No description available.

Many problems in applied mathematics involve simulating the evolution of a system using differential equations with known initial conditions. But what if one records observations and seeks to determine the causal factors which produced them? This is known as an inverse problem. Some prominent inverse problems include data assimilation and parameter recovery, which use partial observations of a system of evolutionary, dissipative partial differential equations to estimate the state of the system and relevant physical parameters (respectively). Recently a set of procedures called nudging algorithms have shown promise in performing simultaneous data assimilation and parameter recovery for the Lorentz equations and the Kuramoto-Sivashinsky equation. This work applies these algorithms and extensions of them to the case of Rayleigh-B\'enard convection, one of the most ubiquitous and commonly-studied examples of turbulent flow. The performance of various parameter update formulas is analyzed through direct numerical simulation. Under appropriate conditions and given the correct parameter update formulas, convergence is also established, and in one case, an analytical proof is obtained.

To control various response biases and rater errors in noncognitive assessment, multidimensional forced choice (MFC) measures have been proposed as an alternative to single-statement Likert-type scales. Historically, MFC measures have been criticized because conventional scoring methods can lead to ipsativity problems that render scores unsuitable for inter-individual comparisons. However, with the recent advent of classical test theory and item response theory scoring methods that yield normative information, MFC measures are surging in popularity and becoming important components of personnel and educational assessment systems. This dissertation presents developments concerning a GGUM-based MFC model henceforth referred to as the GGUM-RANK. Markov Chain Monte Carlo (MCMC) algorithms were developed to estimate GGUM-RANK statement and person parameters directly from MFC rank responses, and the efficacy of the new estimation algorithm was examined through computer simulations and an empirical construct validity investigation. Recently derived GGUM-RANK item information functions and information indices were also used to evaluate overall item and test quality for the empirical study and to give insights into differences in scoring accuracy between two-alternative (pairwise preference) and three-alternative (triplet) MFC measures for future work. This presentation concludes with a discussion of the research findings and potential applications in workforce and educational setting.

No description available.