Browsing by Subject "convergence"
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Item Open Access An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability(IMA Journal of Numerical Analysis, 2007-01-01) Lamba, H; Mattingly, JC; Stuart, AMThe understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.Item Open Access Conditions for Rapid and Torpid Mixing of Parallel and Simulated Tempering on Multimodal Distributions(2007-09-14) Woodard, Dawn BanisterStochastic sampling methods are ubiquitous in statistical mechanics, Bayesian statistics, and theoretical computer science. However, when the distribution that is being sampled is multimodal, many of these techniques converge slowly, so that a great deal of computing time is necessary to obtain reliable answers. Parallel and simulated tempering are sampling methods that are designed to converge quickly even for multimodal distributions. In this thesis, we assess the extent to which this goal is acheived.We give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, meaning that it converges quickly. These conditions are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models in R^M as M increases, and for the mean-field Ising model.We also obtain upper bounds on the convergence rates of parallel and simulated tempering, yielding a single set of sufficient conditions for torpid mixing of both techniques. These conditions imply torpid mixing of parallel and simulated tempering on a normal mixture model with unequal covariances in $\R^M$ as $M$ increases and on the mean-field Potts model with $q \geq 3$, regardless of the number and choice of temperatures, as well as on the mean-field Ising model if an insufficient (fixed) set of temperatures is used. The latter result is in contrast to the rapid mixing of parallel and simulated tempering on the mean-field Ising model with a linearly increasing set of temperatures.Item Open Access Conditions for Rapid Mixing of Parallel and Simulated Tempering on Multimodal Distributions(2009) Woodard, DB; Schmidler, SC; Huber, MWe give conditions under which a Markov chain constructed via parallel or simulated tempering is guaranteed to be rapidly mixing, which are applicable to a wide range of multimodal distributions arising in Bayesian statistical inference and statistical mechanics. We provide lower bounds on the spectral gaps of parallel and simulated tempering. These bounds imply a single set of sufficient conditions for rapid mixing of both techniques. A direct consequence of our results is rapid mixing of parallel and simulated tempering for several normal mixture models, and for the mean-field Ising model.Item Open Access Logic, passion and the problem of convergence.(Interface focus, 2017-06) McShea, Daniel WOur estimate of the likelihood of convergence on human-style intelligence depends on how we understand our various mental capacities. Here I revive David Hume's theory of motivation and action to argue that the most common understanding of the two conventionally recognized components of intelligence-reason and emotion-is confused. We say things like, 'Reason can overcome emotion', but to make this statement meaningful, we are forced to treat reason as a compound notion, as a forced and unhappy mixture of concepts that are incommensurate. An alternative is to parse intelligence in a different way, into two sets of capacities: (i) non-affective capacities, including logic, calculation and problem-solving; (ii) affective capacities, including wants, preferences and cares, along with the emotions. Thus, the question of convergence becomes two questions, one having to do with affective and one with non-affective capacities. What is the likelihood of convergence of these in non-human lineages, in other ecologies, on other worlds? Given certain assumptions, convergence of the non-affective capacities in thinking species seems likely, I argue, while convergence of the affective capacities seems much less likely.Item Open Access Patterns of diversification in the xeric-adapted fern genus myriopteris (Pteridaceae)(Systematic Botany, 2014-01-01) Grusz, AL; Windham, MD; Yatskievych, G; Huiet, L; Gastony, GJ; Pryer, KMStrong selective pressures imposed by drought-prone habitats have contributed to extensive morphological convergence among the 400 + species of cheilanthoid ferns (Pteridaceae). As a result, generic circumscriptions based exclusively on macromorphology often prove to be non-monophyletic. Ongoing molecular phylogenetic analyses are providing the foundation for a revised classification of this challenging group and have begun to clarify its complex evolutionary history. As part of this effort, we generated and analyzed DNA sequence data for three plastid loci (rbcL, atpA, and the intergenic spacer trnG-trnR) for the myriopterid clade, one of the largest monophyletic groups of cheilanthoid ferns. This lineage encompasses 47 primarily North and Central American taxa previously included in Cheilanthes but now placed in the recircumscribed genus Myriopteris. Here, we infer a phylogeny for the group and examine key morphological characters across this phylogeny. We also include a brief discussion of the three well-supported Myriopteris subclades, along with a review of reproductive mode and known ploidy levels for members of this early diverging lineage of cheilanthoid ferns. © Copyright 2014 by the American Society of Plant Taxonomists.Item Open Access Sympatric parallel diversification of major oak clades in the Americas and the origins of Mexican species diversity.(New Phytol, 2017-09-18) Hipp, Andrew L; Manos, Paul S; González-Rodríguez, Antonio; Hahn, Marlene; Kaproth, Matthew; McVay, John D; Avalos, Susana Valencia; Cavender-Bares, JeannineOaks (Quercus, Fagaceae) are the dominant tree genus of North America in species number and biomass, and Mexico is a global center of oak diversity. Understanding the origins of oak diversity is key to understanding biodiversity of northern temperate forests. A phylogenetic study of biogeography, niche evolution and diversification patterns in Quercus was performed using 300 samples, 146 species. Next-generation sequencing data were generated using the restriction-site associated DNA (RAD-seq) method. A time-calibrated maximum likelihood phylogeny was inferred and analyzed with bioclimatic, soils, and leaf habit data to reconstruct the biogeographic and evolutionary history of the American oaks. Our highly resolved phylogeny demonstrates sympatric parallel diversification in climatic niche, leaf habit, and diversification rates. The two major American oak clades arose in what is now the boreal zone and radiated, in parallel, from eastern North America into Mexico and Central America. Oaks adapted rapidly to niche transitions. The Mexican oaks are particularly numerous, not because Mexico is a center of origin, but because of high rates of lineage diversification associated with high rates of evolution along moisture gradients and between the evergreen and deciduous leaf habits. Sympatric parallel diversification in the oaks has shaped the diversity of North American forests.