Applied Mathematics and Computation Seminar 2007-2008

• Apr 1, 2008 to Jun 6, 2008, Fridays at 12:00, in Gilkey 113: Applied Mathematics and Computation Seminar

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.
In 2007-2008 the focus theme includes identification of coefficients, optimization, and inverse problems. We have a series of tutorials and research talks on classical and modern approaches with applications to geosciences and engineering.

Graduate students can sign up for credit as MTH 607, Spring CRN 34730.

Schedule 2004-2005
Schedule 2005-2006
Schedule 2006-2007
Schedule 2007-2008
Current schedule

Schedule Fall and Winter 2007-2008:

• Sep 28, 2007 (Applied Mathematics and Computation Seminar) Adel Faridani, Numerical explorations in helical and fan-beam tomography

  Helical tomography is three-dimensional tomography where the x-ray source moves on a helical curve around the object to be imaged. The associated mathematical problem is to reconstruct a density function from its integrals over some of the lines that intersect this source curve. In recent years theoretically exact inversion formulas for this situation have been found that seem suitable for practical applications. This talks explores some of the foundations for the numerical analysis of algorithms based on such an inversion formula found by A. Katsevich, as well as its relation to 2D fan-beam tomography.

• Oct 5, 2007 (Applied Mathematics and Computation Seminar) Guenter Schneider, OSU Physics, The Density Matrix Renormalization Group

  The Density Matrix Renormalization Group (DMRG) (S. White 1992) is a powerful numerical method for strongly correlated quantum systems. This algorithm has achieved unprecedented accuracy for static, dynamic, and thermodynamic properties of one-dimensional problems but its applications now extend to many areas of physics from condensed matter theory to quantum chemistry to nuclear physics. In this talk I will introduce the method and its mathematical structure, and explore its properties using quantum information theory. I conclude with my own research on time-dependent DMRG and its application to nonequilibrium and transport problems.

• Oct 12, 2007 (Applied Mathematics and Computation Seminar) Fernando Morales, A Darcy-Brinkman Interface System

  The traditional approach for multiscale porous media is to look down from a (global) Darcy flow to a distributed microstructure of slow flow cells. We change our perspective here to look up from Darcy flow to a distribution of fast flow fractures. We describe the exchange of fluid between a porous medium saturated with a slightly compressible viscous fluid and the Stokes flow in a thin adjacent open channel. For the case of channel width close to zero, we show that the width-averaged problem is described by a Brinkman system on the interface.

• Oct 19, 2007 (Applied Mathematics and Computation Seminar) Son-Young Yi, Comparison of numerical methods for unsaturated flow with dynamic capillary pressure

  We consider unsaturated flow models that incorporate dynamic capillary pressures. The presence of dynamic capillary pressure terms changes the original parabolic equations to pseudo-parabolic equations. We systematically exhibit the difficulties associated with numerical approximation of dynamic capillary pressure terms using two classes of methods with many variants: a traditional cell-centered finite difference method (CFDM) and a locally conservative Eulerian-Lagrangian method (LCELM) combined with FD treatment of nonlinear diffusion. We present the results of our numerical simulations.

• Oct 26, 2007 (Applied Mathematics and Computation Seminar) Maria Dragila, OSU Crop and Soil Science, A mechanism of enhanced evaporation from soil cracks

