Optimization governed by stochastic partial differential equations

This thesis provides a rigorous framework for the solution of stochastic elliptic partial differential equation (SPDE) constrained optimization problems. In modeling physical processes with differential equations, much of the input data is uncertain (e.g. measurement errors in the diffusivity coefficients). When uncertainty is present, the governing equations become a family of equations indexed by a stochastic variable. Since solutions of these SPDEs enter the objective function, the objective function usually involves statistical moments. These optimization problems governed by SPDEs are posed as a particular class of optimization problems in Banach spaces. This thesis discusses Monte Carlo, stochastic Galerkin, and stochastic collocation methods for the numerical solution of SPDEs and identifies the stochastic collocation method as particularly useful for the optimization of SPDEs. This thesis extends the stochastic collocation method to the optimization context and explores the decoupling nature of this method for gradient and Hessian computations

A Risk Management Perspective on Statistical Estimation and Generalized Variational Inference

Generalized variational inference (GVI) provides an optimization-theoretic framework for statistical estimation that encapsulates many traditional estimation procedures. The typical GVI problem is to compute a distribution of parameters that maximizes the expected payoff minus the divergence of the distribution from a specified prior. In this way, GVI enables likelihood-free estimation with the ability to control the influence of the prior by tuning the so-called learning rate. Recently, GVI was shown to outperform traditional Bayesian inference when the model and prior distribution are misspecified. In this paper, we introduce and analyze a new GVI formulation based on utility theory and risk management. Our formulation is to maximize the expected payoff while enforcing constraints on the maximizing distribution. We recover the original GVI distribution by choosing the feasible set to include a constraint on the divergence of the distribution from the prior. In doing so, we automatically determine the learning rate as the Lagrange multiplier for the constraint. In this setting, we are able to transform the infinite-dimensional estimation problem into a two-dimensional convex program. This reformulation further provides an analytic expression for the optimal density of parameters. In addition, we prove asymptotic consistency results for empirical approximations of our optimal distributions. Throughout, we draw connections between our estimation procedure and risk management. In fact, we demonstrate that our estimation procedure is equivalent to evaluating a risk measure. We test our procedure on an estimation problem with a misspecified model and prior distribution, and conclude with some extensions of our approach

Differentiating Plasmodium falciparum alleles by transforming Cartesian X,Y data to polar coordinates

<p>Abstract</p> <p>Background</p> <p>Diagnosis of infectious diseases now benefits from advancing technology to perform multiplex analysis of a growing number of variables. These advances enable simultaneous surveillance of markers characterizing species and strain complexity, mutations associated with drug susceptibility, and antigen-based polymorphisms in relation to evaluation of vaccine effectiveness. We have recently developed assays detecting single nucleotide polymorphisms (SNPs) in the <it>P. falciparum </it>genome that take advantage of post-PCR ligation detection reaction and fluorescent microsphere labeling strategies. Data from these assays produce a spectrum of outcomes showing that infections result from single to multiple strains. Traditional methods for distinguishing true positive signal from background can cause false positive diagnoses leading to incorrect interpretation of outcomes associated with disease treatment.</p> <p>Results</p> <p>Following analysis of <it>Plasmodium falciparum </it>dihydrofolate reductase SNPs associated with resistance to a commonly used antimalarial drug, Fansidar (Sulfadoxine/pyrimethamine), and presumably neutral SNPs for parasite strain differentiation, we first evaluated our data after setting a background signal based on the mean plus three standard deviations for known negative control samples. Our analysis of single allelic controls suggested that background for the absent allele increased as the concentration of the target allele increased. To address this problem, we introduced a simple change of variables from customary (<it>X,Y</it>) (Cartesian) coordinates to planar polar coordinates (<it>X </it>= <it>r</it>cos(<it>θ</it>), <it>Y </it>= <it>r</it>sin(<it>θ</it>)). Classification of multidimensional fluorescence signals based on histograms of angular and radial data distributions proved more effective than classification based on Cartesian thresholds. Comparison with known diallelic dilution controls suggests that histogram-based classification is effective for major:minor allele concentration ratios as high as 10:1.</p> <p>Conclusion</p> <p>We have observed that the diallelic SNP data resulting from analysis of <it>P. falciparum </it>mutations is more accurately diagnosed when a simple polar transform of the (<it>X,Y</it>) data into (<it>r,θ</it>) is used. The development of high through-put methods for genotyping <it>P. falciparum </it>SNPs and the refinement of analytical approaches for evaluating these molecular diagnostic results significantly advance the evaluation of parasite population diversity and antimalarial drug resistance.</p

A Multiplex Ligase Detection Reaction-Fluorescent Microsphere Assay for Simultaneous Detection of Single Nucleotide Polymorphisms Associated with Plasmodium falciparum Drug Resistance

Incomplete malaria control efforts have resulted in a worldwide increase in resistance to drugs used to treat the disease. A complex array of mutations underlying antimalarial drug resistance complicates efficient monitoring of parasite populations and limits the success of malaria control efforts in regions of endemicity. To improve the surveillance of Plasmodium falciparum drug resistance, we developed a multiplex ligase detection reaction-fluorescent-microsphere-based assay (LDR-FMA) that identifies single nucleotide polymorphisms (SNPs) in the P. falciparum dhfr (9 alleles), dhps (10 alleles), and pfcrt (3 alleles) genes associated with resistance to Fansidar and chloroquine. We evaluated 1,121 blood samples from study participants in the Wosera region of Papua New Guinea, where malaria is endemic. Results showed that 468 samples were P. falciparum negative and 453 samples were P. falciparum positive by a Plasmodium species assay and all three gene assays (concordance, 82.2%). For P. falciparum infections where the assay for each gene was positive, 2 samples carried resistance alleles for all three genes, 299 carried resistance alleles for dhfr and pfcrt, 131 carried resistance alleles for only one gene (dhfr [n = 40], dhps [n = 1], or pfcrt [n = 90]), and 21 carried only sensitive alleles at all three genes. Mixed-strain infections characterized 100 samples. Overall, 95.4% (432/453) of P. falciparum-infected samples carried at least one allele associated with resistance to Fansidar or chloroquine. In view of the fact that 86.3% (391/453) of P. falciparum-infected samples carried pfcrt mutations, chloroquine is largely ineffective against P. falciparum in Papua New Guinea. Surveillance of additional dhfr and dhps polymorphisms in order to monitor the continued effectiveness of Fansidar is recommended