Seminars

In science and engineering, we regularly face (constrained) polynomial optimization problems (CPOP). That is the task to minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying nonnegativity of real polynomials – a key problem in real algebraic geometry since the 19th century. Since this is a notoriously hard to solve problem (e.g., various NP-complete problems admit a CPOP formulation), one is interested in certificates that imply nonnegativity and are easier to check than nonnegativity itself. Many applications require solving CPOPs that have a sparse structure, i.e., given the n variables and maximal total degree d, just very few of the possible (n+d) terms appear. In this talk, I will give a brief general introduction to polynomial optimization, recall key developments from the last decade regarding certain certificates with advantages in sparse settings, and then give an overview on some of our recent and ongoing work in this area both with a theoretical and an applied focus.

The construction of spline bases over hierarchical T-meshes is
fundamentally important for numerous applications, ranging from
geometric modeling to isogeometric analysis. The case of C^s smooth
splines of degree p = 2s + 1 is particularly attractive, as the
dimension of the resulting spaces is well understood and these
function spaces have proven useful in various contexts. We explore the
generation of these functions as linear combinations of elements from
B-systems, which are collections of tensor-product B-splines
associated with cross vertices. Specifically, we examine the linear
independence of the resulting system of generating functions, thereby
generalizing earlier results by Kang, Xu, Chen, and Deng (Graphical
Models, 2015) to a more general refinement strategy that produces a
broader class of T-meshes. This is joint work with Maodong Pan.