Strong gravitational lensing produces multiple distorted images of a background source when it is closely aligned with a mass distribution along the line of sight. The lensed images provide constraints on the parameters of a model of the lens, and the images themselves can be inverted providing a model of the source. Both of these aspects of lensing are extremely valuable, as lensing depends on the total matter distribution, both luminous and dark. Furthermore, lensed sources are commonly located at cosmological distances and are magnified by the lensing effect. This provides a chance to image sources that would be unobservable when viewed with conventional optics. The semilinear method expresses the source modeling step as a least-squares problem for a given set of lens model parameters. The blurring effect due to the point spread function of the instrument used to observe the lensed images is also taken into account. In general, regularization is needed to solve the source deconvolution problem. We use Krylov subspace methods to solve for the pixelated sources. These optimization techniques, such as the Conjugate Gradient method, provide natural regularizing effects from simple truncated iteration. Using these routines, we are able to avoid the explicit construction of the lens and blurring matrices and solve the least squares source optimization problem iteratively. We explore several regularization parameter selection methods commonly used in standard image deconvolution problems, which lead to previously derived expressions for the number of source degrees of freedom. The parameters that describe the lens density distribution are found by global optimization methods including genetic algorithms and particle swarm optimizers. In general, global optimizers are useful in non-linear optimization problems such as lens modeling due to their parameter space mapping capabilities. However, these optimization methods require many function evaluations and iterative approaches to the least squares problem are beneficial due to the speed advantage that they offer. We apply our modeling techniques to a subset of gravitational lens systems from the Sloan Lens ACS (SLACS) survey, and are able to reliably recover the parameters of the lens mass distribution with both analytical and regularized pixelated sources.

Strong gravitational lensing produces multiple distorted images of a background source when it is closely aligned with a mass distribution along the line of sight. The lensed images provide constraints on the parameters of a model of the lens, and the images themselves can be inverted providing a model of the source. Both of these aspects of lensing are extremely valuable, as lensing depends on the total matter distribution, both luminous and dark. Furthermore, lensed sources are commonly located at cosmological distances and are magnified by the lensing effect. This provides a chance to image sources that would be unobservable when viewed with conventional optics. The semilinear method expresses the source modeling step as a least-squares problem for a given set of lens model parameters. The blurring effect due to the point spread function of the instrument used to observe the lensed images is also taken into account. In general, regularization is needed to solve the source deconvolution problem. We use Krylov subspace methods to solve for the pixelated sources. These optimization techniques, such as the Conjugate Gradient method, provide natural regularizing effects from simple truncated iteration. Using these routines, we are able to avoid the explicit construction of the lens and blurring matrices and solve the least squares source optimization problem iteratively. We explore several regularization parameter selection methods commonly used in standard image deconvolution problems, which lead to previously derived expressions for the number of source degrees of freedom. The parameters that describe the lens density distribution are found by global optimization methods including genetic algorithms and particle swarm optimizers. In general, global optimizers are useful in non-linear optimization problems such as lens modeling due to their parameter space mapping capabilities. However, these optimization methods require many function evaluations and iterative approaches to the least squares problem are beneficial due to the speed advantage that they offer. We apply our modeling techniques to a subset of gravitational lens systems from the Sloan Lens ACS (SLACS) survey, and are able to reliably recover the parameters of the lens mass distribution with both analytical and regularized pixelated sources.

In this article a new branch & bound method is described. It uses an artificial target to improve its bounding capabilities. Therefore the new approach is faster compared to the classical one. It is applied to the stochastic problem of optimal scenario tree reduction. The aspects of global optimization are emphasized here. All necessary components for that problem are developed and some experimental results underline the benefits of the new approach. (author's abstract) / Series: Working Papers on Information Systems, Information Business and Operations

