In this talk, we present the first set of algorithms that can compute the Kreiss constant of a square matrix to arbitrary accuracy under reasonable assumptions.  As famously introduced by H.-O. Kreiss over six decades ago, Kreiss constants inform us about how severely stable systems of ordinary differential (or difference) equations will exhibit transient behavior before they settle down.  Specifically, the Kreiss Matrix Theorem provides theoretically tight upper and lower bounds on the largest magnitude of transient behavior that a given system will attain.  Despite the importance of Kreiss's result, it has been difficult to use the theorem in applications because computing Kreiss constants involves solving continuous optimization problems that are often nonconvex and have multiple local optimizers.  Consequently, to estimate transient behavior, Kreiss constants have historically only been approximated using crude and ad hoc techniques that may not provide even a single digit of accuracy.  By taking advantage of special structure in the optimization problems, we introduce the first globally convergent schemes that compute Kreiss constants to any desired accuracy, which opens to the door to reliable prediction and even minimization of transient behavior in dynamical systems.

About the speaker:
Tim Mitchell is an Associate Professor of Computer Science at CUNY Queens College and Doctoral Faculty at the CUNY Graduate Center interested in numerical analysis, optimization, and scientific computing.  Prior to arriving at CUNY in 2022, he was a research scientist at the Max Planck Institute for Dynamics of Complex Technical Systems (Magdeburg, Germany) working in the Computational Methods in Systems and Control Theory group led by Peter Benner.  He earned his PhD in computer science under the supervision of Michael L. Overton at the Courant Institute of Mathematical Sciences at New York University, where he also stayed on as a postdoc for one year.  He has worked at IBM Thomas J. Watson Research Center (Hawthorne, New York) and double majored in computer science and mathematics at Tufts University.  He is also the author of the open-source software packages GRANSO and ROSTAPACK, which respectively are for solving general nonsmooth, nonconvex constrained optimization problems and computing and optimizing various robust stability measures for dynamical systems.