R-3 Repository :: Browsing by Author "Tapia, Richard A."

Browsing by Author "Tapia, Richard A."

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A clustering algorithm for university admissions
(2007) Reed, Naomi Beth; Tapia, Richard A.
In 2003 The Supreme Court declared that all government funded universities, which choose to consider race in their admissions processes, must utilize a holistic process. A holistic process includes a thorough evaluation of all aspects of each applicant. For larger universities this type of admissions process would be very taxing. A computer scientist from Auburn University created an algorithm, Applications Quest, to handle large quantities of applications in a way that would evaluate applicants holistically with a computational tool. Applications Quest utilizes the Euclidean distance measure, Similarity matrices, Divisive Clustering, and Random Selection. This algorithm produces a diverse admittance class for a university. In this research we simulate this algorithm and run tests with hypothetical Rice University data. Ultimately, we are left with the following question: Can a computational use of arbitrary difference account for human qualities that define certain social phenomena, such as underrepresentation in higher education?
A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization
(1983-05) Fontecilla, Rodrigo; Steihaug, Trond; Tapia, Richard A.
In this paper we develop a general convergence theory for a class of quasi-Newton methods for equality constrained optimization. The theory is set in the framework of the diagonalized multiplier method defined by Tapia and is an extension of the theory developed by Glad. We believe that this framework is flexible and amenable to convergence analysis and generalizations. A key ingredient of a method in this class is a multiplier update. Our theory is tested by showing that a straightforward application gives the best known convergence results for several known multiplier updates. Also a characterization of q-superlinear convergence is presented. It is shown that in the special case when the diagonalized multiplier method is equivalent to the successive quadratic programming approach, our general characterization result gives the Boggs, Tolle and Wang characterization.
A global convergence theory for a class of trust region algorithms for constrained optimization
(1988) El-Alem, Mahmoud Mahmoud; Tapia, Richard A.; Dennis, John E., Jr.
In this research we present a trust region algorithm for solving the equality constrained optimization problem. This algorithm is a variant of the 1984 Celis-Dennis-Tapia algorithm. The augmented Lagrangian function is used as a merit function. A scheme for updating the penalty parameter is presented. The behavior of the penalty parameter is discussed. We present a global and local convergence analysis for this algorithm. We also show that under mild assumptions, in a neighborhood of the minimizer, the algorithm will reduce to the standard SQP algorithm; hence the local rate of convergence of SQP is maintained. Our global convergence theory is sufficiently general that it holds for any algorithm that generates steps that give at least a fraction of Cauchy decrease in the quadratic model of the constraints.
A Global Optimization Method for the Molecular Replacement Problem in X-ray Crystallography
(2002-06) Jamrog, Diane C.; Phillips, George N. Jr.; Tapia, Richard A.; Zhang, Yin
The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein, whose structure is similar to the unknown protein structure that is to be determined, so that at this position the model protein will produce calculated intensities closest to those observed from an X-ray crystallography experiment. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems. We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved a MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.
A global optimization technique for zero-residual nonlinear least-squares problems
(2000) Velazquez Martinez, Leticia; Tapia, Richard A.
This thesis introduces a globalization strategy for approximating global minima of zero-residual least-squares problems. This class of nonlinear programming problems arises often in data-fitting applications in the fields of engineering and applied science. Such minimization problems are formulated as a sum of squares of the errors between the calculated and observed values. In a zero-residual problem at a global solution, the calculated values from the model matches exactly the known data. The presence of multiple local minima is the main difficulty. Algorithms tend to get trapped at local solutions when applied to these problems. The proposed algorithm is a combination of a simple random sampling, a Levenberg-Marquardt-type method, a scaling technique, and a unit steplength. The key component of the algorithm is that a unit steplength is used. An interesting consequence is that this approach is not attracted to non-degenerate saddle points or to large-residual local minima. Numerical experiments are conducted on a set of zero-residual problems, and the numerical results show that the new multi-start strategy is relatively more effective and robust than some other global optimization algorithms.
A historical development of the (n+1)-point secant method
(2007) Papakonstantinou, Joanna Maria; Tapia, Richard A.
Many finite-dimensional minimization problems and nonlinear equations can be solved using Secant Methods. In this thesis, we present a historical development of the (n + 1)-point Secant Method tracing its evolution back before Newton's Method. Many believe the Secant Method arose out of the finite difference approximation of the derivative in Newton's Method. However, historical evidence reveals that the Secant Method predated Newton's Method by more than 3000 years, and it was most commonly referred to as the Rule of Double False Position. The history of the Rule of Double False Position spans a period of several centuries and many civilizations. We describe the Rule of Double False Position and compare and contrast the Secant Method in 1-D with the Regula Falsi Method. We delineate the extension of the 1-D Secant Method to higher dimensions using two viewpoints, the linear interpolation idea and Discretized Newton Methods.
