# DE Seminars: Fall 2011

#### Monday September 19

Title |
Some dynamical systems for supply chains with kanban control |

Speaker |
Thomas Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider a fairly general model of a supply chain (manufacturing system, …) and consider the dynamics under two variants of “kanban control”. The kanban idea (popularized by Toyota) authorizes an order for a supply batch when a buffer drops to its `re-order point’.

For the simplest version it is shown that one has stability (given adequate capacity) if the supplier assignments are consistent with the task dependencies, but an instability example is given with this not holding. On the other hand, a kanban system based on the observation only of final output buffers is stable for the general configuration.

#### Monday October 10

Title |
Mathematical models for DNA looping or cyclization: Monte Carlo simulations and `semiclassical’ approximations |

Speaker |
Robert Manning Department of Mathematics Haverford College |

**Abstract:**

DNA looping is the name given to a biological process in which short segments of DNA (40-400 basepairs) deform to meet some boundary conditions; cyclization is the special case of periodic boundary conditions. Several recent cyclization experiments have revealed special DNA sequences with unusually high flexibility, out of sync with current models. We consider two discrete models of DNA—one that treats the basepairs as rigid and one that treats the individual bases as rigid—with an eye toward new flexibility parameters that may be able to explain the high-flexibility experiments.

A key mathematical quantity to model DNA looping or cyclization is an end-to-end probability density function (pdf) on R^3 x SO(3), the configuration space for the “far end” of the DNA, once the “near end” is fixed by choice of coordinates. Monte Carlo simulations allow direct sampling of this pdf, although the typical values of the pdf are small enough that it can be challenging to get decent statistics. A semiclassical approach, which finds local minimizers of the DNA energy and computes contributions to the pdf based on a quadratic expansion about the minimizer, is much faster, but its accuracy largely unexplored.

We present some preliminary results, including a comparison to classical DNA results of the dependence of cyclization probability on the number of basepairs and the consideration of all possible boundary conditions for “2D looping” of an elastic rod. For these examples, we assess the accuracy of the semiclassical approximation and consider possible improvements.

#### Monday October 31

Title |
Numerical experiments with COMSOL Multiphysics to support mathematical research and teaching |

Speaker |
David Trott Department of Mathematics and Statistics UMBC |

**Abstract:**

COMSOL Multiphysics is a professional grade software package that uses the Finite Element Method (FEM) to numerically solve partial differential equations (PDEs). In this talk, I will discuss the capabilities of COMSOL by highlighting the wide variety of research that is actively being done using this tool both in academia and industry. I will then discuss various numerical studies that I have conducted on the software in the context of verifying that the numerical results are consistent with those expected by the FEM theory.

#### Monday November 7

Title |
Uniform Approximation Property of Implicit Methods for a Family of Stiff Differential Equations |

Speaker |
Subhash Paruchuru Department of Mathematics and Statistics UMBC |

**Abstract:**

Stiff systems are characterized by the presence of multiple time scales where the fast scales are stable. Conventional stability and convergence analysis tend to look at the behavior of a single system. We propose a different type of analysis which captures the important features of both convergence and stability analysis. Our analysis focuses on a method applied to a parametrized family of stiff systems and the uniformity of the convergence across the entire range of parameter values. Using Dini’s theorem we establish that for scalar linear equations, the numerical solutions using Implicit Taylor Methods converge uniformly to the true solution as the step size goes to zero. Furthermore, using the Arzela-Ascoli theorem, we can extend the results to a family of stiff complex systems and a family of non-stiff higher dimensional systems. Finally, we show the uniform convergence of numerical solutions of implicit Euler for a general family of stiff two dimensional systems.

#### Monday November 14

Title |
Effect of Parity on Boundedness of Orbits in Lotka-Volterra Food Chains |

Speaker |
Nicole Massarelli Department of Mathematics and Statistics UMBC |

**Abstract:**

Hairston, Slobodkin, and Smith (HSS) conjectured that top-down forces act on food chains, which opposed the previously accepted theory that bottom-up forces dictate the dynamics of populations. HSS argued that plant life is abundant because carnivores prey on herbivores, which prevents plants from being depleted. From this hypothesis Fretwell inferred that HSS could be applied to food chains with greater or fewer number of trophic levels. The exploitation ecosystem hypothesis (EHH) extends this alternating pattern to a food chain of any length. According to EHH in food chains with an odd number of trophic levels plants are resource limited, which indicates bottom-up forces control the dynamics of the food chain. Alternatively, in food chains with an even number of species plants are limited by the grazing of herbivores. Thus, for an even number of species top-down forces act on the food chain. This implies that the plant population can only increase in an odd level food chain.

#### Monday November 21

Title |
Multigrid solution of a distributed optimal control problem constrained by the Stokes equations |

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

**Abstract:**

Over the last decade has been a growing interest in developing multigrid methods for optimal control problems constrained by partial differential equations. In this work we construct multigrid preconditioners to speed up the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a linear system connecting the state, adjoint, and control variables. Our approach is to eliminate the state and adjoint variables by essentially solving two Stokes systems, and to construct efficient multigrid preconditioners for the Schur-complement of the block associated to the state and adjoint variables. These multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the Stokes system. In particular, the number of conjugate gradient iterations is shown to decrease as the resolution increases, a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems.

#### Monday November 28

Title |
Dynamics a la Gibbs-Appell |

Speaker |
Rouben Rostamian Department of Mathematics and Statistics UMBC |

**Abstract:**

I will begin this talk with an illustration of Newton’s equations of motion (force = mass times acceleration) and point out its shortcomings when they are applied to complex systems. Then I will introduce Lagrange’s formulation of dynamics and show its advantages over Newton’s. Applying Lagrange’s method to even more complex situations (non-holonomic constraints) reveals that Lagrange’s formulation is not a cure-all as it suffers from some of the same inadequacies as Newton’s. Finally, I will introduce a reformulation of the equations of dynamics due to Gibbs and Appell and show that it is ideally suited for handling non-holonomic constraints, such as when one solid object rolls against another.

I will illustrate some of the key concepts through animated demos. You will find a sample animation in http://userpages.umbc.edu/~rostamia/animated-rolling-coin.html.

When necessary, I will track a rolling object’s orientation with the help of quaternions. I talked about the algebra of quaternions in the DE Seminar some time ago but I will keep this presentation independent of that talk.

#### Monday December 5

Title |
Reduced Order Modeling for Parametric Nonlinear PDE Constrained Optimization Problems |

Speaker |
Harbir Antil UMCP |

**Abstract:**

The optimal design of structures and systems described by partial fferential equations (PDEs) often gives rise to large-scale optimization problems, in particular if the underlying system of PDEs represents a multiscale, multiphysics problem. Therefore reduced order modeling techniques such as balanced truncation model reduction, proper orthogonal decomposition, or reduced basis method are used to signicantly decrease the computational complexity while maintaining the desired accuracy of approximation.

We will give a brief introduction to PDE constrained optimization problems and apply model reduction to obtain reduced order optimization problems. However, for the nonlinear or parametrically varying problems, the cost of evaluation of the reduced order models still depend on the size of the full order model. We demonstrate why the standard model reduction alone is insufficient and outline the technique to handle such situation and further extend it to the finite element solutions of nonlinear PDEs and to the solution of shape optimization problems governed by PDEs.

As applications in life sciences, we will be concerned with the optimal design of capillary barriers as part of network of microchannels and reservoirs on surface acoustic wave driven microfluidic biochips. Another application related to numerical relativity will show the effectivity of ideas.