in representing images compactly. Nonetheless, high-order methods have difficulty representing functions containing discontinuities or functions having slow spectral decay in the chosen basis. Certain sensing techniques such as MRI and SAR provide data in terms of Fourier coefficients, and thus prescribe a natural high-order basis. The field of compressed sensing has introduced a set of techniques based on $\ell^1$ regularization that promote sparsity and facilitate working with functions having discontinuities. In this dissertation, high-order methods and $\ell^1$ regularization are used to address three problems: reconstructing piecewise smooth functions from sparse and and noisy Fourier data, recovering edge locations in piecewise smooth functions from sparse and noisy Fourier data, and reducing time-stepping constraints when numerically solving certain time-dependent hyperbolic partial differential equations.
We attempt to analyze the effect of fatigue on free throw efficiency in the National Basketball Association (NBA) using play-by-play data from regular-season, regulation-length games in the 2016-2017, 2017-2018, and 2018-2019 seasons. Using both regression and tree-based statistical methods, we analyze the relationship between minutes played total and minutes played continuously at the time of free throw attempts on players' odds of making an attempt, while controlling for prior free throw shooting ability, longer-term fatigue, and other game factors. Our results offer strong evidence that short-term activity after periods of inactivity positively affects free throw efficiency, while longer-term fatigue has no effect.
We study the interplay between correlations, dynamics, and networks for repeated attacks on a socio-economic network. As a model system we consider an insurance scheme against disasters that randomly hit nodes, where a node in need receives support from its network neighbors. The model is motivated by gift giving among the Maasai called Osotua. Survival of nodes under different disaster scenarios (uncorrelated, spatially, temporally and spatio-temporally correlated) and for different network architectures are studied with agent-based numerical simulations. We find that the survival rate of a node depends dramatically on the type of correlation of the disasters: Spatially and spatio-temporally correlated disasters increase the survival rate; purely temporally correlated disasters decrease it. The type of correlation also leads to strong inequality among the surviving nodes. We introduce the concept of disaster masking to explain some of the results of our simulations. We also analyze the subsets of the networks that were activated to provide support after fifty years of random disasters. They show qualitative differences for the different disaster scenarios measured by path length, degree, clustering coefficient, and number of cycles.
Capacity dimensioning in production systems is an important task within strategic and tactical production planning which impacts system cost and performance. Traditionally capacity demand at each worksystem is determined from standard operating processes and estimated production flow rates, accounting for a desired level of utilization or required throughput times. However, for distributed production control systems, the flows across multiple possible production paths are not known a priori. In this contribution, we use methods from algorithmic game-theory and traffic-modeling to predict the flows, and hence capacity demand across worksystems, based on the available production paths and desired output rates, assuming non-cooperative agents with global information. We propose an iterative algorithm that converges simultaneously to a feasible capacity distribution and a flow distribution over multiple paths that satisfies Wardrop's first principle. We demonstrate our method on models of real-world production networks.
Signaling cascades proliferate signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling cascades, it is not understood why cascades typically show three or four layers. Using singular perturbation theory, Michaelis-Menten type equations are derived for open enzymatic systems. Cascading these equations we demonstrate that the output signal as a function of time becomes sigmoidal with the addition of more layers. Furthermore, it is shown that the activation time will speed up to a point, after which more layers become superfluous. It is shown that three layers create a reliable sigmoidal response progress curve from a wide variety of time-dependent signaling inputs arriving at the cell membrane, suggesting the evolutionary benefit of the observed cascades.