  It has been known for many years that soil cracks enhance drying of the soil matrix. In addition, salt crusting on fracture walls indicates evaporation rates from within fractures that are much higher than can be accommodated with the traditional diffusive venting mechanism. Agricultural field and laboratory experiments over the past 60 years quantified the magnitude of drying in the vicinity of cracks, and under various conditions (e.g., wind, radiative heating, crack aperture). As a result of this work, agricultural practices were developed to ameliorate the negative effects of cracks on crop productivity. However, during those 60 years, no mechanism was suggested for the cause of enhanced drying. Recently we proposed that convection of fracture/crack air is the key mechanism driving the enhanced drying process. Each wall forming the crack maintains thermal boundary conditions with a vertical thermal gradient that oscillates diurnally. Convective venting occurs at night when either the vertical thermal gradient within the crack is significantly unstable (Rayleigh-Bernard instability) or when the density gradient at the atmosphere boundary permits entrainment of cooler (and usually drier) atmospheric air (Rayleigh-Taylor instability), thus generating invasive fingers of cooler atmospheric air. I will present the characteristics of the mechanism, the range of parameters that are realistic for natural settings, and field evidence for the existence of this mechanism in a natural rock fracture. The problem is yet to be solved numerically.

• Nov 2, 2007 (Applied Mathematics and Computation Seminar) Nathan L Gibson, Gradient-based Methods for Optimization. Part I.

  This two part talk will be a tutorial on gradient-based methods, beginning with Newton and Steepest Descent methods, and culminating with Levenberg-Marquardt. Along the way we will discuss variants such as Gauss-Newton, Inexact Newton, and damped Gauss-Newton, and techniques including Line Search Strategies and Trust Regions. Some theory on convergence rates will be presented. However, we will be primarily concerned with comparing and contrasting the various methods, and their particular applicability to least squares objective functions in parameter identification problems. Simple numerical demonstrations will be presented.

• Nov 9, 2007 (Applied Mathematics and Computation Seminar) Nathan L Gibson, Gradient-based Methods for Optimization. Part II.

  This is the second half of a two part tutorial on gradient-based methods, beginning with Newton and Steepest Descent methods, and culminating with Levenberg-Marquardt. Along the way we will discuss variants such as Gauss-Newton, Inexact Newton, and damped Gauss-Newton, and techniques including Line Search Strategies and Trust Regions. Some theory on convergence rates will be presented. However, we will be primarily concerned with comparing and contrasting the various methods, and their particular applicability to least squares objective functions in parameter identification problems. Simple numerical demonstrations will be presented.

• Nov 16, 2007 (Applied Mathematics and Computation Seminar) Stephen Lancaster, OSU Geosciences, Width adjustment to sediment supply in bedrock-incising valleys subject to episodic aggradation: Multiple numerical simulations over geologic time

  In steepland valleys subject to debris flows, episodic deposition typically inundates valley bottoms, but channels nevertheless incise deposits and erode bedrock. The hypothesis explored here is that, while continual fluvial processes evacuate deposits, temporary storage of episodic deposition drives creation of accommodation space through valley width adjustment. Data from three headwater valleys in the Oregon Coast Range show that, where valley width is variable and increases downstream, average valley bottom deposit depth is similar among sites and stationary with respect to contributing area. A simple numerical model of valley cross section evolution couples continual soil production and nonlinear diffusion, constant but contrasting rates of channel incision into deposits and bedrock, and episodic valley bottom-inundating deposition. The model reproduces observed features including flat, deposit-covered valley bottoms and abrupt transitions to steep (often oversteepened) valley sides. Simulations address sensitivity to two dimensionless numbers: (1) transverse slope number describes relative amplitudes of random transverse slopes of deposits and random white noise added to deposit surfaces; and (2) deposition number describes deposition rate relative to evacuation. Simulated valley bottom width increases with deposition and is maximum for deposition number near unity, but as in the data, simulated deposit depth is nearly constant over a large part of the parameter space. Field data and simulation results imply that valley width adjustment is the primary landscape response to changing sediment supply: while valley profile steepening is limited by rock uplift rate, valley widening is limited by the bedrock incision rate and is therefore potentially faster.

