A gradient estimate to a degenerate parabolic equation with a singular absorption term: The global quenching phenomena

We prove global existence of nonnegative solutions to the one dimensional degenerate parabolic problems containing a singular term. We also show the global quenching phenomena for L1 initial datums. Moreover, the free boundary problem is considered in this paper.

The n-tuple network: coping with sub-optimality

The subject of this thesis is the n-tuple net.work (RAMnet). The major advantage of RAMnets is their speed and the simplicity with which they can be implemented in parallel hardware. On the other hand, this method is not a universal approximator and the training procedure does not involve the minimisation of a cost function. Hence RAMnets are potentially sub-optimal. It is important to understand the source of this sub-optimality and to develop the analytical tools that allow us to quantify the generalisation cost of using this model for any given data. We view RAMnets as classifiers and function approximators and try to determine how critical their lack of' universality and optimality is. In order to understand better the inherent. restrictions of the model, we review RAMnets showing their relationship to a number of well established general models such as: Associative Memories, Kamerva's Sparse Distributed Memory, Radial Basis Functions, General Regression Networks and Bayesian Classifiers. We then benchmark binary RAMnet. model against 23 other algorithms using real-world data from the StatLog Project. This large scale experimental study indicates that RAMnets are often capable of delivering results which are competitive with those obtained by more sophisticated, computationally expensive rnodels. The Frequency Weighted version is also benchmarked and shown to perform worse than the binary RAMnet for large values of the tuple size n. We demonstrate that the main issues in the Frequency Weighted RAMnets is adequate probability estimation and propose Good-Turing estimates in place of the more commonly used :Maximum Likelihood estimates. Having established the viability of the method numerically, we focus on providillg an analytical framework that allows us to quantify the generalisation cost of RAMnets for a given datasetL. For the classification network we provide a semi-quantitative argument which is based on the notion of Tuple distance. It gives a good indication of whether the network will fail for the given data. A rigorous Bayesian framework with Gaussian process prior assumptions is given for the regression n-tuple net. We show how to calculate the generalisation cost of this net and verify the results numerically for one dimensional noisy interpolation problems. We conclude that the n-tuple method of classification based on memorisation of random features can be a powerful alternative to slower cost driven models. The speed of the method is at the expense of its optimality. RAMnets will fail for certain datasets but the cases when they do so are relatively easy to determine with the analytical tools we provide.

Problemas de corte : métodos exactos y aproximados para formulaciones mono y multi-objetivo

