Dr. Christof Vermeersch
Research: The (block) Macaulay matrix, a tool to solve systems of multivariate polynomial equations and multiparameter eigenvalue problems
Within the context of this ERC grant, we try to rephrase many system identification and model order reduction problems as systems of multivariate polynomial equations and (rectangular) multiparameter eigenvalue problems, the solutions of which characterize the stationary points of the underlying optimization problem.
One question quickly emerges: “How can we solve both problems accurately?” Some very efficient techniques already exist for systems of multivariate polynomial equations. One of these techniques, via the null/column space of the Macaulay matrix, has been developed by our research group. However, there are not many tools available to tackle (rectangular) multiparameter eigenvalue problems. In our research, we also develop algorithms that solve the (rectangular) multiparameter eigenvalue problems that emerge from these systems theoretical applications. The approach taken to tackle these problems is to go “Back to the Roots” and consider a block extension of the Macaulay matrix from polynomial system solving. By exploiting the structured null/column space of this block Macaulay matrix constructed from the coefficient matrices of the (rectangular) multiparameter eigenvalue problem, we can retrieve the unknown eigentuples. This novel approach sounds easy, but the devil (and research) is in the (numerical/computational) details!
MacaulayLab: Christof Vermeersch is also the maintainer of the related toolbox www.macaulaylab.net.