• Nov 30, 2007 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, Upscaling for linear and nonlinear elliptic problems with finite element methods. Part I

  Upscaling is a process which delivers coefficients for a numerical method at a scale much coarser than the (fine) scale at which data is actually available. Solving at a coarse scale may reduce the computational time, perhaps even by orders of magnitude. Upscaling can be cast as both i) a numerical homogenization procedure, and as ii) a parameter identification method. The outcome depends on what quantity is minimized in the upscaling process, what regularization, and what numerical method are used. In the talk(s) we explore both points of view, and focus on linear and nonlinear elliptic PDEs with applications to flow in porous media. In part I we focus on linear problems and introduce the topic and the basic conforming and nonconforming finite element methods. In part II we present our recent results for non-Darcy flow in porous media obtained with Cristiano Garibotti.

• Jan 11, 2008 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, Upscaling for linear and nonlinear elliptic problems with finite element methods. Part II

  Upscaling is a process which delivers coefficients for a numerical method at a scale much coarser than the (fine) scale at which data is actually available. Solving at a coarse scale may reduce the computational time, perhaps even by orders of magnitude. Upscaling can be cast as both i) a numerical homogenization procedure, and as ii) a parameter identification method. The outcome depends on what quantity is minimized in the upscaling process, what regularization, and what numerical method are used. In the talk(s) we explore both points of view, and focus on linear and nonlinear elliptic PDEs with applications to flow in porous media. In part I we focus on linear problems and introduce the topic and the basic conforming and nonconforming finite element methods. In part II we present our recent results for non-Darcy flow in porous media obtained with Cristiano Garibotti.

• Jan 17, 2008 (Applied Mathematics and Computation Seminar) Jim Dungan Smith, USGS, Edwards' Lecture: Flow, Sediment Transport, and Geomorphic Adjustment in Rivers

  Attendees are encouraged to attend The Fourth Edwards Lecture sponsored by College of Engineering

• Jan 18, 2008 (Applied Mathematics and Computation Seminar) John Singler, OSU Mechanical Engineering, Proper Orthogonal Decomposition-Based Algorithms for Model Reduction and Control of Partial Differential Equations

  Model reduction and control problems for partial differential equations (PDEs) are important for many applications. For linear PDE systems, the solutions to such problems are known to exist, however they are still very challenging to compute accurately. In this talk, we present an overview of this area and also introduce new algorithms for balanced model reduction and control of linear infinite dimensional systems based on proper orthogonal decomposition (POD), an optimal data reconstruction technique. We use POD systematically to provide convergence theory and error bounds. To illustrate our approach, we focus on computing low rank approximate solutions of Lyapunov equations and approximate balanced reduced order models. We present numerical results for a model partial differential equation system.

• Jan 25, 2008 (Applied Mathematics and Computation Seminar) Robert L. Higdon, Timestepping and Multiple Time Scales in Ocean Circulation Models, Part I

  The large-scale dynamics of the ocean include motions that vary on a wide range of time scales. In numerical models of ocean circulation, it is widespread practice to split the fast and slow motions into separate subproblems that are solved by different techniques. These matters will be discussed in two talks in consecutive weeks. The present talk will give an overview of time scales and time splitting and will describe some problems that can arise with multiple time scales and conservation of mass in layered ocean circulation models. Part 2 will describe some issues related to conservation of momentum and will show the results of numerical computations that test two different ways to formulate the momentum equation in the context of a time splitting.

• Feb 1, 2008 (Applied Mathematics and Computation Seminar) Robert L. Higdon, Timestepping and Multiple Time Scales in Ocean Circulation Models, Part II

  This talk is a continuation of Part I. In numerical models of ocean circulation, it is widespread practice to split the fast and slow motions into separate subproblems that are solved by different techniques. The present talk will describe some issues related to conservation of momentum and will show the results of numerical computations that test two different ways to formulate the momentum equation in the context of a time splitting.

• Feb 15, 2008 (Applied Mathematics and Computation Seminar) Rubin Landau, OSU Physics, Computational Physics, An Improved Path for Physics Education?