Limited water supplies for agriculture in the Texas High Plains will require new irrigation technologies and techniques for agriculture to continue in this area. The potential for using one such technology, site-specific irrigation, was evaluated using the Cotton2k crop simulation model. This model and two other simulation models were evaluated for their ability to track water movement and usage over three growing seasons. The models were tested for sites in Lubbock and Hale County, Texas. Cotton2k performed well compared to the other two models on tests of cumulative evapotranspiration and applied water yield relations and equal to the other models for tracking soil water profiles. A global optimization method, simulated annealing, was tested for its ability to spatially calibrate soil water parameters of Cotton2k. The algorithm found multiple parameter sets for the same objective function results. This result runs contrary to expectations for the simulated annealing algorithm, but is possibly from the relationship between available water capacity and crop yield. The annealing algorithm was applied to each sampling point at the Hale County site and improved yield predictions for 32 of 33 points as compared to simulations made with soil textural information alone. The spatially calibrated model was used with historic weather from five seasons to evaluate a site-specific strategy where water was shifted from lower to higher yielding areas of fields. Two irrigation strategies, one with irrigations weekly and one with irrigations applied when 30% of available water was depleted, were tested. With sitespecific management, the weekly interval strategy produced higher yields for two of three water levels, as compared to uniform management. With the soil moisture depletion strategy, site-specific management produced lower yields than uniform management for all three water levels examined. Yield improvement and water savings were also demonstrated for implementing site-specific irrigation when non-producing portions of fields were previously being watered.

Many important traits in plants, animals and humans are quantitative, and most such traits are generally believed to be affected by multiple genetic loci. Standard computational tools for mapping of quantitative traits (i.e. for finding Quantitative Trait Loci, QTL, in the genome) use linear regression models for relating the observed phenotypes to the genetic composition of individuals in an experimental population. Using these tools to simultaneously search for multiple QTL is computationally demanding. The main reason for this is the complex optimization landscape for the multidimensional global optimization problems that must be solved. This thesis describes parallel algorithms, implementations and tools for simultaneous mapping of several QTL. These new computational tools enable genetic analysis exploiting new classes of multidimensional statistical models, potentially resulting in interesting results in genetics. We first describe how the standard, brute-force algorithm for global optimization in QTL analysis is parallelized and implemented on a grid system. Then, we also present a parallelized version of the more elaborate global optimization algorithm DIRECT and show how this can be efficiently deployed and used on grid systems and other loosely-coupled architectures. The parallel DIRECT scheme is further developed to exploit both coarse-grained parallelism in grid systems or clusters as well as fine-grained, tightly-coupled parallelism in multi-core nodes. The results show that excellent speedup and performance can be archived on grid systems and clusters, even when using a tightly-coupled algorithm such as DIRECT. Finally, we provide two distinctly different front-ends for our code. One is a grid portal providing a graphical front-end suitable for novice users and standard forms of QTL analysis. The other is a prototype of an R-based grid-enabled problem solving environment. Both of these front-ends can, after some further refinement, be utilized by geneticists for performing multidimensional genetic analysis of quantitative traits on a regular basis. / eSSENCE

Global dynamic optimization arises in many engineering applications such as parameter estimation, global optimal control, and optimization-based worst-case uncertainty analysis. In branch-and-bound deterministic global optimization algorithms, a major computational bottleneck is generating appropriate lower bounds for the globally optimal objective value. These bounds are typically constructed using convex relaxations for the solutions of dynamic systems with respect to decision variables. Tighter convex relaxations thus translate into tighter lower bounds, which will typically reduce the number of iterations required by branch-and-bound. Subgradients, as useful local sensitivities of convex relaxations, are typically required by nonsmooth optimization solvers to effectively minimize these relaxations. This thesis develops novel techniques for efficiently computing tight convex relaxations with the corresponding subgradients for the solutions of ordinary differential equations (ODEs), to ultimately improve efficiency of deterministic global dynamic optimization. Firstly, new bounding and comparison results for dynamic process models are developed, which are more broadly applicable to engineering models than previous results. These new results show for the first time that in a state-of-the-art ODE relaxation framework, tighter enclosures of the original ODE system's right-hand side will necessarily translate into enclosures for the state variables that are at least as tight, which paves the way towards new advances for bounding in global dynamic optimization. Secondly, new convex relaxations are proposed for the solutions of ODE systems. These new relaxations are guaranteed to be at least as tight as state-of-the-art ODE relaxations. Unlike established ODE relaxation approaches, the new ODE relaxation approach can employ any valid convex and concave relaxations for the original right-hand side, and tighter such relaxations will necessarily yield ODE relaxations that are at least as tight. In a numerical case study, such tightness does indeed improve computational efficiency in deterministic global dynamic optimization. This new ODE relaxation approach is then extended in various ways to further tighten ODE relaxations. Thirdly, new subgradient evaluation approaches are proposed for ODE relaxations. Unlike established approaches that compute valid subgradients for nonsmooth dynamic systems, the new approaches are compatible with reverse automatic differentiation (AD). It is shown for the first time that subgradients of dynamic convex relaxations can be computed via a modified adjoint ODE sensitivity system, which could speed up lower bounding in global dynamic optimization. Lastly, in the situation where convex relaxations are known to be correct but subgradients are unavailable (such as for certain ODE relaxations), a new approach is proposed for tractably constructing useful correct affine underestimators and lower bounds of the convex relaxations just by black-box sampling. No additional assumptions are required, and no subgradients must be computed at any point. Under mild conditions, these new bounds are shown to converge rapidly to an original nonconvex function as the domain of interest shrinks. Variants of the new approach are presented to account for numerical error or noise in the sampling procedure. / Thesis / Doctor of Philosophy (PhD)

