We are examining iteratively regularized methods for solving nonlinear inverse problems. Of particular interest for these types of methods are application problems which are unstable. For these application problems, special methods of numerical analysis are necessary, since classical algorithms tend to be divergent.

A Wald type test with the wrong dispersion matrix is used when the dispersion matrix is not a consistent estimator of the asymptotic covariance matrixof the test statistic. One class of such tests occurs when there are k groups and it is assumed that the population covariance matrices from the k groups are equal, but the common covariance matrix assumption does not hold. The pooled t test, one way AVOVA F test, and one way MANOVA F test are examples of this class. Two bootstrap confidence regions are modified to obtain large sample Wald type tests with the wrong dispersion matrix.

As an alternative to spherical harmonics in modeling the gravity field of the Earth, we built a multiresolution gravity model by employing spherical regularization wavelets in solving the inverse problem, i.e. downward propagation of the gravity signal to the Earth.s surface. Scale discrete Tikhonov spherical regularization scaling function and wavelet packets were used to decompose and reconstruct the signal. We recovered the local gravity anomaly using only localized gravity measurements at the observing satellite.s altitude of 300 km. When the upward continued gravity anomaly to the satellite altitude with a resolution 0.5° was used as simulated measurement inputs, our model could recover the local surface gravity anomaly at a spatial resolution of 1° with an RMS error between 1 and 10 mGal, depending on the topography of the gravity field. Our study of the effect of varying the data volume and altering the maximum degree of Legendre polynomials on the accuracy of the recovered gravity solution suggests that the short wavelength signals and the regions with high magnitude gravity gradients respond more strongly to such changes. When tested with simulated SGG measurements, i.e. the second order radial derivative of the gravity anomaly, at an altitude of 300 km with a 0.7° spatial resolution as input data, our model could obtain the gravity anomaly with an RMS error of 1 ~ 7 mGal at a surface resolution of 0.7° (< 80 km). The study of the impact of measurement noise on the recovered gravity anomaly implies that the solutions from SGG measurements are less susceptible to measurement errors than those recovered from the upward continued gravity anomaly, indicating that the SGG type mission such as GOCE would be an ideal choice for implementing our model. Our simulation results demonstrate the model.s potential in determining the local gravity field at a finer scale than could be achieved through spherical harmonics, i.e. less than 100 km, with excellent performance in edge detection. / text

No description available.

Genome-wide association studies (GWAS) has played an import role in identifying genetic variants underlying human complex traits. However, its success is hindered by weak effect at causal variants and noise at non-causal variants. Penalized regression can be applied to handle GWAS problems. GWAS data has some specificities. Consecutive genetic markers are usually highly correlated due to linkage disequilibrium. This thesis introduces a moving-window penalized method for GWAS which smooths the effects of consecutive SNPs. Simulation studies indicate that this penalized moving window method provides improved true positive findings. The practical utility of the proposed method is demonstrated by applying it to Genetic Analysis Workshop 16 Rheumatoid Arthritis data. Next, the moving-window penalty is applied on generalized linear model. We call such an approach as smoothed lasso (SLasso). Coordinate descent computing algorithms are proposed in details, for both quadratic and logistic loss. Asymptotic properties are discussed. Then based on SLasso, we discuss a two-stage method called MW-Ridge. Simulation results show that while SLasso can provide more true positive findings than Lasso, it has a side-effect that it includes more unrelated random noises. MW-Ridge can eliminate such a side-effect and result in high true positive rates and low false detective rates. The applicability to real data is illustrated by using GAW 16 Rheumatoid Arthritis data. The SLasso and MW-Ridge approaches are then generalized to multivariate response data. The multivariate response data can be transformed into univariate response data. The causal variants are not required to be the same for different response variables. We found that no matter how the causal variants are matched, being fully matched or 60% matched, MW-Ridge can always over perform Lasso by detecting all true positives with lower false detective rates.

