Applied Mathematics and Computation Seminar 2006-2007

The AMC seminar is devoted to general topics in applied mathematics and computation. Each year, we select a focus theme, with the proportion of general to specific topics about 1-1. We welcome speakers and audience of faculty, researchers, and graduate students from mathematics, geosciences, computer science, engineering, atmospheric sciences, and all other disciplines.

Focus theme in 2006-2007: "Uncertainty and multiple scales in modeling and computation".

Graduate students can sign up for credit as MTH 607

Schedule Fall/Winter 2007:

• Sep 29, 2006 (Applied Mathematics and Computation Seminar) Ralph E. Showalter, Brinkman versus Darcy flow in porous media: modeling and analysis

• Oct 6, 2006 (Applied Mathematics and Computation Seminar) Eugene Zhang, OSU Electrical Engineering and Computer Science, Vector Field Simplification and Periodic Orbit Extraction on Surfaces

  Vector fields appear in many science and engineering domains as well as computer graphics. In this talk, I will discuss numerical algorithms that allow the extraction and simplification of topology of a vector field defined on a 3D surface. Such algorithms can be used in computer graphics applications such as computer-generated arts and texture synthesis to remove or reduce the visual artifacts caused by singularities in the input vector field. We also apply our techniques to the visualization of CFD simulation datasets of gas and diesel engines.

• Oct 13, 2006 (Applied Mathematics and Computation Seminar) Gary Egbert, OSU Marine Geology and Geophysics, Introduction to data assimilation and applications

• Oct 20, 2006 (Applied Mathematics and Computation Seminar) Anna Maria Spagnuolo, Oakland University, Reiterated Homogenization and the Double-Porosity Model

  In this talk, I will consider the effects of a hierarchical, multiple layered system of fractures on the flow of a single-phase, slightly compressible fluid through a porous medium. A microscopic flow model is first defined which describes precisely the physics of the flow and the geometry of the fracture system and porous matrix, all of which depend on a positive parameter $\epsilon$ that determines the scale of the various fracture-level thicknesses. I will then show by a rigorous mathematical argument that the unique solution of this microscopic problem converges as $\epsilon\rightarrow 0$ to the solution of a double-porosity model of the global macroscopic flow. The techniques make use of the concept of reiterated homogenization and essentially consist of an adaptation of the methods of extension and dilation operators to the reiterated-homogenization context. Finally, I will show how the porosities and permeability tensor of the porous medium are determined in a precise way by certain physical and geometric features of the microscopic fracture domain, the microscopic matrix blocks, and the interface between them.

• Oct 27, 2006 (Applied Mathematics and Computation Seminar) Belinda Batten, OSU Mechanical Engineering, The design of control systems for Micro Air Vehicles (MAVs)

  Operation of Micro Air Vehicles (MAVs) in cluttered environments at low speed presents significant challenges for design of the control system. In this talk, we will present work done regarding mathematical modeling of bioinspired hair-cell sensors for detection of flow separation over MAV wings. In addition, we will present an approach to sensor and actuator placement and design that utilizes sensitivity analysis applied to distributed parameter systems.

• Nov 3, 2006 (Applied Mathematics and Computation Seminar) Nathan Gibson, Inverse Problem for Distributions of Dielectric Parameters

  In this talk I will present theoretical and computational results for inverse problems involving Maxwell's equations coupled with a general polarization term which includes uncertainty in the dielectric parameters (e.g., relaxation times). I will show our attempts to determine an unknown probability distribution of parameters which describe the dielectric properties of the material. Examples for discrete and continuous distributions will be given.

• Nov 10, 2006 (Applied Mathematics and Computation Seminar) Jerry Brackbill, Particle Solutions, "On Finite Difference Methods for Non-Rectilinear Grids"

• Nov 17, 2006 (Applied Mathematics and Computation Seminar) John W. Lee, A unifying approach to small oscillations of strings, beams, and Sturm-Liouville eigenvalue problems. Part I

• Dec 1, 2006 (Applied Mathematics and Computation Seminar) John W. Lee, A unifying approach to small oscillations of strings, beams, and Sturm-Liouville eigenvalue problems. Part II

• Jan 12, 2007 (Applied Mathematics and Computation Seminar) Vrushali Bokil, Operator Splitting Schemes with Distributed Lagrange Multiplier based Fictitious Domain Methods and Perfectly Matched Layers for Wave Propagation Problems

• Jan 19, 2007 (Applied Mathematics and Computation Seminar) Malte A. Peter, Centre for Industrial Mathematics, University of Bremen, Homogenisation of coupled reaction--diffusion systems inducing an evolution of the microstructure of the porous medium

