# 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, 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: [Spring 2017] | [Fall 2016]

### ➜ Fall 2017

**October 19**

Speaker: Ryan Viertel

**Title: **Quad Meshing, Cross Fields, and the Ginzburg-Landau Theory

**Abstract: ** A generalization of vector fields, referred to as $N$-direction fields or cross fields when $N=4$, has been recently introduced and studied for geometry processing, with applications in quadrilateral (quad) meshing, texture mapping, and parameterization. We make the observation that cross field design for two-dimensional quad meshing is related to the
well-known Ginzburg-Landau problem from mathematical physics. This identification yields a variety of theoretical tools for efficiently computing boundary-aligned quad meshes, with provable guarantees on the resulting mesh, for example, the number of mesh defects and bounds on the defect locations. The procedure for generating the quad mesh is to

(i) find a complex-valued "representation" field that minimizes the Dirichlet energy subject to a boundary constraint,

(ii) convert the representation field into a boundary-aligned, smooth cross field,

(iii) use separatrices of the cross field to partition the domain into four-sided regions, and

(iv) mesh each of these four-sided regions using standard technique.

Under certain assumptions on the geometry of the domain, we prove that this procedure can be used to produce a cross field whose separatrices partition the domain into four-sided regions. To solve the energy minimization problem for the representation field, we use an extension of the Merriman-Bence-Osher (MBO) threshold dynamics method, originally conceived as an algorithm to simulate motion by mean curvature, to minimize the Ginzburg-Landau energy for the optimal representation field. Finally, we demonstrate the method on a variety of test domains.

**October 5**

Speaker: China Mauck

**Title: **Approximating Functions with Standing Acoustic Waves

**Abstract: ** If micro-particles in a reservoir of fluid are neutrally buoyant and less compressible than the fluid, they will cluster around the nodes of a standing acoustic wave. In order to force the particles to cluster in a desired pattern, we search for a standing wave whose nodal set approximates the desired pattern. We use Herglotz wave functions, which are a particular kind of standing waves. In a volume, the best approximation is essentially given by a time reversal experiment with the original function as the source, and the approximation is generally poor. In a plane, the best approximation is essentially a low-pass filter in spatial frequency. These theoretical results are illustrated with numerical experiments.

**September 21**

Speaker: Dong Wang

**Title: **Introduction to Nonuniform Fast Fourier Transform (NUFFT)

**Abstract: ** In this talk, I will mainly follow Leslie Greengard and June-Yub Lee's SIAM review paper
"Accelerating the Nonuniform Fast Fourier Transform" to introduce the nonuniform fast Fourier transform based on fast Gaussian gridding.

**September 14**

Speaker: Akil Narayan

**Title: **Column Subset Selection

**Abstract: ** The problem of selecting a small subset of columns from a large matrix comes up in many computing applications. In such situations one wants to select columns that achive some objective. One such objective that we will use as a central exemplar is finding a subset that forms an accurate low-rank approximation to the full matrix. Our goal will be to give a high-level, non-rigorous description of a few techniques that enjoy popularity today. These techniques include classical numerical-algebraic methods such as rank-revealing QR factorizations, along with more modern
machine-learning-inspired approaches like leverage sampling and group matching methods. Time permitting, we will also discuss methods of a decidedly different flavor originating from functional analysis.

**September 7**

Speaker: Todd Reeb

**Title: **Introduction to Parallel Computing and Scan

**Abstract: **Parallel computing is computation using multiple processing elements to execute multiple instructions simulataneously. In this talk, we will describe the reasons for using parallel computing and the problems that arise with it. We will then discuss the basic tools of parallel computing. Finally we will end the talk by introducing one of the most fundamental parallel algorithms, scan, which gives a solution to the following problem that is faster thatn the naive serial algorithm: Given a list of numbers , compute the partial sums $a_0, a_1, \dots, a_{n-1}$, compute the partial sums
\begin{align*}
s_0 & = a_0 \\
s_1 & = a_0 + a_1 \\
\vdots & \qquad\vdots\qquad\vdots \\
s_{n-1} & = a_0 + a_1 + \dots + a_{n-1}.
\end{align*}

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