# Seminars & Events for PACM IDeAS

##### Solution of the biharmonic equation on regions with corners

The design of microfluidic devices involves the study of fluid dynamics at small length scales. The behavior of such systems is governed by the Stokes equations (biharmonic equation with gradient boundary conditions), due to the dominant influence of lubrication effects. Typically, micro-channels in microfluidic devices manufactured using planar lithographic techniques have nearly rectangular cross sections and thus tend to have many sharp corners. In such cases, reformulating the governing equation as a boundary integral equation is a natural approach, since this reduces the dimensionality of the problem (discretizing the boundaries alone) and permits high order accuracy in complicated geometries.

##### Minimax-optimal semi-supervised regression on unknown manifolds

**Please note different day and location (Tuesday, McDonnell 102A). **In recent years, many semi-supervised regression and classification methods have been proposed. These methods demonstrated empirical success on some data sets, whereas on others the unlabeled data did not appear to help. To analyze semi-supervised learning theoretically, it is often assumed that the data points lie on a low-dimensional manifold. Under this assumption [1] and [2] have shown that classical nonparametric regression methods, using only the labeled data, can achieve optimal rates of convergence. This implies that asymptotically, as the number of labeled points tends to infinity, unlabeled data does not help. However, typical semi-supervised scenarios involve few labeled points, and plenty of unlabeled ones.

##### Sketchy decisions: Low-rank matrix optimization with optimal storage

Convex matrix optimization problems with low-rank solutions play a fundamental role in signal processing, statistics, and related disciplines. These problems are difficult to solve because of the cost of maintaining the matrix decision variable, even though the low-rank solution has few degrees of freedom. This talk presents the first algorithm that provably solves these problems using optimal storage. The algorithm produces high-quality solutions to large problem instances that, previously, were intractable. Joint with Volkan Cevher, Roarke Horstmeyer, Quoc Tran-Dinh, Madeleine Udell, and Alp Yurtsever.

##### Algorithms for Reducing the Computational Burden of cryo-EM

**Please note different day and location (Thursday, McDonnell 102A).** Computational expense has long been a challenge for 3D structure determination in electron cryomicroscopy (cryo-EM). This problem has become more pressing as resolutions have increased and the desire to separate conformational computational variations has required ever-larger datasets. This talk presents two new algorithmic developments which, when coupled with modern GPU hardware, dramatically reduces the computational requirements for cryo-EM. Given low-resolution starting structures, high resolution structures can now be obtained in as little as 10s of minutes on modest desktop workstations. Further, despite the severe non-convexity of the objective function, these new refinement algorithms have shown themselves to be robust to local minima, ena

##### TBA - Afonso Bandeira

**Please note different day and location (Monday, Fine 110.)**

##### Near-optimal bounds for phase synchronization

The problem of phase synchronization is to estimate the phases (angles) of a complex unit-modulus vector from their noisy pairwise relative measurements. It is motivated by applications such as cryo-EM, camera orientation estimation, time synchronization of distributed networks, etc. This problem is usually formulated as a nonconvex optimization problem, and many methods, including semidefinite programming (SDP) and generalized power method (GPM), have been proposed to solve it. Though a simpler problem (Z_2 synchronization) is well understood, a lack of understanding of the properties of the optimization problem, especially statistical dependence, has led to suboptimal results in analysis.