# 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.

Past AMC: [Summer 2018] | [Spring 2018] | [Fall 2017] | [Spring 2017] | [Fall 2016]

### ➜ 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".

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