Direct Methods for Multiobjective Optimization
In multiobjective optimization one considers optimization problems with several competing objective functions. For instance, in engineering, a design often has to be stable and light at the same time. A classical approach to such optimization problems is to formulate suitable parameter-dependent single-objective replacement problems, called scalarizations, such as a weighted sum of the objective functions or the epsilon-constraint scalarization. Then, the parameters are varied and the scalarized problems are solved iteratively.
However, many classes of multiobjective optimization problems have a structure where a scalarization might not be the best approach for an efficient procedure. In this talk, we introduce the basic concepts and some of the classical solution approaches in multiobjective optimization. Thereby, we cover the basic ideas of scalarization-based methods. Then, we present examples of classes of multiobjective optimization problems for which it might be better not to scalarize the problems first. These classes include non-convex continuous problems, convex mixed integer and integer problems, and heterogeneous problems, where one of the objective functions is assumed to be an expensive black-box function while the other objectives are analytically given. We give some of the main ideas behind other solution approaches for them.
Equity analytics: An approach for transforming postsecondary mathematics instruction
This talk introduces the EQUIP observation tool (https://www.equip.ninja) and describes the equity analytics approach to improving mathematics instruction. While mathematics education research provides insight into a variety of new instructional practices, supporting the equitable use of such practices in postsecondary mathematics has remained a challenge. One of the key barriers is that instructors lack practical data about what is happening in their classrooms. The equity analytics approach overcomes this barrier by providing instructors with actionable data that they can use to refine their teaching in an iterative fashion.
In this talk, I draw upon multiple case studies of mathematics faculty members and share the details of the educational transformations they have achieved with the support of equity analytics and EQUIP-based coaching. These cases provide models for what other instructors can hope to achieve in their classrooms. In addition, I provide details of the EQUIP approach so that others can use it for self-study or in coaching contexts.
Data neuroScience: mathematical tools for investigating brain activity from electrophysiological data.Most brain functions are regulated by tiny intercorrelated electrical currents flowing in few specific brain areas. Magneto- and electro-encephalography (M/EEG) are two modern neuroimaging techniques capable of non-invasively recording the electromagnetic field produced outside the scalp by these neural currents. Interpreting the recorded M/EEG data is not straightforward, due for example to source-leakage effects induced by volume conduction, and advanced mathematical techniques are required to estimate the dynamical brain activity that has generated the measured data. A typical workflow of analysis consists of two steps: (i) first the active brain regions and their time-courses are estimated by solving and ill-posed inverse problem (ii) then a proper connectivity metric is compute to quantify the statistical dependencies between the time-courses estimated at different brain locations.
The aim of this talk is to provide some insights on modern data-science tools to automatically estimate brain activity and connectivity from M/EEG data. In detail, in the first part of the talk I will formulate the M/EEG inverse problem within a Bayesian setting and I will present SESAMEEG (SEquential Semi-Analytic Montecarlo Estimation for MEEG ) a Sequential Monte Carlo sampler we developed to obtain fully-automatic parametric representations of brain activity. In the second part of the talk I will focus on brain connectivity. Some recent studies have demonstrated that the classical two step approach previously described may be suboptimal when Tikhonov regularization is used to solve the M/EEG inverse problems. Specifically, unexpected issues in setting the regularization parameter may arise . Motivated by these results I will show how to derive an inverse problem that allows to directly estimate the cross-power spectrum of the neural sources from that of the recorded data. Then I will present a lasso-based approach we developed to obtain sparse estimates of the brain functional networks. Sommariva, S., and Sorrentino, A.. Inverse Problems, 2014.  Vallarino, E., Sommariva, S., Piana, M., and Sorrentino, A. Inverse Problems, 2020.
Scale-resolved computational fluid dynamics: applications and methods
Computational fluid dynamics (CFD) offers a methodological framework for simulating 3D air and liquid flows. During the past two decades, high performance computing has enabled usage of scale-resolving methods to simulate turbulent fluid flows. In such chaotically swirling flows common to many real-life applications, the non-linear interactions of the Navier-Stokes equations lead to multiscale physics. To capture such physical phenomena, numerical solution of the partial differential equations on a fine space-time resolution is required.In the first part of the talk, I will discuss the fundamental methods to solve the Navier-Stokes equation. In the second part of the talk I focus on recent application of finite difference/Fourier methods to simulate airflow and transmission of airborne viruses in indoor environments on a GPU. Finally, In the third part of the talk I discuss a few applications of fluid dynamics including chemically reactive flow and two phase flow using an open source finite volume code.