Stratified nested and related quadrature rules

AbstractThe stratified nested quadrature procedure due to Laurie is discussed together with an alternative computational procedure which leads to the concept of hybrid GKP rules. In the context of the approximation of stratified nested sequences the work of Krogh and Van Snyder on the representation of the GKP rules is considered and a generalisation of this employing hybrid rules of special form is discussed

Uncertainty propagation in neuronal dynamical systems

One of the most notorious characteristics of neuronal electrical activity is its variability, whose origin is not just instrumentation noise, but mainly the intrinsically stochastic nature of neural computations. Neuronal models based on deterministic differential equations cannot account for such variability, but they can be extended to do so by incorporating random components. However, the computational cost of this strategy and the storage requirements grow exponentially with the number of stochastic parameters, quickly exceeding the capacities of current supercomputers. This issue is critical in Neurodynamics, where mechanistic interpretation of large, complex, nonlinear systems is essential. In this paper we present accurate and computationally efficient methods to introduce and analyse variability in neurodynamic models depending on multiple uncertain parameters. Their use is illustrated with relevant example

A geometric approach for the addition of nodes to an interpolatory quadrature rule with positive weights

A novel mathematical framework is derived for the addition of nodes to interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde-matrix describing the relation between the nodes and the weights and can be used to determine all nodes that can be added to an interpolatory quadrature rule with positive weights such that the positive weights are preserved. In the case of addition of a single node, the derived inequalities that describe the regions where nodes can be added or replaced are explicit. It is shown that, depending on the location of existing nodes and moments of the distribution, addition of a single node and preservation of positive weights is not always possible. On the other hand, addition of multiple nodes and preservation of positive weights is always possible, although the minimum number of nodes that need to be added can be as large as the number of nodes of the quadrature rule. Moreover, in this case the inequalities describing the regions where nodes can be added become implicit. An algorithm is presented to explore these regions and it is shown that the well-known Patter

Accelerated Bayesian experimental design for chemical kinetic models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 129-136).The optimal selection of experimental conditions is essential in maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. A general Bayesian framework for optimal experimental design with nonlinear simulation-based models is proposed. The formulation accounts for uncertainty in model parameters, observables, and experimental conditions. Straightforward Monte Carlo evaluation of the objective function - which reflects expected information gain (Kullback-Leibler divergence) from prior to posterior - is intractable when the likelihood is computationally intensive. Instead, polynomial chaos expansions are introduced to capture the dependence of observables on model parameters and on design conditions. Under suitable regularity conditions, these expansions converge exponentially fast. Since both the parameter space and the design space can be high-dimensional, dimension-adaptive sparse quadrature is used to construct the polynomial expansions. Stochastic optimization methods will be used in the future to maximize the expected utility. While this approach is broadly applicable, it is demonstrated on a chemical kinetic system with strong nonlinearities. In particular, the Arrhenius rate parameters in a combustion reaction mechanism are estimated from observations of autoignition. Results show multiple order-of-magnitude speedups in both experimental design and parameter inference.by Xun Huan.S.M