A Journey through the Arabidopsis thaliana Genome: Discovering the Origins of Novel Triterpene Metabolites
(2014-08-27) Castillo-Rivera, Dorianne A; Matsuda, Seiichi P. T.; Hartgerink, Jeffrey D.; Tapia, Richard A.
Plants produce a large variety of natural products, including over 20,000 different triterpenoids. Triterpenoid functions range from roles as membrane sterols and hormones in primary metabolism, to defense compounds in secondary metabolism. Beyond these roles in the plant, triterpenoids have substantial human value as flavors, fragrances, and medicines. This thesis explores the enzymatic formation of triterpenoids through cationic cyclization of a linear precursor oxidosqualene and further metabolism by radical oxidation. Thirteen different oxidosqualene cyclases (OSCs) are encoded by the model plant Arabidopsis thaliana and collectively produce a plethora of triterpene skeletons. This structural diversity of triterpenes was investigated through a comprehensive analysis of the A. thaliana PEN6 product profile. This product profile contained 33 compounds that were found by combining genome mining, heterologous expression in yeast and HSQC analysis. Some of these compounds were novel to Arabidopsis: isoursenol, (13R,14Z,17E)-malabarica-14,17,21-trien-3β-ol, nematocyphol, (20R,S) dammarenediols, Δ8(26)-seco-β-amyrin, and 9αH-Δ8(26)-polypodatetraenol. After the cyclization, triterpenes can be modified by a number of enzymes, including cytochrome P450 monooxygenases (CYP450s). Candidate CYP450s for a given triterpene were identified by gene cluster analysis combined with microarray databases. Heterologous expression of the OSC, ATR2 and CYP450 together with GC-MS and NMR techniques allowed the elucidation of metabolic pathways and structures of a variety of oxygenated triterpenes. These experimental techniques led to the identification of new oxidized metabolites produced by the co-expression of Arabidopsis clusters: THAS1 with CYP708A2 and CYP705A5, PEN1 with CYP705A2 and MRN1 with CYP71A16. The thalianol cluster gave rise to several unexpected side chain and ring oxidized metabolites. This thesis also describes the side chain cleavage of arabidiol by CYP705A1 to give a C19 methyl ketone, and the hydroxylation of an allylic methyl of marneral/marnerol to 23-hydroxymarneral/23-hydroxymarnerol by CYP71A16. This work sheds light on the metabolic fate of some Arabidopsis triterpenes. When applied more generally, this strategy may begin to fill a large knowledge gap in metabolomics and functional genomics.
A Matter of Perspective: Reliable Communication and Coping with Interference with Only Local Views
(2012-09-05) Kao, David; Sabharwal, Ashutosh; Aazhang, Behnaam; Knightly, Edward W.; Tapia, Richard A.; Chiang, Mung
This dissertation studies interference in wireless networks. Interference results from multiple simultaneous attempts to communicate, often between unassociated sources and receivers, preventing extensive coordination. Moreover, in practical wireless networks, learning network state is inherently expensive, and nodes often have incomplete and mismatched views of the network. The fundamental communication limits of a network with such views is unknown. To address this, we present a local view model which captures asymmetries in node knowledge. Our local view model does not rely on accurate knowledge of an underlying probability distribution governing network state. Therefore, we can make robust statements about the fundamental limits of communication when the channel is quasi-static or the actual distribution of state is unknown: commonly faced scenarios in modern commercial networks. For each local view, channel state parameters are either perfectly known or completely unknown. While we propose no mechanism for network learning, a local view represents the result of some such mechanism. We apply the local view model to study the two-user Gaussian interference channel: the smallest building block of any interference network. All seven possible local views are studied, and we find that for five of the seven, there exists no policy or protocol that universally outperforms time-division multiplexing (TDM), justifying the orthogonalized approach of many deployed systems. For two of the seven views, TDM-beating performance is possible with use of opportunistic schemes where opportunities are revealed by the local view. We then study how message cooperation --- either at transmitters or receivers --- increases capacity in the local view two-user Gaussian interference channel. The cooperative setup is particularly appropriate for modeling next-generation cellular networks, where costs to share message data among base stations is low relative to costs to learn channel coefficients. For the cooperative setting, we find: (1) opportunistic approaches are still needed to outperform TDM, but (2) opportunities are more abundant and revealed by more local views. For all cases studied, we characterize the capacity region to within some known gap, enabling computation of the generalized degrees of freedom region, a visualization of spatial channel resource usage efficiency.
