Nonparametric Uncertainty Quantification for Stochastic Gradient Flows

This paper presents a nonparametric statistical modeling method for quantifying uncertainty in stochastic gradient systems with isotropic diffusion. The central idea is to apply the diffusion maps algorithm to a training data set to produce a stochastic matrix whose generator is a discrete approximation to the backward Kolmogorov operator of the underlying dynamics. The eigenvectors of this stochastic matrix, which we will refer to as the diffusion coordinates, are discrete approximations to the eigenfunctions of the Kolmogorov operator and form an orthonormal basis for functions defined on the data set. Using this basis, we consider the projection of three uncertainty quantification (UQ) problems (prediction, filtering, and response) into the diffusion coordinates. In these coordinates, the nonlinear prediction and response problems reduce to solving systems of infinite-dimensional linear ordinary differential equations. Similarly, the continuous-time nonlinear filtering problem reduces to solving a system of infinite-dimensional linear stochastic differential equations. Solving the UQ problems then reduces to solving the corresponding truncated linear systems in finitely many diffusion coordinates. By solving these systems we give a model-free algorithm for UQ on gradient flow systems with isotropic diffusion. We numerically verify these algorithms on a 1-dimensional linear gradient flow system where the analytic solutions of the UQ problems are known. We also apply the algorithm to a chaotically forced nonlinear gradient flow system which is known to be well approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed

RWF rotor-wake-fuselage code software reference guide

The RWF (Rotor-Wake-Fuselage) code was developed from first principles to compute the aerodynamics associated with the complex flow field of helicopter configurations. The code is sized for a single, multi-bladed main rotor and any configuration of non-lifting fuselage. The mathematical model for the RWF code is based on the integration of the momentum equations and Green's theorem. The unknowns in the problem are the strengths of prescribed singularity distributions on the boundaries of the flow. For the body (fuselage) a surface of constant strength source panels is used. For the rotor blades and rotor wake a surface of constant strength doublet panels is used. The mean camber line of the rotor airfoil is partitioned into surface panels. The no-flow boundary condition at the panel centroids is modified at each azimuthal step to account for rotor blade cyclic pitch variation. The geometry of the rotor wake is computers at each time step of the solution. The code produces rotor and fuselage surface pressures, as well as the complex geometry of the evolving rotor wake

Riding the Elephants: The Evolution of World Economic Growth and Income Distribution at the End of the Twentieth Century (1980-2000)

This paper presents estimates of world economic growth for 1970-2000, and changes in the intercountry and interpersonal distribution of world income between 1980 and 2000. These estimates suggest that, while the rate of growth of the world economy slowed in the 1980-2000 period, and average within-country inequality worsened, the distribution of world income among individuals, nevertheless, improved a little. However, that result was wholly due to the exceptional economic performances of China and India. Outside these two countries, the slowdown in world growth was even more dramatic, the distribution of world income unequivocally worsened, and poverty rates remained largely unchanged.world inequality trends; international income distribution, convergence, world poverty trends

All About the Giants: Probing the Influences on Growth and Income Inequality at the End of the 20th Century

This paper presents estimates of world output growth from 1970 to 2000, the distribution of income among countries and persons for the years 1980, 1990 and 2000, and world poverty rates for the same years. It also presents the results of a series of simulation exercises that attempt isolate the effect of particular country and regional experiences on world output growth and changes in global income inequality and poverty. The authors find that rapid growth in China (despite a downward adjustment of official growth estimates) had a powerful impact on the growth of world output in both the 1980s and 1990s, but that negative economic growth in Eastern Europe more than offset that effect in the 1990s. With respect to the distribution of income however, the equalizing effect of China’s rapid growth, despite the contradictory impact of increasing domestic inequality, was dominant through both the 1980s and 1990s. Only India’s influence remained substantial by comparison. Other identifiable events of the period, such as the economic contraction in Eastern Europe and continued economic decline in Africa had little statistical impact. Thus, when the combined influence of these two countries’ above-average growth rates is removed, the improving global distribution of income suggested by all statistical measures becomes one of sharply worsening inequality. The impact of these twocountries is similarly critical with respect to global poverty reduction.