# Past Computational Mathematics and Applications Seminar

The behavior of lipid vesicles is due to a complex interplay between the mechanics of the vesicle membrane, the surrounding fluids, and any external fields which may be present. In this presentation two aspects of vesicle behavior are explored: vesicles in inertial flows and vesicles exposed to electric fields.

The first half of the talk will present work done investigating the behavior of two-dimensional vesicles in inertial flows. A level set representation of the interface is coupled to a Navier-Stokes projection solver. The standard projection method is modified to take into account not only the volume incompressibility of the fluids but also the surface incompressibility of the vesicle membrane. This surface incompressibility is enforced by using the closest point level set method to calculate the surface tension needed to enforce the surface incompressibility. Results indicate that as inertial effects increase vesicle change from a tumbling behavior to a stable tank-treading configuration. The numerical results bear a striking similarity to rigid particles in inertial flows. Using rigid particles as a guide scaling laws are determined for vesicle behavior in inertial flows.

The second half of the talk will move to immersed interface solvers for three-dimensional vesicles exposed to electric fields. The jump conditions for pressure and fluid velocity will be developed for the three-dimensional Stokes flow or constant density Navier-Stokes equations assuming a piecewise continuous viscosity and an inextensible interface. An immersed interface solver is implemented to determine the fluid and membrane state. Analytic test cases have been developed to examine the convergence of the fluids solver.

Time permitting an immersed interface solver developed to calculate the electric field for a vesicle exposed to an electric field will be discussed. Unlike other multiphase systems, vesicle membranes have a time-varying potential which must be taken into account. This potential is implicitly determined along with the overall electric potential field.

It is well known that the trapezoid rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with powerful algorithms all across scientific computing, including double exponential and Gauss quadrature, computation of inverse Laplace transforms, special functions, computational complex analysis, the computation of functions of matrices and operators, rational approximation, and the solution of partial differential equations.

This talk represents joint work with Andre Weideman of the University of Stellenbosch.

We consider the problem of taking a matrix A and finding diagonal matrices D and E such that the rows and columns of B = DAE satisfy some specific constraints. Examples of constraints are that

* the row and column sums of B should all equal one;

* the norms of the rows and columns of B should all be equal;

* the row and column sums of B should take values specified by vectors p and q.

Simple iterative algorithms for solving these problems have been known for nearly a century. We provide a simple framework for describing these algorithms that allow us to develop robust convergence results and describe a straightforward approach to accelerate the rate of convergence.

We describe some of the diverse applications of balancing with examples from preconditioning, clustering, network analysis and psephology.

This is joint work with Kerem Akartunali (Strathclyde), Daniel Ruiz (ENSEEIHT, Toulouse) and Bora Ucar (ENS, Lyon).