Computational Mathematics Seminar

  • Tuesday, December 8, 2009
    10:00 AM, Thackeray 703
     Attou Miloua, Department of Mathematics, University of Pittsburgh
    "TBA"
     Abstract:
    TBA

  • Tuesday, December 1, 2009
    10:00 AM, Thackeray 703
     Noel Walkington, Department of Mathematical Sciences, Carnegie Mellon University
    "Automating DOF Bookkeeping"
     Abstract:
    A finite element code has three distinct components,
    1) Mesh generation (provides a cell complex)
    2) Numerical kernel (constructs element matrices)
    3) Linear solver
    and the "glue" connecting these components are the degrees of freedom. Robust open source meshing and linear solver software is now freely available, and the problem dependent portion of the numerical kernel is typically no more than a page of code. This is not so for the dof bookkeeper; code to manage the dof's for a specific problem is often tedious and time consuming to write.
    In this talk I will discuss the design of a dof bookkeeper. A bookkeeper interacts with the mesh generator (via a cell complex), numerical kernel (via basis functions), and linear solver (via indices for matrices and vectors), so it is necessary to specify interfaces between all of these components and the functionality required of each. I will describe a bookkeeper that I have developed, and illustrate its utility with some classical examples.

  • Tuesday, November 17, 2009
    10:00 AM, Thackeray 703
     Nathaniel Mays, Department of Mathematics, University of Pittsburgh
    "Looking Forward: A Study of Inverse Problems"
     Abstract:
    This talk will consider a variety of inverse problems and a regularization methods for solving them. Regularization methods are needed because typically the inverse operator for the true problem is not continuous. We will consider iterated Tikhonov regularization and modifications of it. The iterated methods are used to obtain higher accuracy in the solution without causing unacceptable amplification of the noise in the data.

  • Tuesday, November 10, 2009
    10:00 AM, Thackeray 703
     Ross Ingram, Department of Mathematics, University of Pittsburgh
    "The Brinkman-Boussinesq model for natural convection in complicated domains"
     Abstract:
    In many applications, fluid velocities are too large to model with Darcy's equation and pore geometry is too complex to model fluid flow with the Navier-Stokes equations. The Brinkman equation is a physically viable and numerically efficient model for these cases. I will discuss two different perspectives of the Brinkman flow model. First, the Brinkman model can be considered an extension of Darcy's equation as a porous media solver. Secondly, the Brinkman equation can be formulated as a penalized Navier-Stokes equation. For the first perspective, I will present a limitation of the Brinkman model via a conditional existence result for steady-state solutions. For the second perspective, in effort to model the fluid/temperature dynamics of pebble bed nuclear reactors, I show that the formally posed (penalized) Brinkman-Boussinesq model is consistent with the Navier-Stokes-Boussinesq equation. An optimal rate of convergence is concluded which will prove central to future numerical analysis of the model.

  • Tuesday, November 3, 2009
    10:00 AM, Thackeray 703
     Wenjun Ying, Department of Mathematical Sciences, Michigan Technological University
    "Multiscale modeling for cardiac electrical dynamics with a space-time adaptive algorithm"
     Abstract:
    Studying cardiac electrical dynamics, the electrical activities of the heart, can help us understand better the underlying mechanisms for some related cardiovascular heart diseases, which kill hundreds of thousands of people in the United States every year. In mathematical biology of the heart, the cardiac electrical dynamics can be modeled by singularly perturbed reaction-diffusion equations, coupled with a set of stiff ordinary differential equations. Using mathematical modeling, hypotheses can be tested and the dynamics can be investigated in ways that cannot be done experimentally or clinically given access to more information about the system. Because electrical waves in the heart involve multiple widely varying scales in both space and time, computer simulation of electrical dynamics have been limited to either small domains or to time durations that are short relative to that observed for realistic arrhythmias. The electrical wave fronts typically occupy only a small fraction of the domain, are very sharp (in space) and change very rapidly (in time) while, in the region away from the wave fronts, the electric potential is much flat and changes slowly. With standard numerical methods on uniform grids, very small mesh parameters and very small timesteps must be used to correctly resolve the fine details of the sharp and rapidly changing wave fronts.
    In this talk, I will present a space and time adaptive mesh refinement algorithm for multiscale modeling of the cardiac electrical dynamics. The adaptive algorithm solves the reaction-diffusion equations with coarse grids and large timesteps in the area where the electric potential is flat and changes slowly. It places fine grids only in the region where the sharp electrical waves are located, and uses small timesteps only in the phases where the action potential changes very rapidly. The number of grid nodes and timesteps used with the adaptive algorithm is to some extent minimized. A novel aspect of the method is that it can be used with non-rectangular elements on domains with complex geometries. Numerical simulations will be presented in two space dimensions (2D) and three space dimensions (3D) demonstrating the performance of the algorithm.