  Chemical processes in porous media are modelled on the pore scale using reaction--diffusion equations. The resulting prototypical systems of coupled linear and nonlinear differential equations are homogenised in the context of periodic media. The talk addresses two aspects: First, different scalings of certain terms of the reaction--diffusion system with powers of the homogenisation parameter are reasonable. The scaling arises from geometrical considerations or from the process itself. Depending on the particular choice of these scaling powers, different systems of equations arise in the homogenisation limit. The resulting models are classified using a unified approach based on two-scale convergence. Second, chemical degradation mechanisms of porous materials often induce a change of the pore geometry. This effect cannot be captured by the standard periodic homogenisation method due to the local evolution of the microscopic domain. A mathematically rigorous approach is suggested which makes use of a transformation of the evolving domain to a periodic reference domain. Two scenarios are considered: First, for given evolution, it is shown that the transformed problem can be homogenised within the context of periodic media and that the homogenised problem can be transformed back to the evolving domain. Second, the additional terms in the transformed problem can also be interpreted physically and related to the process itself. For the prototypical situation where the reaction induces a change of pore-air volume, a model for the additional terms arising from the transformation is suggested, which relates the terms from the transformation to the reaction--diffusion process. The well-posedness of the resulting system of coupled partial and ordinary differential equations and its homogenisation are addressed.

• Jan 26, 2007 (Applied Mathematics and Computation Seminar) , Please attend Colloquium by Dacian Daescu at 3:00.

• Feb 2, 2007 (Applied Mathematics and Computation Seminar) Son-Young Yi, Nonconforming Mixed Finite Element Methods for Linear Elasticity

  In this talk, we present nonconforming, mixed finite element methods based on the Hellinger-Reissner variational principle in which the stress and the displacement are sought simultaneously; both two and three-dimensional elasticity are treated. We show stability and convergence of the methods. Results of numerical computations are also presented.

• Feb 9, 2007 (Applied Mathematics and Computation Seminar) Harry Yeh, OSU Department of Civil Engineering, Jump Condition and Free-Surface Dynamics

  Within the basis of continuum hypothesis, the jump condition of conservation equation is derived. At a material boundary such as air-water interface, the jump condition of conservation of mass yields the kinematic free-surface boundary condition, the jump condition of conservation of linear momentum yields the dynamic free-surface boundary condition. The no-slip boundary condition arises at a material boundary as a consequence of the jump condition for mechanical-energy conservation; note that the no-slip condition is usually an assumed condition, but our analysis demonstrates that it is a necessary condition for the mechanical energy consideration at the interface. A constant magnitude of pressure along the free surface is often imposed in traditional water-wave problems for irrotational fluid motions. If this condition is imposed, the resulting potential-flow solution demands the fictitious energy input from the surroundings, i.e., air, through the free surface. Under a more rigorous free-surface condition such as vanishing stresses on the interface, irrotational flow cannot exist at the free surface and must form a boundary layer In a real-fluid environment, flows in general must be vortical at the free surface. Because of this observation, the jump condition of vorticity equation is examined. It is found that vorticity normal to the interface and vorticity bending caused by the interface curvature play important roles in determination free-surface vortex motions.

• Feb 16, 2007 (Applied Mathematics and Computation Seminar) Robert Lipton, Lousiana State University, Optimal bounds on local stress, strain, and electric fields inside random media

  We investigate local field behavior inside random two-phase media. Over the years a major part of the research work has focused on characterizing the effective transport properties for statistically defined microgeometries. In this lecture we depart from this trend and present new developments that characterize local field behavior inside random composites. Here we present new lower bounds on all higher order moments of local fields inside random media. The bounds are given in terms of the available statistical information describing the microstructure. We show that these bounds are the best possible as they are attained by by several different classes of microstructures including coated confocal ellipsoids, coated spheres, and layered microstructures.

• Feb 23, 2007 (Applied Mathematics and Computation Seminar) John Selker, OSU Department of Biological and Ecological Engineering, The utility and nature of the Bousinesq Equation for description of shallow aquifers

• Mar 2, 2007 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, Peaceman and Thiem well models or how to remove a logarithmic singularity from your numerical solution

  In this expository talk we consider an elliptic PDE whose solution in 2D, due to an "almost" point source term (or line source in 3D) exhibits a singularity. This presents a challenge to numerical methods which try to subtract out that singularity. Alternatively, it is a common practice in reservoir simulation where such singularities are associated with real injection/production wells, to use a so-called well model. We will discuss some such solutions and show how they do NOT carry over to problems with highly heterogeneous coefficients.

• Mar 9, 2007 (Applied Mathematics and Computation Seminar) Ralph E. Showalter, Pseudo-Parabolic Partial Differential Equations

  This class of partial differential equations arises in many applications and has many intimate connections with classical parabolic equations and systems. We describe some of these connections and develop many of the surprising properties of the solutions by very elementary methods.

• Mar 16, 2007 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, Wells continued: large scale computing, geostatistical simulations and optimization

  In this talk we continue our expository material on wells in reservoir simulation, this time focusing on secondary oil and gas recovery using waterflooding. We give overview of the associated PDE models and of difficulties in their numerical approximation. The focus of this talk is on additional levels of complexity encountered when the numerical PDE model is just one element of a complicated system of reservoir characterization and optimization. We show results from large scale computations involving geostatistical simulations and optimization of well placement. In the former we use GSLIB methodology and Monte Carlo simulations, the latter involves a Simulated Annealing algorithm.