A modified augmented Lagrangian merit function, and Q-superlinear characterization results for primal-dual Quasi-Newton interior-point method for nonlinear programming
(1997) Paroda Garcia, Zeferino; Tapia, Richard A.
Two classes of primal-dual interior-point methods for nonlinear programming are studied. The first class corresponds to a path-following Newton method formulated in terms of the nonnegative variables rather than all primal and dual variables. The centrality condition is a relaxation of the perturbed Karush-Kuhn-Tucker condition and primarily forces feasibility in the constraints. In order to globalize the method using a linesearch strategy, a modified augmented Lagrangian merit function is defined in terms of the centrality condition. The second class is the Quasi-Newton interior-point methods. In this class the well known Boggs-Tolle-Wang characterization of Q-superlinear convergence for Quasi-Newton method for equality constrained optimization is extended. Critical issues in this extension are; the choice of the centering parameter, the choice of the steplength parameter, and the choice of the primary variables.
A new global optimization strategy for the molecular replacement problem
(2002) Jamrog, Diane Christine; Zhang, Yin; Phillips, George N., Jr.; Tapia, Richard A.
The primary technique for determining the three-dimensional structure of a protein is X-ray crystallography, in which the molecular replacement (MR) problem arises as a critical step. Knowledge of protein structures is extremely useful for medical research, including discovering the molecular basis of disease and designing pharmaceutical drugs. This thesis proposes a new strategy to solve the MR problem, which is a global optimization problem to find the optimal orientation and position of a structurally similar model protein that will produce calculated intensities closest to those observed from an X-ray crystallography experiment. Improving the applicability and the robustness of MR methods is an important research goal because commonly used traditional MR methods, though often successful, have difficulty solving certain classes of MR problems. Moreover, the use of MR methods is only expected to increase as more structures are deposited into the Protein Data Bank. The new strategy has two major components: a six-dimensional global search and multi-start local optimization. The global search uses a low-frequency surrogate objective function and samples a coarse grid to identify good starting points for multi-start local optimization, which uses a more accurate objective function. As a result, the global search is relatively quick and the local optimization efforts are focused on promising regions of the MR variable space where solutions are likely to exist, in contrast to the traditional search strategy that exhaustively samples a uniformly fine grid of the variable space. In addition, the new strategy is deterministic, in contrast to stochastic search methods that randomly sample the variable space. This dissertation introduces a new MR program called SOMoRe that implements the new global optimization strategy. When tested on seven problems, SOMoRe was able to straightforwardly solve every test problem, including three problems that could not be directly solved by traditional MR programs. Moreover, SOMoRe also solved a MR problem using a less complete model than those required by two traditional programs and a stochastic 6D program. Based on these results, this new strategy promises to extend the applicability and robustness of MR.
A non-linear elliptic problem arising in petroleum engineering
(1971) Mayor de Montricher, Gilbert Franz; Tapia, Richard A.
The problem of determining the temperature distribution of a body heated by radiation is formulated as a variational nonlinear partial differential equation. An existence and uniqueness theorem is proved for this problem. It is shown that Newton's method applied to this nonlinear problem gives a sequence of linear Dirichlet problems. Convergence results for Newton's method are also derived.
A robust choice of the Lagrange multipliers in the successive quadratic programming method
(1994) Cores-Carrera, Debora; Tapia, Richard A.
We study the choice of the Lagrange multipliers in the successive quadratic programming method (SQP) applied to the equality constrained optimization problem. It is known that the augmented Lagrangian SQP-Newton method depends on the penalty parameter only through the multiplier in the Hessian matrix of the Lagrangian function. This effectively reduces the augmented Lagrangian SQP-Newton method to the Lagrangian SQP Newton method where only the multiplier estimate depends on the penalty parameter. In this work, we construct a multiplier estimate that depends strongly on the penalty parameter and we derive a choice for the penalty parameter so that the Hessian matrix, restricted to the null space of the constraints, is positive definite and well conditioned. We demonstrate that the SQP-Newton method with this choice of Lagrange multipliers is locally and q-quadratically convergent.
A study of university timetabling that blends graph coloring with the satisfaction of various essential and preferential conditions
(2004) Redl, Timothy Anton; Dean, Nathaniel; Tapia, Richard A.