The primary objective of this thesis is to develop and implement a global optimization algorithm to solve a class of nonconvex programming problems, and to test it using a collection of engineering design problem applications.The class of problems we consider involves the optimization of a general nonconvex factorable objective function over a feasible region that is restricted by a set of constraints, each of which is defined in terms of nonconvex factorable functions. Such problems find widespread applications in production planning, location and allocation, chemical process design and control, VLSI chip design, and numerous engineering design problems. This thesis offers a first comprehensive methodological development and implementation for determining a global optimal solution to such factorable programming problems. To solve this class of problems, we propose a branch-and-bound approach based on linear programming (LP) relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials, coordinated with a {em Reformulation-Linearization Technique} (RLT). The initial stage of the lower bounding step generates a tight, nonconvex polynomial programming relaxation for the given problem. Subsequently, an LP relaxation is constructed for the resulting polynomial program via a suitable RLT procedure. The underlying motivation for these two steps is to generate a tight outer approximation of the convex envelope of the objective function over the convex hull of the feasible region. The bounding step is thenintegrated into a general branch-and-bound framework. The construction of the bounding polynomials and the node partitioning schemes are specially designed so that the gaps resulting from these two levels of approximations approach zero in the limit, thereby ensuring convergence to a global optimum. Various implementation issues regarding the formulation of such tight bounding problems using both polynomial approximations and RLT constructs are discussed. Different practical strategies and guidelines relating to the design of the algorithm are presented within a general theoretical framework so that users can customize a suitable approach that takes advantage of any inherent special structures that their problems might possess. The algorithm is implemented in C++, an object-oriented programming language. The class modules developed for the software perform various functions that are useful not only for the proposed algorithm, but that can be readily extended and incorporated into other RLT based applications as well. Computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. It is shown that, for all the test problems, a very competitive computational performance is obtained. In most cases, the LP solution obtained for the initial node itself provides a very tight lower bound. Furthermore, for nine of these fifteen problems, the application of a local search heuristic based on initializing the nonlinear programming solver MINOS at the node zero LP solution produced the actual global optimum. Moreover, in finding a global optimum, our algorithm discovered better solutions than the ones previously reported in the literature for two of these test instances. / Master of Science

The goal of this thesis is to explore, improve and implement some advanced modern computational methods in statistics, focusing on applications in genetics. The thesis has three major directions. First, we study likelihoods for genetics analysis of experimental populations. Here, the maximum likelihood can be viewed as a computational global optimization problem. We introduce a faster optimization algorithm called PruneDIRECT, and explain how it can be parallelized for permutation testing using the Map-Reduce framework. We have implemented PruneDIRECT as an open source R package, and also Software as a Service for cloud infrastructures (QTLaaS). The second part of the thesis focusses on using sparse matrix methods for solving linear mixed models with large correlation matrices. For populations with known pedigrees, we show that the inverse of covariance matrix is sparse. We describe how to use this sparsity to develop a new method to maximize the likelihood and calculate the variance components. In the final part of the thesis we study computational challenges of psychiatric genetics, using only pedigree information. The aim is to investigate existence of maternal effects in obsessive compulsive behavior. We add the maternal effects to the linear mixed model, used in the second part of this thesis, and we describe the computational challenges of working with binary traits. / eSSENCE

