Computational Mathematics Seminar


  • Tuesday, September 23, 2008
    3:00 pm, Thackeray 703
    Alexander Lozovskiy, University of Pittsburgh,
    Noise generated by turbulent flow - Lighthill's acoustic analogy and its computational implementation.
    Abstract:
    We derive in intuitive and physical manner the Lighthill equation, and present the FEM implementation, with error estimates.

  • Tuesday, September 30, 2008
    1:00 pm, Thackeray 704
    Zhangxin Chen, University of Calgary,
    Beyond CSS, SAGD, and VAPEX: Reservoir Modeling and Simulation for Heavy Oil and Oil Sands
    Abstract:
    Heavy oil and oil sands are important hydrocarbon resources that are destined to play an increasingly critical role in the oil supply of the entire world. The oil sands in Alberta contain two (2) trillion barrels of oil. The important question is: How much of this oil is recoverable and what methods can be used? In this presentation, the speaker will present some of the recovery methods for heavy oil and oil sands: CSS (cyclic steam stimulation), SAGD (steam assisted gravity drainage), and VAPEX (vapor extraction). The differences between these methods and conventional recovery methods will be emphasized, and some open challenging problems in heavy oil and oil sands reservoir modeling and simulation will be discussed.

  • Thursday, October 2, 2008
    2:00 pm, Thackeray 704
    Konstantin Lipnikov, LANL,
    Mimetic finite difference method for solving PDEs on polygonal and polyhedral meshes
    Abstract:
    Various approaches to extend finite element methods to non-traditional elements (pyramids, polyhedra, etc) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry analysis. The mimetic finite difference (MFD) method has many similarities with the low-order finite element method. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build elemental stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for complex elements. Existence (but not uniqueness!) of real basis functions corresponding to stiffness and mass matrices is used only in the convergence analysis.

  • Tuesday, October 7, 2008
    3:00 pm, Thackeray 703
    Nate Mays, University of Pittsburgh,
    Defect Correction with Tikhonov
    Abstract:
    We will take a look at using the Defect Correction Method applied to Tikhonov regularization to come up with an iterated Tikhonov method. This will allow us to choose the Tikhonov parameter for stability and then iterate for accuracy.

  • Friday, October 31, 2008
    2:00 pm, Thackeray 703
    Christopher Colburn, Mechanical and Aerospace Engineering, UCSD,
    EnVE: A consistent hybrid ensemble/variational estimation strategy for environmental turbulent flows
    Abstract:
    Chaotic systems are characterized by long-term unpredictability. Existing methods designed to estimate and forecast such systems, such as Extended Kalman filtering (a "sequential" or "incremental" matrix-based approach) and 4DVar (a "variational" or "batch" vector-based approach), are essentially based on the assumption that Gaussian uncertainty in the initial state, state disturbances, and measurement noise lead to uncertainty of the state estimate at later times that is well described by a Gaussian model. This assumption is not valid in chaotic systems with appreciable uncertainties. A new method is thus proposed that combines the speed and LQG optimality of a sequential-based method, the non-Gaussian uncertainty propagation of an ensemble-based method, and the favorable smoothing properties of a variational-based method. This new approach, referred to as Ensemble Variational Estimation (EnVE), is a natural extension of the Ensemble Kalman and 4DVar algorithms. EnVE is a hybrid method leveraging sequential preconditioning of the batch optimization steps, simultaneous backward-in-time marches of the system and its adjoint (eliminating the checkpointing normally required by 4DVar), a receding-horizon optimization framework, and adaptation of the optimization horizon based on the estimate uncertainty at each iteration. If the system is linear, EnVE is consistent with the well-known Kalman filter, with all of its well-established optimality properties. The strength of EnVE is its remarkable effectiveness in highly uncertain nonlinear systems, in which EnVE consistently uses and revisits the information contained in recent observations with batch (that is, variational) optimization steps, while consistently propagating the uncertainty of the resulting estimate forward in time.

  • Tuesday, November 4, 2008
    3:00 pm, Thackeray 703
    Benjamin Ganis, University of Pittsburgh,
    Implementation of a Mortar Mixed Finite Element Method using a Multiscale Flux Basis with Applications to Stochastic Collocation of Flow in Porous Media
    Abstract:
    In the first half of this talk, I describe an alternate implementation of the mortar mixed finite element method, which is computationally more efficient in some cases. This implementation forms a multiscale flux basis, containing discrete Green's functions for mortar degrees of freedom. In the second half of this talk, I introduce the stochastic collocation method which is used to quantify uncertainty for flow in porous media in which permeability is stochastic. In the non-stationary case, I show how the multiscale flux basis implementation can be used to make this algorithm computationally more efficient by several orders of magnitude.



  • Tuesday, November 11, 2008
    3:00 pm, Thackeray 703
    John Burkardt, Interdisciplinary Center for Applied Mathematics, Virginia Tech,
    SPARSE GRIDS - Extracting Information in High Dimensions
    Abstract:
    The relentless development of mathematical techniques and computing power has made it possible to pose integral problems in abstract spaces of very high dimension. Approximate methods that work well in lower dimensions can become impractical, unreliable, or inefficient as the dimension is increased. Product grids, in particular, fail spectactularly in high dimensions, and yet these grids have many attractive properties. They allow us to specify particular rules in each dimension, they can be constructed to take advantage of nesting, and they can be made to have desirable exactness properties. Smolyak's sparse grid construction provides a way to retain the desirable properties of a product grid, while avoiding the catastrophic explosion in the number of points needed to define a quadrature rule. We will examine the algorithm that is used to develop a sparse grid. We will discuss the kinds of problems for which a Smolyak sparse grid can be expected to compete or outperform the Monte Carlo method and its relatives. We will discuss some software that makes it possible to use a sparse grid as though it were just another quadrature rule.



    • Last updated: September 16, 2008 by Catalin Trenchea