Abstract

In a variety of disciplines including atmospheric, oceanic, hydrologic and environmental sciences, large numerical simulations have become an essential tool for understanding the physical processes, synthesizing data, and for prediction. A key problem for modeling these dynamical systems is how to deal with uncertainties and error in the models’ representation of key physical processes. This talk will introduce some of the recent ensemble-based data assimilation approaches such as the use of an ensemble Kalman filter for Simultaneous State and Parameter Estimation (SSPE) in the treatment and quantification of model error and uncertainties. Applications of SSPE to a variety of phenomena ranging from the atmospheric boundary layer transport, air-sea fluxes and a physically based land-surface hydrologic model will be presented.

Biography

Prof. Zhang’s research interests include atmospheric dynamics and predictability, data assimilation, tropical cyclones, gravity waves and regional-scale climate. He earned his B.S. and M.S. in meteorology from Nanjing University, China in 1991 and 1994, respectively, and his Ph.D. in atmospheric science in 2000 from North Carolina State University. He has authored/co-authored over 150 peer reviewed journal publications that have a total of more than 3300 citations. He has received numerous awards for his research and service.

Wear is the process of material removal under abrasion. Multi-material systems are characterized by non-uniform wear-rates that are leading to non-planar profiles under polishing. Some recent iterative models have predicted with success the well established experimental fact, that under an initially uniform pressure load, multi-component wearing surfaces reach asymptotically a steady-state profile that continues to recess at a constant rate. In a first part, a continuous model is derived from the original iterative scheme, and extrapolated to a mathematical, abstract, but versatile framework. It is shown that the steady-state can be computed directly by solving a time-independent elliptic partial differential equation, providing a substantial computational gain against the first iterative scheme. In a second part, this formulation is used to apply modern shape optimization methods to optimize wear performance of bi-material systems. Several objectives for systems undergoing wear are identified and formalized with shape derivatives. As an example, the unit-cell of 2D multi-material composites is optimized using a level-set based method. A minimum feature size must be taken into account to avoid the convergence of minimizing design sequences toward composite materials. To address this issue, a variant of a the level-set topology optimization method is developed . Through the use of a single updating equation, this scheme conveniently enforces volume equality constraints, controls the complexity of design features with a perimeter penalization, and nucleates material inclusions with the use of the topological gradient. Keywords: topology and shape optimization, wear, multi-composite surface, level-set methods.

A stochastic optimization methodology is formulated for computing energy-optimal paths from among time-optimal paths of autonomous vehicles navigating in a dynamic flow field. To set up the energy optimization, the relative vehicle speed and headings are considered to be stochastic, and new stochastic Dynamically Orthogonal (DO) level-set equations that govern their stochastic time-optimal reachability fronts are derived. Their solution provides the distribution of time-optimal reachability fronts and corresponding distribution of time-optimal paths. An optimization is then performed on the vehicle’s energy-time joint distribution to select the energy-optimal paths for each arrival time, among all stochastic time-optimal paths for that arrival time. The accuracy and efficiency of the DO level-set equations for solving the governing stochastic level-set reachability fronts are quantitatively assessed, including comparisons with independent semi-analytical solutions. The stochastic DO level-set equations is then extended to account for uncertainties in the flow field. Time-optimal planning is completed in a wind-driven barotropic quasi-geostrophic stochastic double-gyre ocean circulation (these stochastic flow fields are simulated using our DO Navier Stokes equations). Energy-optimal missions are studied in wind-driven barotropic quasi-geostrophic double-gyre circulations, and in realistic data-assimilative re-analyses of multiscale coastal ocean flows. The latter re-analyses are obtained from multi-resolution 2-way nested primitive-equation simulations of tidal-to-mesoscale dynamics in the Middle Atlantic Bight and Shelbreak Front region. The effects of tidal currents, strong wind events, coastal jets, and shelfbreak fronts on the energy-optimal paths are illustrated and quantified. Results showcase the opportunities for longer-duration missions that intelligently utilize the ocean environment to save energy, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.

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.

Abstract:

When humans or robots operate in complex dynamic environments, the planning of paths and the collection of observations are basic, indispensable problems. In the oceanic and atmospheric environments, the concurrent use of mobile sensing platforms in unmanned missions is growing very rapidly. Opportunities for a paradigm shift in the science of autonomy involve fundamental theories to optimally collect information, learn, collaborate and make decisions under uncertainty while persistently adapting to and utilizing the dynamic environment. To address such pressing needs, this thesis derives governing equations and develops rigorous methodologies for optimal path planning and optimal sampling using collaborative swarms of autonomous mobile platforms. The application focus is the coastal ocean where currents can be much larger than platform speeds, but the fundamental results also apply to other dynamic environments.

We first undertake a theoretical synthesis of minimum-time control of vehicles operating in general dynamic flows. Using various ideas rooted in non-smooth calculus, we prove that an unsteady Hamilton-Jacobi equation governs the forward reachable sets in any type of Lipschitz-continuous flow. Next, we show that with a suitable modification to the Hamiltonian, the results can be rigorously generalized to perform time-optimal path planning with anisotropic motion constraints and with moving obstacles and unsafe `forbidden’ regions. We then derive a level-set methodology for distance-based coordination of swarms of vehicles operating in minimum time within strong and dynamic ocean currents. The results are illustrated for varied fluid and ocean flow simulations. Finally, the new path planning system is applied to swarms of vehicles operating in the complex geometry of the Philippine Archipelago, utilizing realistic multi-scale current predictions from a data-assimilative ocean modeling system.

In the second part of the thesis, we derive a theory for adaptive sampling that exploits the governing nonlinear dynamics of the system and captures the non-Gaussian structure of the random state fields. Optimal observation locations are determined by maximizing the mutual information between the candidate observations and the variables of interest. We develop a novel Bayesian smoother for high-dimensional continuous stochastic fields governed by general nonlinear dynamics. This smoother combines the adaptive reduced-order Dynamically-Orthogonal equations with Gaussian Mixture Models, extending linearized Gaussian backward pass updates to a nonlinear, non-Gaussian setting. The Bayesian information transfer, both forward and backward in time, is efficiently carried out in the evolving dominant stochastic subspace. Building on the foundations of the smoother, we then derive an efficient technique to quantify the spatially and temporally varying mutual information field in general nonlinear dynamical systems. The globally optimal sequence of future sampling locations is rigorously determined by a novel dynamic programming approach that combines this computation of mutual information fields with the predictions of the forward reachable set. All the results are exemplified and their performance is quantitatively assessed using a variety of simulated fluid and ocean flows.

The above novel theories and schemes are integrated so as to provide real-time computational intelligence for collaborative swarms of autonomous sensing vehicles. The integrated system guides groups of vehicles along predicted optimal trajectories and continuously improves field estimates as the observations predicted to be most informative are collected and assimilated. The optimal sampling locations and optimal trajectories are continuously forecast, all in an autonomous and coordinated fashion.

Thesis Committee:

1. Professor Pierre F. J. Lermusiaux (pierrel@mit.edu), Associate Professor, Department of Mechanical Engineering (Chair)
2. Dr. Franz S. Hover (hover@mit.edu), Senior Lecturer, Department of Mechanical Engineering
3. Professor Henrik Schmidt (henrik@mit.edu), Professor, Department of Mechanical and Ocean Engineering
4. Professor Youssef M. Marzouk (ymarz@mit.edu), Associate Professor, Department of Aeronautics and Astronautics