Prof. Christian Lessig
Assistant Professor, Otto-von-Guerick-Universität Magdeburg
Divergence Free Polar Wavelets
Friday, May 18, 2018, 11:00am to 12:00pm | Room 1-150
Divergence free vector fields play an important role in many systems in science and engineering, making their analysis and numerical representation important in fields ranging from climate science to medical imaging. We introduce a tight frame of divergence free wavelets that resolves the different scales and orientations that occur in these vector fields. Our wavelets are thereby divergence free in the ideal, analytic sense, have an intuitive correspondence to natural phenomena, closed form expressions in frequency and space, a multi-resolution structure, and fast transforms. Our construction also provides well defined directional selectivity that, among other things, models the behavior of solenoidal vector fields in the vicinity of boundaries. With suitable window functions, this provides (up to a logarithmic factor) an optimal approximation rate for piecewise continuous divergence-free vector fields in two dimensions. We demonstrate the numerical practicality and efficiency of our construction for the representation of solenoidal vector fields and the simulation of the Navier-Stokes equation.
Christian Lessig is an assistant professor at Otto-von-Guerick-Universität Magdeburg, Germany. Prior to that, he held postdoctoral positions at Technische Universität Berlin and the California Institute of Technology. He obtained his Ph.D at the University of Toronto. His research is at the intersection of computer science, applied mathematics and mathematical physics and broadly aims at the efficient numerical simulation of dynamical systems..