Constructing a satisfactory conflict-free semester-long timetable of courses and creating a similarly satisfactory conflict-free timetable for end-of-semester final examinations are two closely related and often difficult problems that colleges and universities face each semester. We discuss the relevance of such timetabling problems as a natural and practical application of graph coloring, and develop a mathematical and computational model for solving university timetabling problems using techniques of graph coloring that incorporates the satisfaction of both "essential" timetabling conditions (i.e., conditions or constraints that must be satisfied in order to produce a legal or feasible timetable) as well as suggested "preferential" timetabling conditions (i.e., additional conditions or constraints that need not necessarily be satisfied to produce a legal or legitimate timetable, but if satisfied may very well produce a more "acceptable" timetable for students and/or faculty members). We discuss in detail the step-by-step process that is taken to implement our timetabling-by-graph-coloring procedure, from the assembling of university course data, to creating a course conflict graph based on the assembled data, to coloring the conflict graph, to transforming this coloring to a conflict-free timetable, to finally assigning courses to classrooms. Once a conflict-free timetable of courses has been constructed, we present ways in which such a course timetable for a particular semester can be used to construct a conflict-free timetable of final examinations. Our model also considers a number of sociological scheduling concerns and preferences addressed by university registrars, faculty, staff, and students. Computational results, obtained by the author using actual data provided by Rice University and the University of St. Thomas, are documented.
Advanced computational techniques for incompressible/compressible fluid-structure interactions
(2005) Kumar, Vinod; Tapia, Richard A.; Barrera, Enrique V.
Fluid-Structure Interaction (FSI) problems are of great importance to many fields of engineering and pose tremendous challenges to numerical analyst. This thesis addresses some of the hurdles faced for both 2D and 3D real life time-dependent FSI problems with particular emphasis on parachute systems. The techniques developed here would help improve the design of parachutes and are of direct relevance to several other FSI problems. The fluid system is solved using the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation for the Navier-Stokes equations of incompressible and compressible flows. The structural dynamics solver is based on a total Lagrangian finite element formulation. Newton-Raphson method is employed to linearize the otherwise nonlinear system resulting from the fluid and structure formulations. The fluid and structural systems are solved in decoupled fashion at each nonlinear iteration. While rigorous coupling methods are desirable for FSI simulations, the decoupled solution techniques provide sufficient convergence in the time-dependent problems considered here. In this thesis, common problems in the FSI simulations of parachutes are discussed and possible remedies for a few of them are presented. Further, the effects of the porosity model on the aerodynamic forces of round parachutes are analyzed. Techniques for solving compressible FSI problems are also discussed. Subsequently, a better stabilization technique is proposed to efficiently capture and accurately predict the shocks in supersonic flows. The numerical examples simulated here require high performance computing. Therefore, numerical tools using distributed memory supercomputers with message passing interface (MPI) libraries were developed.
An abstract analysis of differential semblance optimization
(1994) Gockenbach, Mark Steven; Symes, William W.; Tapia, Richard A.
Differential Semblance Optimization (DSO) is a novel way of approaching a class of inverse problems arising in exploration seismology. The promising feature of the DSO method is that it replaces a nonsmooth, highly nonconvex cost function (the Output Least-Squares (OLS) objective function) with a smooth cost function that is amenable to standard (local) optimization algorithms. The OLS problem can be written abstractly as a partially linear least-squares problem with linear constraints. The DSO objective function is derived from the associated quadratic penalty function. It is shown that one way to view the DSO objective function is as a regularization of a function that is dual (in a certain sense) to the OLS objective function. By viewing the DSO problem as a perturbation of this dual problem, this method can be shown to be effective. In particular, it is demonstrated that, under suitable assumptions, the DSO method defines a parameterized path of minimizers converging to the desired solution, and that for certain values of the parameter, standard optimization techniques can be used to find a point on the path. The predictions of the theory are motivated and illustrated on two simple model problems for seismic velocity inversion, the plane wave detection problem and the "layer-over-half-space" problem. It is shown that the theory presented in this thesis extends the existing theory for the plane wave detection problem.
An introduction to linear algebra: A curricular unit for pre-calculus students
(1995) Anthony, Tamara Lynn; Tapia, Richard A.
Matrices are important mathematical tools that facilitate the process of organizing and manipulating data. In this work, the matrix operations of addition, subtraction, scalar multiplication, and matrix multiplication are built logically from the intuition of the students and their knowledge of real numbers. From this knowledge, the concepts of inverses, determinants, and consistency and inconsistency of linear systems of equations are formed. Interesting applications of matrices in the areas of Markov chains, curve fitting, and eigenpairs are included and are not beyond the comprehension of pre-calculus students when they are presented carefully. Pre-calculus students can also appreciate many of the numerical challenges that can be encountered when real-world problems are solved; therefore, we include a discussion of some of these topics.
