Bulletin of the American Physical Society
Annual Meeting of the APS Four Corners Section
Volume 60, Number 11
Friday–Saturday, October 16–17, 2015; Tempe, Arizona
Session K11: Emergent Phenomena II |
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Chair: Sara Walker, Arizona State University Room: PSH151 |
Saturday, October 17, 2015 1:12PM - 1:24PM |
K11.00001: Classifying and Quantifying Nonlinearities in Mathematical Models Alexander Shumway, Mark Transtrum Models with many parameters describing complex systems are ubiquitous in the scientific disciplines. Though they model phenomena as varied as gene expression, complex materials, engineered systems, and machine learning, these models share similar statistical properties. This motivates study of complex models as a whole as opposed to analysis of each one individually. In particular, I work to quantify and classify the parameter nonlinearities of arbitrary models, as it is these nonlinearities that give models interesting behavior. In this talk I will introduce several complex models and show how they are statistically similar. I will further introduce the mathematical framework for quantifying nonlinearity and show with specific examples how analyzing nonlinearities leads to a deep understanding of the model. I will also discuss potential applications and future avenues for this work. [Preview Abstract] |
Saturday, October 17, 2015 1:24PM - 1:36PM |
K11.00002: Stochastic Search in 1D: Optimizing the search time Christy Contreras, Sidney Redner We investigate the efficiency of several search strategies in one dimension. First we simulate an unbiased stochastic search with multiple searchers. We consider an infinitely long one-dimensional line where N searchers are launched from an initial position x $=$ L at t $=$ 0 in an attempt to reach a target at x $=$ 0. We simulate this process computationally for multiple searchers to find the optimal number of searchers to minimize the search cost and compare our computational results to the analytical results from Meerson and Redner, 2014. We find that the distribution of the search cost follows a power law distribution on a log-log scale. Secondly, we consider a biased search with one biased random walker that is reset to its starting point with rate r and the direction of the bias alternates with every reset. We simulate the case without diffusion computationally for various rates to find the optimum resetting rate r* and compare our results with analytical results. Lastly, we consider a resetting search with a fixed diffusion and vary the magnitude of the bias in both directions to determine how the minimum search time corresponding to each optimal resetting rate is affected as the magnitude of the bias increases. [Preview Abstract] |
Saturday, October 17, 2015 1:36PM - 1:48PM |
K11.00003: Quantifying Model Error using a Kriging Process Malachi Tolman Uncertainty Quantification (UQ) is an important and rapidly growing field with applications across many disciplines. Since all mathematical models employ approximations, they represent physical reality with varying degrees of fidelity. An important component of UQ is quantifying the inherent inaccuracy of the model. This error usually manifests itself as a systematic bias in the parameter estimates and model predictions. A Kriging process is a type of meta-model designed to account for model bias by introducing a few hyper-parameters. We use information geometry to explore the statistical properties of this hyper-parameter space. We show that the potential for estimating model error depends on the amount of experimental noise present. [Preview Abstract] |
Saturday, October 17, 2015 1:48PM - 2:00PM |
K11.00004: Data Fitting in Oscillatory and Chaotic Models Ben Francis, Mark Transtrum Nonlinear systems are common in physics, biology, and other fields. These systems often exhibit oscillatory or chaotic behavior. Models of these systems often involve many parameters that must be fit to data, usually by least squares. Finding a good fit is often challenging because the cost function may have many local minima. We show that in many cases the problem is exacerbated by having more data. To alleviate this difficulty, we propose a novel similarity measure of oscillatory behavior based on a kernel density estimate of the system's phase space density. The new cost surface is typically characterized by a single basin. We demonstrate our method on two model: the Fitzhugh-Nagumo model of neuronal spiking and the Lorenz attractor. [Preview Abstract] |
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