• #### Speaker: Alex Townsend (Cornell University) - http://www.math.cornell.edu/~ajt/

When: Tue, November 14, 2017 - 3:30pm
Where: Kirwan Hall 3206

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Abstract: A classical technique for computing with functions on the sphere and disk is to "double up" the domain, leading to regularity preserving approximants. We synthesize this with new techniques for constructing low rank function approximations to develop a whole collection of fast and adaptive algorithms for sphere and disk computations that are accurate to machine precision. Applications include vector calculus, the solution of PDEs, and the long-time simulation of active biological fluids. This is joint work with Heather Wilber and Grady Wright from Boise State University.
• #### Speaker: James Adler (Tufts University Department of Mathematics) - https://jadler.math.tufts.edu/

When: Tue, December 5, 2017 - 3:30pm
Where: Kirwan Hall 3206

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Abstract: Liquid crystals are substances that possess mesophases with properties intermediate between liquids and crystals, existing at different temperatures or solvent concentrations. We consider two types of liquid crystals in this talk, nematic and cholesteric, which are formed by rod-like molecules that self-assemble into an ordered structure, such that the molecules tend to align along a preferred orientation. The preferred average direction at any point in a domain is known as the director and is taken to be of unit length at every point in the domain. In addition to their inherent structure, which is affected both by the self-assembly and boundary affects, most liquid crystals are responsive to electric fields and may be compelled to arrange their structure in new ways by the presence of these fields. Simulating this realignment from all of these factors is crucial for understanding many applications where such switching is useful, such as energy generators and actuators.

The main focus of this talk will be on the computational modeling of equilibrium configurations for liquid crystals influenced by elastic and electric effects. Thus, the method targets minimization of the system free energy based on the so-called Frank-Oseen free-energy model, subject to the unit-length constraint of the director. We consider an energy-minimization finite-element approach to discretize the constrained optimization problem along with Newton's method to linearize the system. We are able to show that solutions to the intermediate discretized linearizations exist generally and are unique under certain assumptions. Numerical experiments are performed for problems with a range of Frank elastic constants as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous Frank constants and efficiently resolves configurations resulting from complicated boundary conditions relevant in ongoing research. Additionally, we present techniques to handle situations in which multiple equilibrium states can arise. This involves the incorporation of multilevel and nonlinear deflation techniques to solve the resulting linear systems.

• #### Speaker: Mert Gurbuzbalaban (Rutgers University) - https://mert.lids.mit.edu

When: Tue, December 12, 2017 - 3:30pm
Where: Kirwan Hall 3206

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Abstract:

Many of the emergent technologies and systems including infrastructure systems (communication, transportation and energy systems) and decision networks (sensor and robotic networks) require rapid processing of large data and comprise dynamic interactions that necessitate robustness to small errors, disturbances or outliers.
Motivated by large-scale data processing in such systems, we first consider additive cost convex optimization problems (where each component function of the sum represents the loss associated to a data block) and propose and analyze novel incremental gradient algorithms which process component functions sequentially and one at a time, thus avoiding costly computation of a full gradient step. We focus on two randomized incremental methods, Stochastic Gradient Descent (SGD) and Random Reshuffling (RR), which have been the most widely used optimization methods in machine learning practice since the fifties. The only difference between these two methods is that RR samples the component functions without-replacement whereas SGD samples with-replacement. Much empirical evidence suggested that RR is faster than SGD, although no successful attempt has been made to explain and quantify this discrepancy for a long time. We provide the first theoretical convergence rate result of O(1/k2s) for any s in (1/2,1) (and O(1/k2) for a bias-removed novel variant) with probability one for RR showing its improvement over Î©(1/k) rate of SGD and highlighting the mechanism for this improvement. Our result relies on a detailed analysis of deterministic incremental methods and a careful study of random gradient errors. We then consider deterministic incremental gradient methods with memory and show that they can achieve a much-improved linear rate using a delayed dynamical system analysis.

In the second part, we focus on large-scale continuous-time and discrete-time linear dynamical systems that model various interactions over complex networks and systems. There are a number of different metrics that can be used to quantify the robustness of such dynamical systems with respect to input disturbance, noise or error. Some key metrics are the H-infinity norm and the stability radius of the transfer matrix associated to the system. Algorithms to compute these metrics exist, but they are impractical for large-scale complex networks or systems where the dimension is large and they do not exploit the sparsity patterns in the network structure. We develop and analyze the convergence of a novel scalable algorithm to approximate both of the metrics for large-scale sparse networks. We then illustrate the performance of our method on numerical examples and discuss applications to design optimal control policies for dynamics over complex networks and systems.