Image restoration problems deal with images in which information has been degraded by blur or noise. In practice, the blur is usually caused by atmospheric turbulence, motion, camera shake, and several other mechanical or physical processes. In this study, we present two regularization algorithms for the image deblurring problem. We first present a new method based on solving a regularized least-squares (RLS) problem. This method is proposed to find a near-optimal value of the regularization parameter in the RLS problems. Experimental results on the non-blind image deblurring problem are presented. In all experiments, comparisons are made with three benchmark methods. The results demonstrate that the proposed method clearly outperforms the other methods in terms of both the output PSNR and structural similarity, as well as the visual quality of the deblurred images. To reduce the complexity of the proposed algorithm, we propose a technique based on the bootstrap method to estimate the regularization parameter in low and high-resolution images. Numerical results show that the proposed technique can effectively reduce the computational complexity of the proposed algorithms. In addition, for some cases where the point spread function (PSF) is separable, we propose using a Kronecker product so as to reduce the computations. Furthermore, in the case where the image is smooth, it is always desirable to replace the regularization term in the RLS problems by a total variation term. Therefore, we propose a novel method for adaptively selecting the regularization parameter in a so-called square root regularized total variation (SRTV). Experimental results demonstrate that our proposed method outperforms the other benchmark methods when applied to smooth images in terms of PSNR, SSIM and the restored image quality. In this thesis, we focus on the non-blind image deblurring problem, where the blur kernel is assumed to be known. However, we developed algorithms that also work in the blind image deblurring. Experimental results show that our proposed methods are robust enough in the blind deblurring and outperform the other benchmark methods in terms of both output PSNR and SSIM values.

The Sherman Morrison iteration method is developed to solve regularized least squares problems. Notions of pivoting and splitting are deliberated on to make the method more robust. The Sherman Morrison iteration method is shown to be effective when dealing with an extremely underdetermined least squares problem. The performance of the Sherman Morrison iteration is compared to classic direct methods, as well as iterative methods, in a number of experiments. Specific Matlab implementation of the Sherman Morrison iteration is discussed, with Matlab codes for the method available in the appendix. / Master of Science

ABSTRACT Background: Alcohol use disorders (AUD) affect health and wellbeing, and have broad societal costs (Bouchery, Harwood, Sacks, Simon, & Brewer, 2011; Rehm et al., 2009; Sudhinaraset, Wigglesworth, Takeuchi, & Tsuker, 2016). While treatments have existed for decades, they are limited in success and expensive to administer. As such, understanding which factors best predict who will benefit most from treatment remains a laudable goal. Prior attempts to predict factors associated with positive treatment outcome are limited by methodology including statistical methods that lead to poor predictive power in new samples. This study aims to use a data-driven approach to clarify the predictors of AUD treatment success (Objective 1) accompanied by a theory-driven analysis assessing the mediation of treatment outcomes through psychological distress (Objective 2). Methods: One hundred forty-five patients seeking treatment for alcohol use problems at the Day One Intensive Outpatient Treatment Program (part of UVM Medical Center) between June 2011 and June 2012 were examined. Variables were extracted through chart review and were categorized using the Bronfenbrenner Ecological Model. First, 20% of the sample was set-aside for model testing, and the remaining 80% was used in an Elastic Net Regularized linear regression, with 10-fold cross validation. Models were tested on the set-aside sample to yield estimates of out-of-sample prediction and repeated models were compared to ensure generalizability. Next, a theoretical model was tested examining a model of psychological distress mediating the relationship between individual predictors and treatment outcome. Results: The models developed from the Elastic Net Regularization approach demonstrated consistency in model strength (mean=0.32, standard deviation=0.03) with models ranging from 14 to 31 included variables. Across the models, 15 variables occurred in >75% of the models, and an additional 7 variables were included in 25% - 75% of the models. Some of the strongest predictors included treatment non-compliance (β=-0.92), ASI Alcohol Composite (β=0.63), treatment dosage (β =-0.36), and readiness to change (β=-0.95). The results of the theory-driven mediation analysis demonstrated several strong direct predictors of outcome frequency of alcohol use, including readiness to change (β=-0.59), initial frequency of alcohol use (β=0.27), and access to a primary care physician (β=-2.20). The theoretical model found that none of the mediation pathways (testing psychological variables) were significantly different from the direct models. Conclusions: This study used both data-driven and theory-driven methods to examine factors affecting treatment of AUDs. The application of data-driven methods provided several predictors of outcome that can guide treatment efforts within Day One IOP treatment, as well as generalized to other abstinence-based treatment settings. For example, focusing on treatment attendance and using motivational interviewing to enhance readiness to change are methods supported by this study. Demographic variables that have been shown to predict treatment outcome in small studies, without cross-validation were not identified by the elastic net regression (e.g., age and gender). It is suspected that this is due to model overfitting in prior studies supporting the importance of using generalizable statistical methods to understand predictors of treatment outcome. This notion is supported by the results of the theory-driven model, which did not yield a strong model of treatment success. Taken together, the results support the use of strong analytic techniques which will guide theory in the future.