Los problemas de corte y empaquetado son una familia de problemas de optimización combinatoria que han sido ampliamente estudiados en numerosas áreas de la industria y la investigación, debido a su relevancia en una enorme variedad de aplicaciones reales. Son problemas que surgen en muchas industrias de producción donde se debe realizar la subdivisión de un material o espacio disponible en partes más pequeñas. Existe una gran variedad de métodos para resolver este tipo de problemas de optimización. A la hora de proponer un método de resolución para un problema de optimización, es recomendable tener en cuenta el enfoque y las necesidades que se tienen en relación al problema y su solución. Las aproximaciones exactas encuentran la solución óptima, pero sólo es viable aplicarlas a instancias del problema muy pequeñas. Las heurísticas manejan conocimiento específico del problema para obtener soluciones de alta calidad sin necesitar un excesivo esfuerzo computacional. Por otra parte, las metaheurísticas van un paso más allá, ya que son capaces de resolver una clase muy general de problemas computacionales. Finalmente, las hiperheurísticas tratan de automatizar, normalmente incorporando técnicas de aprendizaje, el proceso de selección, combinación, generación o adaptación de heurísticas más simples para resolver eficientemente problemas de optimización. Para obtener lo mejor de estos métodos se requiere conocer, además del tipo de optimización (mono o multi-objetivo) y el tamaño del problema, los medios computacionales de los que se dispone, puesto que el uso de máquinas e implementaciones paralelas puede reducir considerablemente los tiempos para obtener una solución. En las aplicaciones reales de los problemas de corte y empaquetado en la industria, la diferencia entre usar una solución obtenida rápidamente y usar propuestas más sofisticadas para encontrar la solución óptima puede determinar la supervivencia de la empresa. Sin embargo, el desarrollo de propuestas más sofisticadas y efectivas normalmente involucra un gran esfuerzo computacional, que en las aplicaciones reales puede provocar una reducción de la velocidad del proceso de producción. Por lo tanto, el diseño de propuestas efectivas y, al mismo tiempo, eficientes es fundamental. Por esta razón, el principal objetivo de este trabajo consiste en el diseño e implementación de métodos efectivos y eficientes para resolver distintos problemas de corte y empaquetado. Además, si estos métodos se definen como esquemas lo más generales posible, se podrán aplicar a diferentes problemas de corte y empaquetado sin realizar demasiados cambios para adaptarlos a cada uno. Así, teniendo en cuenta el amplio rango de metodologías de resolución de problemas de optimización y las técnicas disponibles para incrementar su eficiencia, se han diseñado e implementado diversos métodos para resolver varios problemas de corte y empaquetado, tratando de mejorar las propuestas existentes en la literatura. Los problemas que se han abordado han sido: el Two-Dimensional Cutting Stock Problem, el Two-Dimensional Strip Packing Problem, y el Container Loading Problem. Para cada uno de estos problemas se ha realizado una amplia y minuciosa revisión bibliográfica, y se ha obtenido la solución de las distintas variantes escogidas aplicando diferentes métodos de resolución: métodos exactos mono-objetivo y paralelizaciones de los mismos, y métodos aproximados multi-objetivo y paralelizaciones de los mismos. Los métodos exactos mono-objetivo aplicados se han basado en técnicas de búsqueda en árbol. Por otra parte, como métodos aproximados multi-objetivo se han seleccionado unas metaheurísticas multi-objetivo, los MOEAs. Además, para la representación de los individuos utilizados por estos métodos se han empleado codificaciones directas mediante una notación postfija, y codificaciones que usan heurísticas de colocación e hiperheurísticas. Algunas de estas metodologías se han mejorado utilizando esquemas paralelos haciendo uso de las herramientas de programación OpenMP y MPI.

Generating unconstrained two-dimensional non-guillotine cutting patterns by a Recursive Partitioning Algorithm

In this study, a dynamic programming approach to deal with the unconstrained two-dimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. The method is able to find the optimal cutting pattern of a large number of pro blem instances of moderate sizes known in the literature and a counterexample for which the approach fails to find known optimal solutions was not found. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method. Journal of the Operational Research Society (2012) 63, 183-200. doi: 10.1057/jors.2011.6 Published online 17 August 2011

Solução do problema de corte bidimensional de peças retângulares tipo não-guilhotinado usando simulated annealing

Pós-graduação em Engenharia Elétrica - FEIS

Decomposition and reformulation of integer linear programming problems

This thesis deals with an investigation of Decomposition and Reformulation to solve Integer Linear Programming Problems. This method is often a very successful approach computationally, producing high-quality solutions for well-structured combinatorial optimization problems like vehicle routing, cutting stock, p-median and generalized assignment . However, until now the method has always been tailored to the specific problem under investigation. The principal innovation of this thesis is to develop a new framework able to apply this concept to a generic MIP problem. The new approach is thus capable of auto-decomposition and autoreformulation of the input problem applicable as a resolving black box algorithm and works as a complement and alternative to the normal resolving techniques. The idea of Decomposing and Reformulating (usually called in literature Dantzig and Wolfe Decomposition DWD) is, given a MIP, to convexify one (or more) subset(s) of constraints (slaves) and working on the partially convexified polyhedron(s) obtained. For a given MIP several decompositions can be defined depending from what sets of constraints we want to convexify. In this thesis we mainly reformulate MIPs using two sets of variables: the original variables and the extended variables (representing the exponential extreme points). The master constraints consist of the original constraints not included in any slaves plus the convexity constraint(s) and the linking constraints(ensuring that each original variable can be viewed as linear combination of extreme points of the slaves). The solution procedure consists of iteratively solving the reformulated MIP (master) and checking (pricing) if a variable of reduced costs exists, and in which case adding it to the master and solving it again (columns generation), or otherwise stopping the procedure. The advantage of using DWD is that the reformulated relaxation gives bounds stronger than the original LP relaxation, in addition it can be incorporated in a Branch and bound scheme (Branch and Price) in order to solve the problem to optimality. If the computational time for the pricing problem is reasonable this leads in practice to a stronger speed up in the solution time, specially when the convex hull of the slaves is easy to compute, usually because of its special structure.

