Applied Math Collective

Applied Math Collective was initiated by my advisors and Fernando Guevara Vasquez. The aim is to provide an informal platform where the speaker discusses general-interest "SIAM review"-style applied math papers, led by either faculty or graduate student. We meet Thursdays at 4pm in LCB 222, when the Department Colloquium does not have a speaker. Please contact me if you would like to attend or give a talk so that I can add you to the mailing list.

➜ Spring 2017 (organised by Todd Reeb)

February 16
Speaker: Todd Reeb
Title: Fast Multiple Methods
Abstract: Fast Multipole Methods (FMM) originally referred to numerical methods for the fast evaluation of all $N^2$ pairwise interactions between $N$ objects, e.g. finding the electrostatic potentials and forces on each of $N$ point charges in the plane, but FMM have grown to encompass evaluation of the matrix-vector product $Ax$ where $A$ is a dense $N\times N$ matrix arising from an integral operator in almost linear time (as opposed to the usual quadratic time). In this talk, we will describe the classical FMM algorithm due to Greengard and Rokhlin. This talk is based on the following survey by Per-Gunnar Martinsson "Fast Multipole Methods".

February 23
Speaker: Christel Hohenegger
Title: Life at Low Reynolds Number
Abstract: Displacement of solid bodies on small scales in a fluid is dominated by the effects of viscous forces and, as a result, is subject to strong constraints. Purcell's statement of the scallop theorem delimitates the types of swimmer designs which are not effective on small scales. We explain how the theorem arises and discuss ways nature goes around the constraints for locomotion purposes. The discussion is based on the review article "Life around the scallop theorem" by E. Lauga and on the original paper by Purcell "Life at low Reynolds number".

March 2
Speaker: Chee Han Tan
Title: A Simple Model for Cloaking
Abstract: Cloacking is just a jargon for making objects invisible under certain imaging techniques, one example being electromagnetic imaging known as Electrical Impedance Tomography (EIT). Instead of using Maxwell's equation which is more realistic in general, we present a simplified model using the Laplace's equation and illustrate the main idea behind cloaking an object in a unit ball in $R^2$. This talk is based on the SIAM-review article by Kurt Bryan and Tanya Leise: "Impedance Imaging, Inverse Problems, and Harry Potter's Cloak".

April 13
Speaker: Akil Narayan
Title: Introduction to the Reduced Basis Method for Parameterized Partial Differential Equations
Abstract: The numerical solution of PDE models can be very expensive when the geometry is complicated, the solution has fine-scale structure, or when the PDE operator is sensitive to model inputs. Adding to the difficulty is that many realistic PDEs have tunable model parameters, hence the solution is dependent on these parameters. The end-goal of many simulations is optimization or averaging the solution over parameter val;ues: this requires onerous computation of the expensive PDE solution for numerous parameter values. The reduced basis method (RBM) is one of many "model order reduction" strategies that aims to mitigate the cost of numerical solutions to parameterized PDEs in the many-query context described above. We will briefly describe the basic mathematical and numerical techniques for RBM algorithms, and will give a high-level discussion of the advantages and disadvantages of this approach. Reference (presentation slides): B. Stamm, "Introduction to the certified reduced basis method".