis considers exactly soluble models of quantum many-body systems. The first part of this thesis extends the family of many-body spin models for which we can find a freefermion solution. A solution method that was developed for a specific free-fermion model is generalized in such a way that allows application to a broader class of many-body spin system than was previously known to be free. Models which admit a solution via this method are characterized by a graph theory invariants: in brief it is shown that a quantum spin system has an exact description via non-interacting fermions if its frustration graph is claw-free and contains a simplicial clique. The second part of this thesis gives an explicit example of how the usefulness of exactly soluble models can extend beyond the solution itself. This chapter pertains to the calculation of the topological entanglement entropy in topologically ordered loop-gas states. Topological entanglement entropy gives an understanding of how correlations may extend throughout a system. In this chapter the topological entanglement entropy of two- and three-dimensional loop-gas states is calculated in the bulk and at the boundary. We obtain a closed form expression for the topological entanglement in terms of the anyonic theory that the models support

    Symmetric separable convex resource allocation problems with structured disjoint interval bound constraints

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    Motivated by the problem of scheduling electric vehicle (EV) charging with a minimum charging threshold in smart distribution grids, we introduce the resource allocation problem (RAP) with a symmetric separable convex objective function and disjoint interval bound constraints. In this RAP, the aim is to allocate an amount of resource over a set of nn activities, where each individual allocation is restricted to a disjoint collection of mm intervals. This is a generalization of classical RAPs studied in the literature where in contrast each allocation is only restricted by simple lower and upper bounds, i.e., m=1m=1. We propose an exact algorithm that, for four special cases of the problem, returns an optimal solution in O((n+m−2m−2)(nlog⁡n+nF))O \left(\binom{n+m-2}{m-2} (n \log n + nF) \right) time, where the term nFnF represents the number of flops required for one evaluation of the separable objective function. In particular, the algorithm runs in polynomial time when the number of intervals mm is fixed. Moreover, we show how this algorithm can be adapted to also output an optimal solution to the problem with integer variables without increasing its time complexity. Computational experiments demonstrate the practical efficiency of the algorithm for small values of mm and in particular for solving EV charging problems.Comment: 20 pages, 4 figure
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