  Evidence of the need for a change in physics education will be presented. It will be argued that computational physics provides a broader, more balanced and more flexible education than the traditional physics major. A survey of all computational science programs in the USA will be presented, as will be details of the BS in Computational Physics program at Oregon State University and of the educational materials developed. It will be proposed that presentation of physics within a computational problem-solving paradigm is a more effective and efficient way to teach physics than the traditional one.

• Feb 22, 2008 (Applied Mathematics and Computation Seminar) Ronald B. Guenther, Modeling Shocks Underground

  This investigation arose from an effort to destroy mines in the near shore region by blowing up the mines in question. The technique in question will be described, the problems that arose in our investigation will be explored, and finally the shock model will be presented.

• Feb 29, 2008 (Applied Mathematics and Computation Seminar) Ralph E. Showalter, A Dam Problem

  The formulation and existence theory is presented for a system modeling diffusion of a slightly compressible fluid through a partially saturated poroelastic medium. Nonlinear effects of density, saturation, porosity and permeability variations with pressure are included, and the seepage surface is determined by a variational inequality on the boundary.

• Mar 14, 2008 (Applied Mathematics and Computation Seminar) Dr. Konstantin Lipnikov, Los Alamos National Laboratory, Mimetic finite difference method for diffusion problems

  ABSTRACT. The mimetic finite difference (MFD) method mimics important properties of physical and mathematical models. As a result, conservation laws, solution symmetries, and the fundamental identities of the vector and tensor calculus are held for discrete models. The MFD method retains these attractive properties for full tensor coefficients and arbitrary polygonal and polyhedral meshes which may include non-convex and degenerate elements. Modeling with polygonal and polyhedral meshes has a number of advantages. For subsurface flows, such meshes allow to describe accurately small, detailed structures such as tilted layers, pinch-outs, irregular inclusions, etc. The polygonal and polyhedral meshes cover the space more efficiently than simplicial meshes which eventually reduces the number of discrete unknowns without lose of accuracy. The locally refined meshes with hanging nodes are particular examples of polygonal and polyhedral meshes with degenerate elements. Such meshes are used frequently to improve resolution in the regions of interest, such as moving fluid fronts, sharp solution variations, etc. The MFD method works for all these meshes. For a linear diffusion problem, I'll discuss important details of the MFD method. In particular, the MFD method gives a rich family of schemes with equivalent properties. For simplicial meshes, I'll show how the MFD method is related to a finite element method. For polygonal and polyhedral meshes, I'll show how new multi-point flux approximation methods can be derived and analyzed using the MFD framework. I'll also compare the MFD method with the Kuznetsov-Repin finite element method.

Schedule Spring 2007-2008:

• Apr 4, 2008 (Applied Mathematics and Computation Seminar) Vrushali Bokil, Finite Difference and Finite Element Methods for Maxwell's Equations: A Tutorial

  In this talk, I will give an introduction to the direct numerical approximation of Maxwell's equations, with emphasis on finite difference and finite element methods in the time domain. In 1966 Kane Yee from Lawrence Livermore National Laboratory proposed a finite difference scheme for the time-domain Maxwell's equations which is now well known as the Finite Difference Time Domain (FDTD) or Yee scheme. The FDTD scheme has become a standard in computational electromagnetics. In the 1980's J.C. Nedelec popularized a family of mixed finite elements called edge elements for the numerical simulation of Maxwell's equations. Thereafter many different finite elements methods, as well as finite volume and finite difference methods, have been constructed. On structured, Cartesian grids many of these schemes share similarities to the original Yee FDTD scheme. I will talk about the numerical aspects of these methods including their stability, dispersion and accuracy properties. The talk is aimed at graduate students who have a background in numerical analysis of partial differential equations.

