Partial Differential Equations Commons^™

Some Proofs Regarding Minami Estimates And Local Eigenvalue Statistics For Some Random Schrödinger Operator Models, Samuel Herschenfeld

Theses and Dissertations--Mathematics

We provide three proofs on different, but related models in the field of random Schrödinger operators. All three results are motivated by the desire to extend results and techniques on eigenvalue statistics or Minami estimates (an essential ingredient Poisson eigenvalue statistics).

Chapters 2 and 4 are explorations of the only two known techniques for proving Minami estimates for continuum Minami estimates. In Chapter 2, we provide an alternative and simplified proof of Klopp that holds in d = 1. Chapter 4 is an application of the techniques of Dietlein and Elgart to prove a Minami estimate for finite rank lattice …

New Numerical Approximations Of Geological Processes In Heterogeneous Systems Using Radial Basis Functions, Nadun Lakshitha Dissanayake Kulasekera Mudiyanselage

Dissertations, Master's Theses and Master's Reports

This dissertation includes four chapters. A brief description of each chapter is organized as follows. The first chapter provides an introduction to the RBF method. The chapter follows the historical progression of the Radial Basis Function (RBF) method while outlining the method’s advantages and disadvantages. A brief introduction about RBF interpolation, the RBF-FD method, and how to use it to solve PDEs is provided. Chapter 2 introduces a novel computationally efficient RBF-FD algorithm to solve the groundwater flow equation in the presence of an active well. We show that our method analytically handles the singularities in the PDE caused by …

Inference Of Surface Velocities From Oblique Time Lapse Photos And Terrestrial Based Lidar At The Helheim Glacier, Franklyn T. Dunbar Ii

Graduate Student Theses, Dissertations, & Professional Papers

Using time dependent observations derived from terrestrial LiDAR and oblique
time-lapse imagery, we demonstrate that a Bayesian approach to glacial motion es-
timation provides a concise way to incorporate multiple data products into a single
motion estimation procedure effectively producing surface velocity estimates with
an associated uncertainty. This approach brings both improved computational effi-
ciency, and greater scalability across observational time-frames when compared to
existing methods. To gauge efficacy, we apply these methods to a set of observa-
tions from the Helheim Glacier, a critical actor in contemporary mass loss trends
observed in the Greenland Ice Sheet. We find that …

Improving The Temporal Accuracy Of Turbulence Models And Resolving The Implementation Issues Of Fluid Flow Modeling, Kyle J. Schwiebert

Dissertations, Master's Theses and Master's Reports

A sizeable proportion of the work in this thesis focuses on a new turbulence model, dubbed ADC (the approximate deconvolution model with defect correction). The ADC is improved upon using spectral deferred correction, a means of constructing a higher order ODE solver. Since both the ADC and SDC are based on a predictor-corrector approach, SDC is incorporated with essentially no additional computational cost. We will show theoretically and using numerical tests that the new scheme is indeed higher order in time than the original, and that the benefits of defect correction, on which the ADC is based, are preserved.

The …

A New Mathematical Theory For The Dynamics Of Large Tumor Populations, A Potential Mechanism For Cancer Dormancy & Recurrence And Experimental Observation Of Melanoma Progression In Zebrafish, Adeyinka A. Lesi

Dissertations and Theses

Cancer, a family of over a hundred disease varieties, results in 600,000 deaths in the U.S. alone. Yet, improvements in imaging technology to detect disease earlier, pharmaceutical developments to shrink or eliminate tumors, and modeling of biological interactions to guide treatment have prevented millions of deaths. Cancer patients with initially similar disease can experience vastly different outcomes, including sustained recovery, refractory disease or, remarkably, recurrence years after apparently successful treatment. The current understanding of such recurrences is that they depend on the random occurrence of critical mutations. Clearly, these biological changes appear to be sufficient for recurrence, but are they …

Deterministic And Statistical Methods For Inverse Problems With Partial Data, Yanfang Liu

Dissertations, Master's Theses and Master's Reports

Inverse problems with partial data have many applications in science and engineering. They are more challenging than the complete data cases since the lack of data increases ill-posedness and nonlinearity. The use of only deterministic or statistical methods might not provide satisfactory results. We propose to combine the deterministic and statistical methods to treat such inverse problems. The thesis is organized as follows.

In Chapter 1, we briefly introduce the inverse problems and their applications. The classical deterministic methods and Bayesian inversion are discussed. The chapter is concluded with a summary of contributions.

Chapter 2 considers the reconstruction of the …