Speaker: Alex Kaltenbach (Technical University of Berlin) - https://alexkaltenbach.github.io/ When: Tue, September 1, 2026 - 3:30pm Where: Kirwan Hall 3206
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Abstract: Nonsmooth convex variational problems arise in many applications, including the p-Dirichlet problem, obstacle problems, total variation minimization, elastoplastic torsion, and optimal insulation. In this talk, we use Fenchel duality to develop a systematic approach to three central aspects of their numerical treatment: a posteriori error analysis, a priori error analysis, and numerical optimization.
We first introduce the infinite-dimensional primal and dual problems together with the corresponding duality relations, optimality conditions, and reconstruction formulas. Based on continuous primal-dual error identities, we obtain exact and fully computable a posteriori error estimators that can be used for adaptive mesh refinement.
We then transfer this duality structure to the discrete level. To this end, we exploit discrete integration-by-parts and orthogonality relations between the nonconforming Crouzeix–Raviart and Raviart–Thomas finite element spaces. The resulting discrete primal-dual error identities provide explicit error control and, together with suitable discretely admissible approximations of the primal and dual solutions, yield a priori convergence rates.
Finally, we derive a proximity-based semismooth Newton method from the discrete primal-dual optimality conditions. By treating the primal and dual variables simultaneously, the method is more robust than canonical purely primal or purely dual semismooth Newton methods.
We conclude with an outlook on subgradient flows, optimal control problems in digital-twin-based structural health monitoring, and proximity-certified warm-start operators for classical iterative solvers.
Speaker: Sudeshna Bhattacharjee (University of Maryland) - https://sites.google.com/view/sudeshnabhattacharjee/home When: Wed, September 2, 2026 - 2:00pm Where: Kirwan Hall 1311
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Abstract: The KPZ fixed point is a Markov process on the space of upper semi-continuous functions. It is the conjectured universal scaling limit of the height function evolution for models in the KPZ universality class and has been shown to be such for many solvable (and even some unsolvable) models. The directed landscape provides a coupling for the growth of the KPZ fixed point starting from all initial conditions. Under this coupling, starting from an initial condition, the forward evolution of the KPZ fixed point can be described by a variational problem involving the directed landscape. It is an interesting question to characterize all eternal solutions of the KPZ fixed point (i.e. functions defined for both forward and backward times and satisfying the variational formula at all times). In this talk we give a full characterization of these eternal solutions and discuss their structure.
We show that all the eternal solutions are supremum of the well known eternal solutions, called the Busemann functions. Furthermore, we show that there is a homeomorphism between the space of all eternal solutions and a certain space of upper semi-continuous functions on a metric space. We also study the geometric properties of these eternal solutions. In particular, the eternal solutions have a representation as patching up countably many Busemann functions along certain interfaces. These interfaces have interesting geometric properties. For example, we see that going forward in time these interfaces coalesce, but no three interfaces can coalesce at the same point.
The talk is based on joint works with Ofer Busani and Evan Sorensen.