SIAM_SEAS_2012
Florida State Participation

On Saturday and Sunday, 24/25 March 2012, the 2012 SIAM SEAS meeting will be held at the University of Alabama in Huntsville.

A group from the Department of Scientific Computing at Florida State University will be attending the meeting. In particular, our departmental chair, Max Gunzburger, will be delivering one of the plenary addresses.

One large group will drive up in the Department of Scientific Computing van:

1. Evan Bollig (contributed talk: "Multi-GPU Solutions of Hyperbolic and Elliptic PDEs with RBF-FD")
2. John Burkardt
3. Xi Chen (minisymposium, chaired by Steve Henke)
4. Marta D'Elia (contributed talk)
5. Steve Henke (chair and speaker, minisymposium on peridynamics)
6. Michal Palczewski (contributed talk)
7. Mauro Perego (contributed talk)
8. Dong Sun (minisymposium, chaired by Leo Rebholz, Clemson University)
9. Guannan Zhang (minisymposium, chaired by Lili Ju, South Carolina State University)

Steve Henke will chair a minisymposium titled "Peridynamics: Material modeling without derivatives". The minisymposium description is:

Peridynamics is a recently-developed reformulation of solid mechanics that is non-local and avoids spatial derivatives, using an integral approach instead. Thus, it is suitable for modeling phenomena involving discontinuities, including cracks and fractures. In this minisymposium, speakers will address analytical and computational issues pertinent to peridynamics or related non-local models.

• Numerical Methods for the Nonlocal Peridynamics Model,
  Xi Chen, Florida State University;
  Numerical prediction of crack growth and damage are long-standing problems in computational mechanics. The difficulties inherent in these problems arise from the basic incompatibility of cracks with the partial differential equations that are used in the classical theory of solid mechanics. The peridynamic model attempts to unite the mathematical modeling of continuous media, cracks, and particles within a single framework. Relevant numerical experiments will be conducted to show the effectiveness of the proposed methods.
• Convergence and Scaling of a Peridynamic Diffusion Equation In Multiple Dimensions,
  Steve Henke, Florida State University;
  We present a nonlocal, derivative-free mathematical model that contains classical diffusion, fractional diffusion, and one-dimensional peridynamics as special cases. In this model, the dimensionality, interaction strength, and non-local extent are assumed to be controlled quantities. We solve the resulting equations using a meshfree method, and profile its accuracy and convergence properties as the controlled parameters and quadrature points are adjusted.
• Well-posedness of the Linear Peridynamic Model of Continuum Mechanics,
  Tadele Mengesha, Pennsylvania State University,
  We present a mathematical analysis of the basic equations of continuum mechanics given in the peridynamic (PD) formulation. This work focuses on the linear bond-based PD model for homogeneous materials where the corresponding integral PD operator allow a sign changing kernel. We analyze this operator and the function spaces associated with it. We prove the well-posedness of both the equilibrium equations, given as nonlocal boundary value problems with volume constraints, and the Cauchy problem of the time dependent equations of motion. (This is a joint work with Qiang Du.)
• Finite Element Methods for Peridynamics: A Tale of Many Quadratures,
  Miro Stoyanov, Oak Ridge National Laboratory;
  Peridynamics (PD) models are very robust in predicting both continuous and discontinuous phenomena, however, the numerical PD methods are not. We take a Finite Element approach with robust mixed continuous and discontinuous function basis in 1-D and 2-D. However, the numerical scheme requires approximation of double integrals over very unusual domains, for example arbitrary intersections of disks with triangles. The choice of quadrature is critical and convergent rules have to be built "on-the-fly".

Dong Sun will participate in a minisymposium, organized by Leo Rebhola and Hyesuk Lee, on "Numerical Methods for Incompressible Flow Problems (Parts I-II)".