he focus of this research has been devoted to study the interaction between two or more self-propelled toroidal swimmers in Stokes flow by applying the method of regularized Stokeslets and also study the effect of a nearby wall to the movement of a helical ring by using the method of regurlarized Stokeslets with images. In the study of the interaction between two or more toroidal swimmers, we interpret these as three-dimensional, zero Reynolds number analogues of finite vortex dipoles in an ideal fluid. Then, we examine the stability of relative equilibria that can form for these swimmers when they are initially placed in tandem or abreast. In addition, we examine the dynamics of the torus when a spherical cell body is placed at its center. This gives us an insight into the mechanical role of the transverse flagellum of dinoflagellates. Moreover, we show that the torus with a sphere moves more efficiently than one without. Lastly, we model the transverse flagellum of a dinoflagellate as a helical ring and study the effect of a nearby wall on its movement. The numerical results show that the wall baffles the movement of the helical ring, which is consistent with the phenomenon of sperm accumulation near surfaces. / acase@tulane.edu

It is well established that non-equilibrium flows cannot properly be described by traditional hydrodynamics, namely, the Navier-Stokes-Fourier (NSF) equations. Such flows occur, for example, in micro-electro-mechanical systems (MEMS), and ultra vacuum systems, where the dimensions of the devices are comparable to the mean free path of a gas molecule. Therefore, the study of non-equilibrium effects in gas flows is extremely important. The general interest of the present study is to explore boundary value problems for moderately rarefied gas flows, with an emphasis on numerical solutions of the regularized 13--moment equations (R13). Boundary conditions for the moment equations are derived based on either phenomenological principles or on microscopic gas-surface scattering models, e.g., Maxwell's accommodation model and the isotropic scattering model. Using asymptotic analysis, several non-linear terms in the R13 equations are transformed into algebraic terms. The reduced equations allow us to obtain numerical solutions for multidimensional boundary value problems, with the same set of boundary conditions for the linearized and fully non-linear equations. Some basic flow configurations are employed to investigate steady and unsteady rarefaction effects in rarefied gas flows, namely, planar and cylindrical Couette flow, stationary heat transfer between two plates, unsteady and oscillatory Couette flow. A comparison with the corresponding results obtained previously by the DSMC method is performed. The influence of rarefaction effects in the lid driven cavity problem is investigated. Solutions obtained from several macroscopic models, in particular the classical NSF equations with jump and slip boundary conditions, and the R13--moment equations are compared. The R13 results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Direct Simulation Monte Carlo (DSMC) method. Flow and heat transfer in a bottom heated square cavity in a moderately rarefied gas are investigated using the R13 and NSF equations. The results obtained are compared with those from the DSMC method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results including flow patterns in fair agreement with DSMC in the transition regime, which the conventional Navier-Stokes-Fourier equations are not able to capture. / Graduate / 0548 / anirudh@uvic.ca