Este trabalho apresenta o dimensionamento otimizado de redes pressurizadas de distribuição de água em regime permanente para áreas de topografia relativamente plana. Além dos critérios tradicionais de dimensionamento hidráulico, o resultado ótimo é aquele com menor custo global, ou seja, onde a soma dos custos de implantação e de operação é mínimo. Para a determinação dos resultados, as equações que envolvem a perda de carga nos circuitos e vazões em cada nó foram solucionadas através da programação não linear com emprego de métodos matriciais. As principais variáveis analisadas foram os diferentes materiais das tubulações (PEAD, PVC e Ferro Fundido), tipos de superfície (terra, concreto, paralelepípedo e asfalto), locação da rede (passeio, viário pavimentado e sem pavimentação), tarifas de energia elétrica para concessionárias de água, vida útil usuais para o sistema de bombeamento e taxa de juros ao longo da operação do sistema. Os resultados obtidos através da metodologia proposta demonstraram que as principais variáveis em relação ao custo referencial unitário foram o custo da pressurização inicial, custo da tubulação, além do custo de remoção e recomposição de viário em pavimento asfáltico. / This work presents the optimal design of pressurized networks of water distribution in steady state flow to areas of relatively flat topography. In addition to the traditional hydraulic criteria for design, the optimal outcome is that with lower overall cost, in other words, where the sum of the costs of implementation and operation is minimal. To obtain the results, the equations that involving headloss in the circuits and flows at each node were solved by nonlinear programming with the use of matrix methods. The main variables studied were the different materials of pipes (HDPE, PVC and Cast Iron), surface types (clay, concrete and asphalt paving), network location (walk, paved and unpaved road), electricity tariffs for water utilities, normal life for the pumping system and interest rates along the system operation. The results obtained by the proposed methodology showed that the main variables in relation to the unit cost were the cost of initial pressurization, cost of the pipe, besides the cost of removal and restoration of roads in asphalt pavement.

Este trabalho apresenta o dimensionamento otimizado de redes pressurizadas de distribuição de água em regime permanente para áreas de topografia relativamente plana. Além dos critérios tradicionais de dimensionamento hidráulico, o resultado ótimo é aquele com menor custo global, ou seja, onde a soma dos custos de implantação e de operação é mínimo. Para a determinação dos resultados, as equações que envolvem a perda de carga nos circuitos e vazões em cada nó foram solucionadas através da programação não linear com emprego de métodos matriciais. As principais variáveis analisadas foram os diferentes materiais das tubulações (PEAD, PVC e Ferro Fundido), tipos de superfície (terra, concreto, paralelepípedo e asfalto), locação da rede (passeio, viário pavimentado e sem pavimentação), tarifas de energia elétrica para concessionárias de água, vida útil usuais para o sistema de bombeamento e taxa de juros ao longo da operação do sistema. Os resultados obtidos através da metodologia proposta demonstraram que as principais variáveis em relação ao custo referencial unitário foram o custo da pressurização inicial, custo da tubulação, além do custo de remoção e recomposição de viário em pavimento asfáltico. / This work presents the optimal design of pressurized networks of water distribution in steady state flow to areas of relatively flat topography. In addition to the traditional hydraulic criteria for design, the optimal outcome is that with lower overall cost, in other words, where the sum of the costs of implementation and operation is minimal. To obtain the results, the equations that involving headloss in the circuits and flows at each node were solved by nonlinear programming with the use of matrix methods. The main variables studied were the different materials of pipes (HDPE, PVC and Cast Iron), surface types (clay, concrete and asphalt paving), network location (walk, paved and unpaved road), electricity tariffs for water utilities, normal life for the pumping system and interest rates along the system operation. The results obtained by the proposed methodology showed that the main variables in relation to the unit cost were the cost of initial pressurization, cost of the pipe, besides the cost of removal and restoration of roads in asphalt pavement.