• Two kinds of 2nd order IMEX methods for Stokes-Darcy system,
  Dong Sun, Florida State University;
  This is a joint work with Wenbin Chen, Max Gunzburger and Xiaoming Wang. We study two kinds of 2nd order Implicit-Explicit (IMEX) methods for Stokes-Darcy system. One is the Backward Differentiation Formula (BDF) and the other is the Adams-Moulton-Bashforth (AMB). We prove unconditional and long-time stability for both schemes. Error estimates are discussed and numerical examples are implemented to present the efficiency of the two schemes.

Guannan Zhang will participate in a minisymposium, organized by Lili Ju and Xinfeng Liu, on "Recent Advances in Numerical PDEs and Computational Biology (Parts I-III)".

• Error Analysis of a Stochastic Collocation Method for Parabolic Partial Differential Equations with Random Input Data,
  Guannan Zhang, Florida State University;
  Integration factor methods are a class of "exactly linear part" time discretization methods. Recently, a class of efficient implicit integration factor (IIF) methods were developed for solving systems with both stiff linear and nonlinear terms, arising from spatial discretization of time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. The tremendous challenge in applying IIF temporal discretization for PDEs on high spatial dimensions is how to evaluate the matrix exponential operator efficiently. For spatial discretization on unstructured meshes to solve PDEs on complex geometrical domains, how to efficiently apply the IIF temporal discretization was open. In this talk, I will present our results in solving this problem by applying the Krylov subspace approximations to the matrix exponential operator. Then we apply this novel time discretization technique to discontinuous Galerkin (DG) methods on unstructured meshes for solving reaction-diffusion equations. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method in resolving the stiffness of the DG spatial operator for reaction-diffusion PDEs. Application of the method to mathematical models in pattern formation shall be shown.

Contributed talks include:

• Multi-GPU Solutions of Hyperbolic and Elliptic PDEs with RBF-FD, Evan Bollig, Florida State University,
  Radial Basis Function Finite Difference (RBF-FD) is a generalized FD scheme that functions on unstructured grids, has stability for large time-steps and competitive accuracy compared to other state-of-the-art methods. We present an ongoing effort to develop fast and efficient parallel implementations of RBF-FD for hyperbolic and elliptic PDEs in geophysics. This work targets Keeneland, an NSF funded GPU cluster with over three hundred GPUs.
• A Bayesian approach to data assimilation in hemodynamics, Marta D'Elia, Florida State University,
  In this talk we discuss a Bayesian data assimilation technique for hemodynamics. Our method is formulated as a control problem where a misfit between data and velocity is minimized subject to the Navier-Stokes equations. The resulting probability density function of the velocity is used to derive its statistical estimators and confidence regions. We present numerical results on 2D and 3D approximations of blood vessels, we compare statistical and deterministic estimators and we draw velocity confidence regions.
• A Continuous Model for Gene Flow, Michal Palczewski, Florida State University,
  Gene flow is a particularly interesting problem in population genetics. A common model for gene flow is discrete randomly mating populations exchanging migrants each generation. Inferences on such a model are currently done by Markov Chain Monte Carlo. Current methods simulate discrete migration events in order to see how well genetic data matches a model and its parameters. There is little power to infer these events individually, and they are a nuisance parameter. What I propose is a new model of migration. In this model Gene Flow is modeled in a continuous matter using a non homogeneous Poisson process. Individual lineages no longer belong to a population but instead have a probability associated with each population. We hope that by turning this discrete problem into a continuous one we will be able to work with larger problems and more complicated scenarios.
• Numerical solution of ice sheet dynamics, Mauro Perego, Florida State University,
  Ice-sheet dynamics play a significant role in Climatology. Several models, characterized by different complexity and accuracy, have been proposed for describing ice-sheet dynamics. In this work we present a parallel finite element implementations of some of them. We compare the results obtained using these different models on Greenland and Antarctica ice-sheets. Also, we explore different strategies for the solution of the resulting non linear system, with particular care to the scalability of the solution.