Three-Dimensional Operational Calculi for Nonlocal Evolution Boundary Value Problems

Иван Христов Димовски, Юлиан Цанков Цанков - Построени са директни операционни смятания за функции u(x, y, t), непрекъснати в област от вида D = [0, a] × [0, b] × [0, ∞). Наред с класическата дюамелова конволюция, построението използва и две некласически конволюции за операторите ∂2x и ∂2y. Тези три едномерни конволюции се комбинират в една тримерна конволюция u ∗ v в C(D). Вместо подхода на Я. Микусински, основаващ се на конволюционни частни, се развива алтернативен подход с използване на мултипликаторните частни на конволюционната алгебра (C(D), ∗).

Two-dimensional supersonic nonlinear Schrodinger flow past an extended obstacle

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schroumldinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear ""ship-wave"" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures

This paper addresses the development of a hybrid-mixed finite element formulation for the quasi-static geometrically exact analysis of three-dimensional framed structures with linear elastic behavior. The formulation is based on a modified principle of stationary total complementary energy, involving, as independent variables, the generalized vectors of stress-resultants and displacements and, in addition, a set of Lagrange multipliers defined on the element boundaries. The finite element discretization scheme adopted within the framework of the proposed formulation leads to numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the equilibrium boundary conditions. This formulation consists, therefore, in a true equilibrium formulation for large displacements and rotations in space. Furthermore, this formulation is objective, as it ensures invariance of the strain measures under superposed rigid body rotations, and is not affected by the so-called shear-locking phenomenon. Also, the proposed formulation produces numerical solutions which are independent of the path of deformation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions compared with those obtained using the standard two-node displacement/ rotation-based formulation.

Simulated annealing applied to the irregular rotational placement of shapes over containers with fixed dimensions

This work deals with the problem of minimizing the waste of space that occurs on a rotational placement of a set of irregular two dimensional polygons inside a two dimensional container. This problem is approached with an heuristic based on simulated annealing. Traditional 14 external penalization"" techniques are avoided through the application of the no-fit polygon, that determinates the collision free area for each polygon before its placement. The simulated annealing controls: the rotation applied, the placement and the sequence of placement of the polygons. For each non placed polygon, a limited depth binary search is performed to find a scale factor that when applied to the polygon, would allow it to be fitted in the container. It is proposed a crystallization heuristic, in order to increase the number of accepted solutions. The bottom left and larger first deterministic heuristics were also studied. The proposed process is suited for non convex polygons and containers, the containers can have holes inside. (C) 2009 Elsevier Ltd. All rights reserved.

Finite element modelling of three-dimensional convection problems in fluid-saturated porous media heated from below

We use the finite element method to model three-dimensional convective pore-fluid flow in fluid-saturated porous media when they are heated from below. In particular, we employ the particle-tracking technique to mimic the trajectories of particles in three-dimensional fluid flow problems. The related numerical results demonstrated that: (1) The progressive asymptotic approach procedure, which was previously developed for the finite element modelling of two-dimensional convective pore-fluid flow problems, is equally applicable to the finite element modelling of three-dimensional convective pore-fluid flow in fluid-saturated porous media heated from below. (2) The perturbation of gravity at different planes has a significant effect on the pattern of three-dimensional convective pore-fluid flow and therefore, may influence the pattern of orebody formation and mineralization in three-dimensional hydrothermal systems. Copyright (C) 2001 John Wiley & Sons, Ltd.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems

In this work, we present the solution of a class of linear inverse heat conduction problems for the estimation of unknown heat source terms, with no prior information of the functional forms of timewise and spatial dependence of the source strength, using the conjugate gradient method with an adjoint problem. After describing the mathematical formulation of a general direct problem and the procedure for the solution of the inverse problem, we show applications to three transient heat transfer problems: a one-dimensional cylindrical problem; a two-dimensional cylindrical problem; and a one-dimensional problem with two plates.