# DE Seminars: Spring 2011

#### Monday March 14

Title |
Can you differentiate inside an integral?Parametric sensitivity analysis for stochastic chemical kinetics |

Speaker |
Muruhan Rathinam Department of Mathematics and Statistics UMBC |

**Abstract:**

Stochastic dynamical systems in continuous time with discrete integer states appear in many applications including population dynamics, intracellular chemical kinetics and queueing theory. These examples are often approximated by ODEs. Parametric sensitivity analysis of such systems is relatively easy when modeled by ODEs, but is particularly challenging when modeled by discrete state stochastic dynamical systems.

We describe three different approaches and some of the mathematical issues encountered in an ongoing research project.

#### Monday March 28

Title |
Vitamin D: Mathematical Modeling to Better Immunity |

Speaker |
Brad Peercy Department of Mathematics and Statistics UMBC |

**Abstract:**

Vitamin D is created when we are exposed to sunlight and vitamin D can be supplemented in our diet. Though vitamin D acts in several primary ways, how it can affect our immune response is of significant interest. We develop a mathematical model to address the question of which mechanism (if any) generates significant levels of the active form of vitamin D leading to genetic response in a type of white blood cell. After initial steady state analysis, we reduce the high-dimensional system to an equivalent low-dimensional system including a seemingly novel application of the rapid buffering approximation.

#### Monday April 4

Title |
Training Students for Skills in CS&E |

Speaker |
Matthias Gobbert Department of Mathematics and Statistics UMBC |

**Abstract:**

Recent years have seen a dramatic shift in how research in the applied sciences including mathematics and statistics is conducted. On the technical side, computational tools are more heavily used than ever before. But additionally, the expectations for presentation, writing, and other ‘soft’ skills of undergraduate and graduate students are growing. This seminar will report on several presentations given at a minisymposium at the SIAM Conference on Computational Science & Engineering in February/March 2011 that discussed many ideas for how to improve the preparation of students for this new environment. The slides of these presentations are available at http://www.math.umbc.edu/~gobbert/presentations.

#### Tuesday April 5

Title |
Dispersal and Recurrence in Disease Dynamics: The Case of Influenza |

Speaker |
Carlos Castillo-Chavez Department of Mathematics Arizona State University |

**Abstract:**

The role of mathematical models in the study of disease dynamics has a long and distinguished history that goes back to three physicians: Daniel Bernoulli, Sir Ronald Ross, and A. G. MacKendrick. Why were mathematical models introduced? This question will be addressed by revisiting early and recent applications of contagion models. Some of the mechanisms responsible for recurrent epidemic outbreaks (influenza being the underlying disease) or what appear to be periodic outbreaks over short- and long-time scales will be identified. The role of movement (dispersal) on disease dynamics in the context of topics that include for example, the deliberate release of biological agents, will be discussed.

#### Monday April 11

Title |
Methods for obtaining high-energy mode-locking |

Speaker |
Edwin Ding Department of Applied Mathematics University of Washington |

**Abstract:**

Since its first proposed use in the early 90’s, the ring cavity fiber laser mode-locked by utilizing the nonlinear polarization technique has become one of the most reliable and compact sources for robust ultra-short optical pulses. However, due to the limitations in the energy output, these fiber lasers have lagged well behind the solid-state lasers in the key performance parameters. In this talk I will present recent developments in achieving high-energy pulses in a ring cavity laser that is passively mode-locked by a series of waveplates and a polarizer. Specifically, I will show how the multi-pulsing instability can be circumvented in favor of bifurcating to higher-energy single pulses by appropriately adjusting the group-velocity-dispersion in the fiber and the waveplate/polarizer settings in the saturable absorber. The findings may be used as a practical guideline for designing high-power lasers since the theoretical model relate directly to the experimental ! settings. I will also talk about challenges in obtaining high-energy pulses with multi-mode fibers.