Schedule Spring 2007:

• Apr 6, 2007 (Applied Mathematics and Computation Seminar) Julia Jones, OSU Geosciences, Modelling time to extinction of ecological communities in stream networks

  Collaborating authors: Jorge Ramirez, and Sean Moore

• Apr 13, 2007 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, A-posteriori error estimate framework in finite element and finite difference methods

  When using numerical computation as a scientific tool, one has to include validation and verification as an internal part of the computing process. One of the elements of that process is assessment of the computational error: that is, the difference between the analytical solution and the numerical solution. Without knowing the analytical solution this seems like an impossible task; however, the theory of a-posteriori error estimators and the practice of error indicators comes to the rescue. The former has been developed mainly for finite elements and is deeply rooted in functional analysis. The latter is easy to apply especially for finite difference methods but is not as firmly based on theory. In the talk we give an overview of theoretical and practical issues as well as present a technical proof of an estimator for mixed finite element methods.

• Apr 20, 2007 (Applied Mathematics and Computation Seminar) Edward C. Waymire, A Rate of Convergence for the Lagrangian Averaged Regularization of 3d Incompressible Navier-Stokes Equations

  The Lagrangian Averaged Navier-Stokes equations is a regularization, depending on a parameter $alpha ge 0$, of Navier-Stokes equations ($alpha = 0$) designed as a turbulence model in place of Navier-Stokes for numerical computations. While regularizations of Navier-Stokes date back to Leray, the LANSalpha model is regarded as a physically more realistic model than its predecessors in this area. In this talk we will describe how to use a combination of probabilistic and functional/harmonic analysis to obtain (small ball) global existence, uniqueness and a rate of convergence as $alpha o 0$. This is based on joint work with Larry Chen, Ronald Guenther, Enrique Thomann at OSU, and Sun-Chul Kim, Chung Ang University.

• Apr 27, 2007 (Applied Mathematics and Computation Seminar) Ralph E. Showalter, The Darcy and Stokes Systems

  This is an exposition of the basic theory of these two central systems which describe fluid flow in very different situations. The development emphasizes the similarities in the formulation of the basic initial-boundary-value problems, existence, and regularity of the solution.

• May 9, 2007 (Applied Mathematics and Computation Seminar) Professor Jim Douglas, Jr., Purdue University, Euler-Lagrangian Methods in the Simulation of Two-Phase Flow in Heterogeneous Porous Media

  The lecture will present the application of a locally-conservative Eulerian- Lagrangian method that has been discussed by Dr. Son-Young Yi for semi- linear parabolic equations to multiphase flow in porous media. After a brief reminder of her presentation and of the history of the derivation of the method, I shall apply a version of the method to two-phase flow in fractally inhomogeneous porous media. Several experimental results will be shown to indicate the convergence of the method under mesh refinement and its overall validity.

• May 18, 2007 (Applied Mathematics and Computation Seminar) Roger Samelson, College of Oceanic and Atmospheric Sciences, Some aspects of geophysical fluid modeling

  Numerical modeling of the coastal ocean has progressed from first-order linear wave equations thirty years ago, to three-dimensional nonlinear numerical simulations today. The range of scales is vast, and the simulation problem presents many challenges. For the prediction problem, uncertainties in initial conditions and forcing motivate the consideration of forecast ensembles, which are widely used in operational numerical weather prediction systems. This in turn leads to consideration of the mechanisms of disturbance growth and instability in time-dependent fluid flows. In this context, a new method (due to recent COAS Ph.D. Christopher Wolfe) for obtaining multiple Lyapunov vectors for large systems is discussed.

• May 25, 2007 (Applied Mathematics and Computation Seminar) Mart Oostrom, Andy Ward, Mark White, Pacific Northwest National Laboratory, Subsurface Transport Over Multiple Phases (STOMP) presentation: I. Experimentation and Numerical Simulation. II. Heterogeneity and Inverse Modeling. III. Numerical Simulation of Natural Gas Hydrate Production.

• Jun 1, 2007 (Applied Mathematics and Computation Seminar) Enrique A. Thomann, A mathematical model for carbon sequestration by forest ecosystems

  Modeling the amount of carbon sequestered in a forested landscape under different disturbance regimes is a problem of contemporary interest due to its management and policy implications. In this presentation, a summary of the main aspects of a mathematical framework that captures some the essential aspects of numerical simulation models of carbon sequestration will be presented. The mathematical framework is flexible enough to allow for different disturbance regimes affecting the stands (e.g., different interarrival times of disturbances, amounts of carbon released by disturbance, etc). Some consequences that can be drawn from this analysis and open problems will also be discussed. This is joint work with Nam Ngo (Math MS OSU June 2006), Mark Harmon (Forestry OSU), benefited from discussion with Ed Waymire, and originated from problems presented as part of the IGERT in Ecosystem Informatics.

• Jun 8, 2007 (Applied Mathematics and Computation Seminar) Robert L. Higdon, Multiple Scales in Ocean Circulation Models

  The dynamics of the ocean involve wide ranges of space scales, time scales, and rates of mixing. These multiple scales must be taken into account in numerical models of the circulation of the ocean. This talk will give an overview of these physical scales and then describe some numerical issues related to multiple time scales.