Consensus problems have attracted significant attention in the control community over the last decade. They act as a rich source of new mathematical problems pertaining to the growing field of cooperative and distributed control. The talk focuses on consensus problems whose underlying state-space is not a linear space, but instead a highly symmetric nonlinear space such as the circle and several other relevant generalizations. A geometric approach is shown to highlight the connection between several fundamental models of consensus, synchronization, and coordination, to raise significant global convergence issues not captured by linear models, and to be relevant for a number of engineering applications, including the design of coordinated motions in the plane or in the three-dimensional space. Finally, nonlinear considerations on the original consensus problem defined in the positive orthant shed light on the special role of nonquadratic Lyapunov functions in this framework.
Rodolphe Sepulchre
Rodolphe Sepulchre is Professor and Chair of the department of Electrical Engineering and Computer Science at the University of Liege, Belgium. He received his engineering degree and Ph.D. degree in applied mathematics from the University of Louvain, Belgium, in 1990 and 1994 respectively. He held various research and visiting positions at UCSB (1994-1996), Princeton (2002-2003), and Mines Paris-Tech (2009-2010).
His main research interests include nonlinear control systems, optimization on matrix manifolds with applications to large-scale problems in linear algebra, and applications of dynamical systems, in particular in the area of rhythmic and collective behavior.
He currently serves as an associate editor for the journals SIAM J. of Control and Optimization, Mathematics of Control, Signals, and Systems, Journal of Nonlinear Science, and Systems and Control Letters. He is a fellow of the IEEE and the recipient of the 2008 IEEE CSS Ruberti young researcher prize.
Location
Distinguished Lecture Program
Talk Title: Consensus in Nonlinear Spaces
Talk Title: Rank-preserving optimization on the cone of positive semidefinite matrices: a geometric approach
The talk is an introduction to a recent computational framework for optimization over the set of fixed rank positive semidefinite matrices. The foundation is geometric and the motivation is algorithmic, with a bias towards low-rank computations in large-scale problems. Special attention is given to two quotient riemannian geometries that are rooted in classical matrix factorizations and that lead to rank-preserving efficient computations in the cone of symmetric positive definite matrices. The field of applications is vast, and the talk surveys recent developments that illustrate the potential of the approach in large-scale computational problems encountered in control, optimization, and machine learning. The talk is introductory and requires no particular background in riemannian geometry.
Talk Title: Algorithmic challenges in an information-rich age
Modern scientific exploration is increasingly based on distributed technologies. Those include extremely large telescopes made of thousand individual mirrors, sensor networks sharing hundreds of spatially distributed measurements in the ocean, microarrays allowing for the simultaneous measurement of thousands gene expressions in a single cell, or the simultaneous acquisition of thousands of diffusion tensors (one per voxel) in brain imaging.
Distributed technologies overcome fundamental hardware limitations at the price of formidable computational challenges. Those challenges raise new algorithmic questions that involve a mix of statistical, machine learning, and optimization tools.
This non-technical talk illustrates some of these challenges and open questions through a journey across three concrete research projects, suggesting that geometry plays a fundamental role in making those computational problems tractable.