#### Monday April 18

Title |
Fractional Diffusion Limits for Kinetic Equations |

Speaker |
Antoine Mellet Department of Mathematics UMCP |

**Abstract:**

The asymptotic behavior of the solutions of a kinetic equation, such as the linear Boltzmann equation, depends strongly on the properties of the collision operators. I will discuss situations in which the lack of spectral gap for this operator leads to anomalous transport behavior.

#### Monday April 25

Title |
Insight Into Spontaneous Recurrent Calcium Waves in a 3-D Cardiac Cell Based on Analysis of a 1-D Deterministic model |

Speaker |
Zana Coulibaly Department of Mathematics and Statistics UMBC |

**Abstract:**

Spontaneous calcium sparks, under certain conditions, can lead to propagation of a self-initiated calcium wave in a heart cell. It is a concern that self-initiated calcium wave propagation in heart cells can in turn lead to irregular heart beats, which can potentially cause cardiac Arrhythmia. Studying a model of this phenomenon, such as we do with a system of coupled reaction-diffusion equations with stochastic release, at the cellular level with sub-cellular refinement requires computationally intensive long-time simulations. Previous studies showed that wave propagations (without recovery), are sensitive to certain parameters involved in the 3-D model. To gain insight into the parameter set that may lead the model to display behaviors that are biophysically acceptable and experimentally relevant, we perform a parameter analysis based on a 1-D deterministic version of the model. This analysis led us to determine a range of parameters that when used in the 3-D model generate spontaneous recurrent calcium waves with recovery!

#### Monday May 2

Title |
Multigrid solution of a distributed optimal control problem constrained by a semilinear elliptic PDE |

Speaker |
Jyoti Saraswat Department of Mathematics and Statistics UMBC |

**Abstract:**

This study focuses on the numerical solution of a large-scale, distributed optimal control problem constrained by semilinear elliptic PDEs. As is often the case for high-resolution (large-scale) problems related to PDEs, distributed optimal control problems face the additional challenge of the control size also increasing with resolution. This normally results in an increased number of optimization iterations in addition to the usual challenges related to resolving the large-scale PDE constraints.

Multigrid algorithms have long been used for solving large-scale problems traditionally associated with elliptic PDEs, however, it is well known that for the control problems under scrutiny classical multigrid algorithms are not applicable. In this work we have developed efficient multigrid methods specifically targeting problems with elliptic-like constraints. More precisely, since gradients and Hessian-vector multiplications can be computed using adjoint methods, we use Newton’s method for solving the optimization problem, and then specially designed multigrid methods are applied to precondition the Hessian of the cost functional. It is shown that the quality of the resulting multigrid preconditioner increases at an optimal rate with respect to the mesh-size, as in the case of optimal control of linear elliptic PDEs.

#### Monday May 9

Title |
Multi-physics domain decomposition methods for Stokes-Darcy model |

Speaker |
Xiaoming He Missouri University of Science and Technology |

**Abstract:**

The Stokes-Darcy model arises in many interesting real world applications, including groundwater flows in karst aquifers, interaction between surface and subsurface flows, industrial filtrations, oil reservoir in vuggy porous medium, and so on. This model describes the free flow of a liquid by the Stokes or Navier-Stokes equation and the confined flow in a porous media by the Darcy equation; the two flows are coupled through interface conditions. For the problems mentioned, the resulting coupled Stokes-Darcy model has higher fidelity than either the Darcy or Stokes systems on their own. However, coupling the two constituent models leads to a very complex system.

This presentation discusses multi-physics domain decomposition methods for solving the coupled Stokes-Darcy system. Robin boundary conditions based on the physical interface conditions are utilized to decouple the Stokes and Darcy parts of the system. A parallel iterative domain decomposition method is first constructed for the steady state Stokes-Darcy model with the Beavers-Joseph interface condition. Then two parallel non-iterative domain decomposition methods are proposed for the time-dependent Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. Numerical examples are presented to illustrate the features of these methods and verify the theoretical results.