• Apr 11, 2008 (Applied Mathematics and Computation Seminar) Vrushali Bokil, Perfectly Matched Layers for Computational Electromagnetics

  The effective modeling of electromagnetic waves on unbounded domains by numerical techniques, such as the finite difference or the finite element method, is dependent on the particular absorbing boundary condition used to truncate the computational domain. In 1994, J. P. Berenger created the perfectly matched layer (PML) technique for the reflection-less absorption of electromagnetic waves in the time domain. The PML is an absorbing layer that is placed around the computational domain of interest in order to attenuate outgoing radiation. Berenger showed that his continuous PML model allowed perfect transmission of electromagnetic waves across the interface of the computational domain regardless of the frequency, polarization or angle of incidence of the waves. The waves are then attenuated exponentially with respect to depth into the absorbing layers.

  The properties of the continuous PML model have been studied extensively and are well documented. Since its original inception in 1994, the PML technique has extended its applicability to areas other than computational electromagnetics such as acoustics and elasticity. In this talk, I will describe the original split field PML technique of Berenger as well as some of the popular modifications of the original method. The mathematical and numerical aspects of the problem will be discussed. Finite difference and mixed finite element methods will be applied to the discretization of the PML in the time domain. The discrete PML will be used to demonstrate wave propagation on unbounded domains in two dimensions.

• Apr 18, 2008 (Applied Mathematics and Computation Seminar) Kagan Tumer, OSU Mechanical Engineering, Control and Coordination in Distributed Autonomous Systems: A Collectives Approach

  The number of autonomous systems composed of many interacting computational agents has exploded over the last decade. Spurred by the ever increasing size, interconnectivity and complexity of systems on one hand and the miniaturization and affordability of computing power on the other, new paradigms to controlling such systems are emerging. Coordinating thousands of computational agents in dynamic and stochastic environments, an idea that a mere decade ago would have been outlandish, is not only possible, but imperative today. Indeed, the technological bottlenecks today stem from the lack of mathematics and algorithms to autonomously coordinate such systems rather than difficulties associated with building them.
  This talk directly addresses this issue. In particular, it focuses on how to design and coordinate such autonomous systems through ``collectives'' (a set of learning agents that optimize a system level objective through pursuing their own local objectives). The main challenge in this approach is in deriving the local objectives that when reached by the agents, lead to good system level behavior. Successful applications of collectives include controlling multiple robots/autonomous vehicles; managing air traffic; and coordinating thousands of nano or micro computing devices.

• Apr 25, 2008 (Applied Mathematics and Computation Seminar) Son-Young Yi, Numerical methods for saddle point problems (TALK CANCELLED DUE TO ILLNESS)

  Large linear systems in saddle point form arise in many applications throughout computational science and engineering. The mixed finite element methods in fluid and solid mechanics are typical examples of saddle point problems. The indefiniteness of their matrices makes it hard to solve the systems. In this talk, we will present various numerical solution techniques for this type of systems, with an emphasis on iterative methods and preconditioning techniques for large and sparse problems. This expository talk will be useful to graduate students as an introduction to this rich and important subject.

• May 2, 2008 (Applied Mathematics and Computation Seminar) John Schmitt, OSU Mechanical Engineering, Stabilization of Reduced Order Models of Legged Locomotion

  Reduced order models have been utilized to accurately model the steady locomotion dynamics of a variety of running animals and have been used as a target for control of higher dimensional robotic implementations. Tuned appropriately, these models exhibit passively stable periodic gaits when utilizing a fixed leg touch-down angle protocol to determine foot placement. However, incorporating a similar leg touch-down protocol into a spatial spring loaded inverted pendulum model yields only unstable gaits, suggesting that changes in the leg touch-down angle in response to perturbations are important for stability. Additionally, while each template models the leg dynamics by an energy-conserving spring, insects and animals have structures that dissipate, store and produce energy during a stance phase. Recent investigations into the spring-like properties of limbs, as well as animal response to drop-step perturbations, suggest that animals use their legs to manage energy storage and dissipation, and that this management is important for gait stability. In this presentation, we modify the planar reduced order locomotion models to include changes in leg touch-down angle and energy. We introduce leg touch-down angle variations through a simple feedback control law based upon the previous leg touch-down and lift-off angles. Energy variations in the sagittal plane model are incorporated via a clock-driven leg actuation protocol that varies the force-free leg length during the stance phase, yet maintains qualitatively correct force and velocity profiles. In contrast to the partially asymptotically stable gaits identified in previous analyses, we find that incorporating energy and leg angle variations in this manner enables the system to recover from perturbations similar to those that might be encountered during locomotion over rough terrain.

