# DE Seminars: Fall 2007

#### Saturday September 15

Title |
An H^{2}-compatible Finite Element Solver for the Navier-Stokes Equations |

Speaker |
Ana Maria Soane Department of Mathematics and Statistics UMBC |

**Abstract:**

Liu, Liu, and Pego have developed an unconditionally stable algorithm for finite element discretization of the Navier-Stokes equations in the setting of the H^{2} Sobolev spaces (Stability and convergence of efficient Navier-Stokes solvers via a commutator estimate, Comm. Pure & Appl. Math, 2007). The algorithm enforces conservation of mass without a need for the inf-sup condition that is common in H^{1} formulations. In domains with reentrant corners the solution of the Navier-Stokes equations need not be in H^{2}, therefore the algorithm needs to be modified. We have modified the algorithm by recasting it in weighted Sobolev spaces where appropriate weights at reentrant cornerscompensate for the solution’s singularities. The numerical results obtained this way show good agreement with those obtained by the usual H^{1} methods.

#### Monday October 1

Title |
Switching control of transport on a graph |

Speaker |
Tom Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider a hybrid system (mixed state: discrete and continuous) for which the continuous dynamics model quasilinear transport on a graph and the discrete dynamics model a sequence of control actions, switching between available `modes’ of the system. After commenting on the modelling, we show the well- posedness of the system (with switching as data), existence of an optimal control, and “hybrid well-posedness” of a feedback implementation.

#### Monday October 8

Title |
Solving the Stochastic Steady-State Diffusion Problem using Multigrid |

Speaker |
Darran Furnival Department of Mathematics University of Maryland, College Park |

**Abstract:**

Abstract: It is known that multigrid is an optimal method for solving the linear system that results from application of the finite element method to the steady-state diffusion problem, insomuch as its convergence rate is independent of the spatial discretization. In this talk the method of polynomial chaos (a stochastic finite element method) is applied to the stochastic steady-state diffusion problem and a multigrid algorithm is applied to solve the resulting linear system. It will be shown, theoretically and experimentally, that the convergence rate is independent of the spatial discretization, and also, under appropriate scaling of the stochastic basis functions, independent of the stochastic parameters.

#### Monday October 15

Title |
Numerical Continuation for Atomic and Molecular Fluids |

Speaker |
Kelly Dickson Department of Mathematics North Carolina State University |

**Abstract:**

Numerical continuation is the process of solving systems of nonlinear parameter dependent equations for various parameter values. Continuation studies are often helpful in understanding changes in a physical or natural system. In particular, as a fluid changes density or temperature, one can observe crucial transitions in structural and thermodynamic properties. In this talk, I present new integral equation theory for atomic and molecular fluids developed at the Institute for Molecular Design at the University of Houston. Further, I discuss a new implementation of this theory in the context of numerical continuation using Trilinos, a software framework developed at Sandia National Labs.

#### Monday October 29

Title |
Maxwell’s Equations in biperiodic structures |

Speaker |
Aurelia Minut Department of Mathematics Unites States Naval Academy |

**Abstract:**

In this talk, we present L^{p} estimates for the solutions of Maxwell’s Equations with source term. We consider Maxwell’s equations in a domain that is composed of two different materials. Such an estimate is employed to solve a nonlinear optics problem. This is joint work with G. Bao and Z. Zhou.

#### Monday November 5

Title |
Stochastic dynamics of flagellar growth |

Speaker |
Muruhan Rathinam Department of Mathematics and Statistics UMBC |

**Abstract:**

We study a stochastic model for the movement of intraflagellar transporters which play a crucial role in flagellar growth. The detailed model is a discrete state vector valued Markov process occurring in continuous time. First the detailed stochastic model is compared and contrasted with a simple scalar ordinary differential equation (ODE) model of flagellar growth. Numerical simulations reveal that the steady state mean value of the stochastic model is well approximated by the ODE model. We also derive a continuous state scalar valued Markov process model in the form of a stochastic differential equation (SDE) as a “diffusion approximation” of the higher dimensional discrete model. We show how this nonlinear SDE may be further approximated by a linear additive noise model which enables one to calculate the variance analytically. The accuracy of the linear noise model is compared with the numerical simulation results of the detailed model. This is on going work with collaborator Dr Hana El-Samad and the undergraduate research student Yuriy Sverchkov.

#### Monday November 12

Title |
Phase approximations of coupled neural oscillators |

Speaker |
Tim Kiemel Department of Kinesiology University of Maryland, College Park |

**Abstract:**

Neural circuits called central pattern generators (CPGs) in the spinal cord are responsible for producing rhythmic behaviors such as walking in humans or swimming in fish. CPGs consist of coupled nonlinear neural oscillators and are often modeled by coupled phase oscillators. Phase models are accurate approximations when coupling is weak. However, coupling between neural oscillators in CPGs is often strong, raising the question of whether phase models are an useful approximation. We address this question by considering two types of phase models: standard first-order phase models and second-order phase models. Second-order phase models allow us to understand how phase lags between oscillators begin to change as coupling strength increases. As such, they may serve as a bridge between the simplicity of standard phase models and the more complicated reality of systems of coupled neural oscillators.

#### Monday December 3

Title |
Towards a Model of Sensory Feedback Loop in the Locomotion CPG of the Lamprey |

Speaker |
Kathleen Hoffman Department of Mathematics and Statistics UMBC |

**Abstract:**

Swimming in the lamprey is generated by neural circuits called central pattern generators (CPGs) that signal a muscle contraction or extension. A wave a muscular activation occurs down the body of the lamprey, propelling it through the water. The CPGs can be modeled by a chain of coupled nonlinear neural oscillators. In this talk, I will focus on two different models of the CPG: connectionist models and a phase models. I will discuss a `random’ coupling strategy for connecting the oscillators, which limits to the analogous deterministic connections. I will further discuss the role of proprioceptive sensor, called edge cells in locomotion, and describe some of the biological experiments and mathematical challenges in understanding this closed loop system.

#### Monday December 10

Title |
An interlaced time stepping method for stiff systems of Stochastic Differential Equations |

Speaker |
Ioana Cipcigan Department of Mathematics and Statistics UMBC |

**Abstract:**

We present a time-stepping strategy for solving stiff systems of stochastic differential equations( SDEs), by interlacing an implicit Euler time step with a sequence of explicit Euler time steps. Besides the well known problem of stability for stiff systems for both ODEs and SDEs, stochasticity induces another problem in stiff systems, that is, the failure in computing the distribution correctly. The reason for this is that, for stiff SDE systems, the fluctuations off the slow manifold might not be negligible in size, and an implicit scheme will dampen all these fluctuations and therefore will fail in computing the variance( and higher order moments) correctly. In addition, even when the stability is maintained, the Explicit Euler method overestimates the variance of the stationary distribution, while the Implicit Euler method underestimates it. To recover the variance we simply take an implicit time step and then m number of explicit time steps. Here we study suitably chosen test problems and we determine the optimal number m of explicit time steps such that the asymptotic variance computed by this interlaced method is the closest to the variance of the stationary distribution of the exact solution. It turns out that m depends on the problem parameters as well as the implicit and explicit time steps, and we provide a formula for computing this. Finally, we illustrate the method on some stiff SDE systems and we present the convergence results for these systems.