Bayesian decision-theoretic method and semi-parametric approach with applications in clinical trial designs and longitudinal studies
(2013-11-25) Jiang, Fei; Lee, J. Jack; Cox, Dennis D.; Scott, David W.; Ma, Yanyuan; Tapia, Richard A.
The gold of biostatistical researches is to develop statistical tools that improves human health or increases understanding of human biology. One area of the studies focuses on designing clinical trials to find out if new drugs or treatments are efficacious. The other area focuses on studying diseases related variables, which gives better understanding of the diseases. The thesis explores these areas from both theoretical and practical points of views. In addition, the thesis develop statistical devices which improve the existing methods in these areas. Firstly, the thesis proposes a Bayesian decision-theoretic group sequential – adaptive randomization phase II clinical trial design. The design improves the trial efficiency by increasing statistical power and reducing required sample sizes. The design also increases patients’ individual benefit, because it enhances patients’ opportunities of receiving better treatments. Secondly, the thesis develops a semiparametric restricted moment model and a score imputation estimation for survival analysis. The method is more robust than the parametric alternatives. In addition to data analysis, the method is used to design a seamless phase II/III clinical trial. The seamless phase II/III clinical trial design shortens the durations between phase II and III studies, and improves the efficiency of the traditional designs by utilizing additional short term information for making decisions. Finally, the thesis develops a partial linear time varying semi-parametric single-index risk score model and a fused B-spline/kernel estimation for longitudinal data analysis. The method models confounder effects linearly. In addition, it uses a nonparametric nonlinear function, namely the single-index risk score, to model the effects of interests. The fused B-spline/kernel technique estimates both the parametric and nonparametric components consistently. The methodology is applied to study the onsite of Huntington’s disease in determining certain time varying covariate effects on the disease risk.
Block Coordinate Descent for Regularized Multi-convex Optimization
(2013-09-16) Xu, Yangyang; Yin, Wotao; Tapia, Richard A.; Zhang, Yin; Baraniuk, Richard G.
This thesis considers regularized block multi-convex optimization, where the feasible set and objective function are generally non-convex but convex in each block of variables. I review some of its interesting examples and propose a generalized block coordinate descent (BCD) method. The generalized BCD uses three different block-update schemes. Based on the property of one block subproblem, one can freely choose one of the three schemes to update the corresponding block of variables. Appropriate choices of block-update schemes can often speed up the algorithm and greatly save computing time. Under certain conditions, I show that any limit point satisfies the Nash equilibrium conditions. Furthermore, I establish its global convergence and estimate its asymptotic convergence rate by assuming a property based on the Kurdyka-{\L}ojasiewicz inequality. As a consequence, this thesis gives a global linear convergence result of cyclic block coordinate descent for strongly convex optimization. The proposed algorithms are adapted for factorizing nonnegative matrices and tensors, as well as completing them from their incomplete observations. The algorithms were tested on synthetic data, hyperspectral data, as well as image sets from the CBCL, ORL and Swimmer databases. Compared to the existing state-of-the-art algorithms, the proposed algorithms demonstrate superior performance in both speed and solution quality.
Comparison of Two Sets of First-order Conditions as Bases of Interior-Point Newton Methods for Optimization with Simple Bounds
(2001-06) Jamrog, Diane C.; Tapia, Richard A.; Zhang, Yin
In this paper, we compare the behavior of two Newton interior-point methods derived from two different first-order necessary conditions for the same nonlinear optimization problem with simple bounds. One set of conditions was proposed by Coleman and Li; the other is the standard KKT set of conditions. We discuss a perturbation of the CL conditions for problems with one-sided bounds and the difficulties involved in extending this to problems with general bounds. We study the numerical behavior of the Newton method applied to the systems of equations associated with the unperturbed and perturbed necessary conditions. Preliminary numerical results for convex quadratic objective functions indicate that, for this class of problems, Newton's method based on the perturbed KKT formulation appears to be the most robust.
Computational science: Identifying and explaining the mathematical computational methods used by the TI-83 calculator
(2002) Barra, Emily Caroline; Tapia, Richard A.
This paper looks in detail at the algorithms used by the Texas Instruments TI-83 calculator to calculate the square root of x, the solution to Ax = b and the trigonometric functions sine and cosine. What is exciting about these algorithms is that for various reasons, TI uses techniques that are not necessarily well-known or frequently used by today's numerical analysis community. I have written the explanations of these algorithms in such a way that high school math teachers as well as bright high school students searching for enrichment may use them.

Browsing by Author "Tapia, Richard A."

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