• May 9, 2008 (Applied Mathematics and Computation Seminar) Kenneth Kennedy, (Student presentation): Application of the 2DSVD to Gappy Images

  The 2DSVD is presented with an application to incomplete data. It has been shown that the SVD (with many other names) may be used to approximate missing data in images. The techniques are extended to the 2DSVD and comparisons are made between the computational efficiency, storage, and results from the SVD and the 2DSVD.

• May 9, 2008 (Applied Mathematics and Computation Seminar) Viviane Klein, (Student presentation, part II) A-posteriori error estimate for FE methods with non-smooth coefficients

  An a-posteriori estimate by Bernardi/Verfuerth as well as an interesting regularity result for solutions to problems with non-smooth coefficients in fractional order Sobolev spaces will be presented. An example demonstrating the application will be shown.

• May 16, 2008 (Applied Mathematics and Computation Seminar) John W. Lee, Continuation Methods

  A gentle introduction to continuation methods for solving problems when it may be difficult to know where to start -- rough translation finding good initial guesses when such are not obvious. Examples drawn from algebraic equations or systems of such equations and boundary value problems for ODEs.

• May 23, 2008 (Applied Mathematics and Computation Seminar) Malgorzata Peszynska, Numerical multiscale methods:overview

  Direct numerical discretization of stationary and/or transient PDEs with multiscale coefficients requires solution of huge linear systems or very stiff ODEs. The associated complexity can be handled in some cases by use of various multigrid and multilevel solvers.
  On the other hand, in the last few years a plethora of multiscale numerical methods based on finite elements have been proposed for various applications. These include the heterogeneous multiscale method, the variational multiscale method, and the multiscale FE (with overlapping variants) as well as subgrid or mortar methods. Most are designed to compute the macroscale average solution, and some are able to recover next order effects. Some methods work best for periodic coefficients, and some can be extended to handle any coefficients including random. Some assume scale separation, and some use special test functions in the classical or mixed variational formulation. Finally, some methods can be naturally applied to transient and nonlinear problems. In the talk we give an overview of main ideas and issues as well as discuss some open problems in various applications.

• May 30, 2008 (Applied Mathematics and Computation Seminar) Son-Young Yi, Numerical methods for saddle point problems

  Large linear systems in saddle point form arise in many applications throughout computational science and engineering. The mixed finite element methods in fluid and solid mechanics are typical examples of saddle point problems. The indefiniteness of their matrices makes it hard to solve the systems. In this talk, we will present various numerical solution techniques for this type of systems, with an emphasis on iterative methods and preconditioning techniques for large and sparse problems. This expository talk will be useful to graduate students as an introduction to this rich and important subject.

• Jun 6, 2008 (Applied Mathematics and Computation Seminar) Scott Clark, (Senior Thesis Presentation) Finite Element Modeling of Uncertain Interfaces

  Modeling of uncertainty within computational mathematics is an active field with many diverse applications. We consider solving partial differential equations with finite element method where some of the data of the problem is uncertain. We explore modeling of uncertain but piecewise constant coefficients of elliptic boundary value problems. The domain of the PDE is bounded and is decomposed into subdomains dependent on the values of coefficients. Either the value or the interface between the subdomains is uncertain. We are interested in understanding the influence of the uncertainty of the coefficients upon the fluxes through the boundaries. We present the model problem and the approaches we used.