  • Tuesday, October 27, 2009
    10:00 AM, Thackeray 703
     William Layton, Department of Mathematics, University of Pittsburgh
    "Evolve then Filter"
     Abstract:
    This talk will consider filter based stabilization for solving evolution equations.

  • Tuesday, October 20, 2009
    10:00 AM, Thackeray 703
     Jeffrey Connors, Department of Mathematics, University of Pittsburgh
    "Decoupled time stepping algorithms for fluid-fluid interaction"
     Abstract:
    A model of two incompressible, Newtonian fluids coupled across a common interface is studied. The nonlinearity of the coupling condition exacerbates the problem of decoupling the fluid calculations in each subdomain, a natural parallelization strategy employed in current climate models. Specialized partitioned time stepping methods will be discussed which decouple the discrete fluid equations without sacri cing stability and maintaining convergence. The results of the numerical analysis are presented and computational tests are shown, demonstrating the robustness of these methods and their possible advantages over alternative approaches.

  • Tuesday, October 6, 2009
    10:00 AM, Thackeray 703
     Adélia Sequeira, Department of Mathematics, Technical University of Lisbon
    "Fluid-Structure and geometrical multiscale models of the vascular system"
     Abstract:
    Mathematical modeling and numerical simulation can provide an invaluable tool for the interpretation and analysis of the circulatory system functionality, in both physiological and pathological situations. However, this is still an incredibly challenging problem.
    Blood flow in arteries is characterized by travelling pressure waves due to the interaction of blood with the vessel wall. In order to capture these phenomena, complex fluid-structure interaction (FSI) problems must be considered, coupling physiologically meaningful models for both the blood and the vessel wall. From the theoretical point of view this is extremely difficult because of the high non-linearity of the problem and the low regularity of the displacement of the fluid-structure interface. So far, existence results have been obtained only in simplified cases. From the numerical point of view, the use of partitioned schemes which solve iteratively the fluid and the structure sub-problems, supplied with suitable transmission conditions, is difficult to handle in hemodynamic problems, due to the large added mass effect.
    In this talk we introduce some basic differential models for the description of blood flow in the circulatory system. We extend to a generalized Newtonian shear-thinning blood flow model, existing results for 3D FSI problems with Newtonian fluids, using a fully implicit coupled algorithm. We discuss the geometrical multiscale approach to simulate the reciprocal interactions between local and systemic hemodynamics. It consists in coupling a hierarchy of models with different levels of complexity, namely 3D FSI models, 1D hyperbolic and 0D or lumped parameter models. Finally, we present numerical results, illustrating the effectiveness of the couplings and the geometrical multiscale strategy.

  • Tuesday, September 22, 2009
    10:00 AM, Thackeray 703
     Alexander Lozovskiy, Department of Mathematics at University of Pittsburgh
    "Prediction of the aerodynamical noise via Lighthill analogy in DNS and LES"
     Abstract:
    The prediction of the aerodynamical noise has been one of the fundamental engineering problems. The Lighthill analogy will be presented as non-homegeneous wave equation describing the generation of sound in the turbulent regions of the flow. The time-dependent FEM numerical scheme for DNS of the Lighthill analogy will be presented and studied, as well as possible ways for computing the sound power inside the so-called far field. For LES we'll present two subgrid scale models for Lighthill tensor and study the computational schemes in both cases.

  • Friday, August 21, 2009
    11:00 AM, Thackeray 323
    Peter Binev, Department of Mathematics at University of South Carolina
    "Adaptive Methods in Learning Theory"
    Abstract:
    Clouds of points appear in several practical problems. One important application of adaptive approximation is to find a function that represents well the values given by the point clouds and at the same time has minimal complexity. From geometrical point of fitting (minimal) surfaces to a given point cloud data. Typically, the data is corrupted by noise which requires use of statistical methods. Thus, the approach to solve the problem could be viewed as a melding of ~{\em deterministic}~ and ~{\em stochastic}~ ideas. Many of the solution methods are part of Learning Theory which attempts to find the relationships in given data set in order to make predictions on future data. We can consider the point cloud as a collection of independent random samplings from an unknown probability distribution $\rho$. The goal is to find a reliable approximation to different numerical characteristics of the point cloud without any a priori assumptions about $\rho$. One particularly important problem is to approximate the regression function $f_\rho : x\rightarrow y$. For example, consider data points $(x,y)$ from which $x$ is easy to collect and $y$ is a value we want to know. The function $f_\rho(x)$ gives the expected value $y$ (the average outcome) for a given $x$.


    Last updated: September 18, 2009