From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure

Quantifying the Statistical Impact of GRAPPA in fcMRI Data with a Real-Valued Isomorphism

The interpolation of missing spatial frequencies through the generalized auto-calibrating partially parallel acquisitions (GRAPPA) parallel magnetic resonance imaging (MRI) model implies a correlation is induced between the acquired and reconstructed frequency measurements. As the parallel image reconstruction algorithms in many medical MRI scanners are based on the GRAPPA model, this study aims to quantify the statistical implications that the GRAPPA model has in functional connectivity studies. The linear mathematical framework derived in the work of Rowe , 2007, is adapted to represent the complex-valued GRAPPA image reconstruction operation in terms of a real-valued isomorphism, and a statistical analysis is performed on the effects that the GRAPPA operation has on reconstructed voxel means and correlations. The interpolation of missing spatial frequencies with the GRAPPA model is shown to result in an artificial correlation induced between voxels in the reconstructed images, and these artificial correlations are shown to reside in the low temporal frequency spectrum commonly associated with functional connectivity. Through a real-valued isomorphism, such as the one outlined in this manuscript, the exact artificial correlations induced by the GRAPPA model are not simply estimated, as they would be with simulations, but are precisely quantified. If these correlations are unaccounted for, they can incur an increase in false positives in functional connectivity studies

The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question that we recently were able to prove. While Gopalan et al. considered CNF(S)-formulas with constants, we here look at two important variants: CNF(S)-formulas without constants, and partially quantified formulas. For the diameter and the st-connectivity question, we prove dichotomies analogous to those of Gopalan et al. in these settings. While we cannot give a complete classification for the connectivity problem yet, we identify fragments where it is in P, where it is coNP-complete, and where it is PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.Comment: superseded by chapter 3 of arXiv:1510.0670

Development of boldness and docility in yellow-bellied marmots

Peer reviewedPostprin

Forest Stand Structure and Primary Production in relation to Ecosystem Development, Disturbance, and Canopy Composition

Temperate forests are complex ecosystems that sequester carbon (C) in biomass. C storage is related to ecosystem-scale forest structure, changing over succession, disturbance, and with community composition. We quantified ecosystem biological and physical structure in two forest chronosequences varying in disturbance intensity, and three late successional functional types to examine how multiple structural expressions relate to ecosystem C cycling. We quantified C cycling as wood net primary production (NPP), ecosystem structure as Simpson’s Index, and physical structure as leaf quantity (LAI) and arrangement (rugosity), examining how wood NPP-structure relates to light distribution and use-efficiency. Relationships between structural attributes of biodiversity, LAI, and rugosity differed. Development of rugosity was conserved regardless of disturbance and composition, suggesting optimization of vegetation arrangement over succession. LAI and rugosity showed significant positive productivity trends over succession, particularly within deciduous broadleaf forests, suggesting these measures of structure contain complementary, not redundant, information related to C cycling