Path-planning has many applications, ranging from self-driving cars to flying drones, and to our daily commute to work. Path-planning for autonomous underwater vehicles presents an interesting problem: the ocean flow is dynamic and unsteady. Additionally, we may not have perfect knowledge of the ocean flow. Our goal is to develop a rigorous and computationally efficient methodology to perform path-planning in uncertain flow fields. We obtain new stochastic Dynamically Orthogonal (DO) Level Set equations to account for uncertainty in the flow field. We first review existing path-planning work: time-optimal path planning using the level set method, and energy-optimal path planning using stochastic DO level set equations. We build on these methods by treating the velocity field as a stochastic variable and deriving new stochastic DO level set equations. We use the new DO equations to simulate a simple canonical flow, the stochastic highway. We verify that our results are correct by comparing to corresponding Monte Carlo results. We explore novel methods of visualizing the results of the equations. Finally we apply our methodology to an idealized ocean simulation using Double-Gyre flows.

Abstract

I will discuss a nonparametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution of the backward Kolmogorov equation of the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to evolve the probability distribution of non-trivial dynamical systems with equation-free modeling. I will also discuss nonparametric filtering methods, leveraging the diffusion forecast in Bayesian framework to initialize the forecasting distribution given noisy observations.

Bio

John Harlim is an associate professor of mathematics and meteorology at the Pennsylvania State University. He received Ph.D. in Applied Mathematics and Scientific Computation from University of Maryland in 2006. His research interests is applied mathematics related to data-driven estimation and prediction problems; this includes filtering multiscale dynamical systems, stochastic parameterization, uncertainty quantification, diffusion maps, non-parametric modeling.

Co-hosted with Prof. Themis Sapsis.

Abstract

The development of numerical weather prediction was one of the great scientific and computational achievements of the last century. Computer models that approximate solutions of the partial differential equations that govern fluid flow and a comprehensive global observing network are two components of this prediction enterprise. An essential third component is data assimilation, the computational method that combines observations with predictions from previous times to produce initial conditions for subsequent predictions. The best present-day numerical weather prediction systems have evolved over decades and feature model-specific assimilation systems built with nearly a person century of effort.

This talk describes the design of a community software facility for ensemble Kalman filter data assimilation, the Data Assimilation Research Testbed (DART). DART can produce high-quality weather predictions but can also be used to build a comprehensive forecast system for any prediction model and observations. DART forecast systems must be inexpensive to implement and must run efficiently on computing platforms ranging from laptops to the largest available supercomputing. A description of the basic ensemble Kalman filter algorithm is followed by a discussion of algorithmic enhancements, in particular localization of observation impacts and inflation of prior ensembles that are essential for efficient implementations for large prediction models. Several example applications in geosciences will be used to examine additional capabilities of modern ensemble prediction systems.

Bio of the speaker:

Jeffrey Anderson’s research career has spanned two decades and has been focused by the common theme to improve predictions of the earth’s atmosphere. He has made research contributions in theoretical geophysical fluid dynamics, seasonal prediction, predictability, ensemble prediction and ensemble data assimilation. His accomplishments in software engineering, applied mathematics and statistics have been directly in support of his goal to improve prediction.

Abstract We provide an overview of the evolution of the so-called hybridizable discontinuous Galerkin (HDG) methods. We motivate the introduction of the methods and describe the main ideas of their development within the framework of steady-state diffusion. We then describe the status of their application to other problems of practical interest. A significant part of this material is joint work with N.C. Nguyen and J. Peraire, from MIT.

Abstract

Ocean models typically employ the hydrostatic assumption because, for most problems of interest, vertical inertia is orders of magnitude smaller than horizontal inertia, thereby validating the assumption that vertical pressure variability arises purely from hydrostatics. This ultimately implies that horizontal scales of motion are much larger than vertical scales, and hence that the hydrostatic approximation is valid to simulate processes with large horizontal scales relative to the depth. The primary advantage of the hydrostatic assumption is that it eliminates computation of the nonhydrostatic pressure which can increase the computation time of typical oceanic calculations by one order of magnitude. Although most processes of interest in the ocean are hydrostatic, internal gravity waves exist over a wide range of horizontal scales and hence internal gravity waves with relatively short wavelengths are nonhydrostatic.

The primary physical effect of the nonhydrostatic pressure in internal gravity waves is frequency dispersion which causes waves of different frequencies to travel at different speeds. However, numerical errors can lead to erroneous numerical dispersion that mimics the effect of the nonhydrostatic pressure. In order for this numerical dispersion to be smaller than the physical nonhydrostatic dispersion, the horizontal grid resolution must be smaller than the relevant vertical depth scale, which can be the depth of the mixed layer. This can impose a significant computational overhead in 3D numerical simulations of internal gravity waves in the coastal ocean. The cost associated with the horizontal grid resolution requirement can be alleviated by assuming that a bulk of the internal wave energy in the ocean propagates as low-mode internal gravity waves. These waves are well-represented through use of a reduced number of isopycnal layers which follow the density surfaces, as opposed to use of many fixed vertical coordinates. The result can be a reduction in computational cost by up to two orders of magnitude. However, while isopcynal-coordinate models are commonly used in the ocean modeling community, none are nonhydrostatic.

In this presentation I will discuss nonhydrostatic modeling of internal gravity waves with an emphasis on development of a nonhydrostatic, isopycnal-coordinate ocean model to accurately and efficiently simulate nonhydrostatic internal gravity waves in the coastal ocean. I will discuss the model and associated approximations and present results of test cases to demonstrate model efficiency in comparison to standard z-level or fixed vertical-coordinate techniques.

Bio of the speaker:

Oliver Fringer is associate professor in the Department of Civil and Environmental Engineering at Stanford University, where he has been since 2003. He received his BSE from Princeton University in Aerospace Engineering and then received an MS in Aeronatics and Astronautics, followed by a PhD in Civil and Environmental Engineering, both from Stanford University. His research focuses on the application of numerical models and parallel computing to the study of laboratory- and field-scale environmental flows to understand the physics of salt and sediment transport in lakes and estuaries, internal waves and mixing, and turbulence in rivers. Dr. Fringer received the ONR Young Investigator award in 2008 and was awarded the Presidential Early Career Award for Scientists and Engineers in 2009.