Theory and algorithms for optimization in function spaces

Optimization under uncertainty and risk

Design and control problems for physical systems are typically posed with uncertain data. Optimizing an expected value is often not enough: one needs to control the tail of the distribution of outcomes. I work on the analytical foundations of risk-averse PDE-constrained optimization, including existence, optimality conditions and stability with respect to the underlying probability law. I also develop algorithms that remain efficient in infinite dimensions.

Examples include the epi-regularization of risk measures, primal–dual and interior-point methods for risk minimization, and the asymptotic analysis of Monte Carlo (sample average) approximations, much of it in long-standing collaboration with Drew P. Kouri (Sandia National Laboratories).

Structure-preserving numerical methods

Pointwise bounds, obstacles and other inequality constraints appear throughout mechanics, physics and design. Proximal Galerkin and the latent variable proximal point (LVPP) method, developed with Brendan Keith and coauthors, enforce such constraints exactly at the discrete level by working with a latent variable inside a Bregman proximal point iteration. The result is a family of finite element solvers whose iteration counts are largely insensitive to mesh refinement.

The same ideas underlie SiMPL, a simple first-order method for density-based topology optimization, and recent extensions to phase-field fracture and multi-material design. Proximal Galerkin received the 2025 Langtangen Prize. The methods are implemented in FEniCS, Firedrake and MFEM.

Variational analysis and optimal control

Many control and design problems have a nonsmooth structure: the state is defined by a variational inequality, a complementarity system or the solution of a lower-level optimization problem. I use tools from variational analysis, such as generalized derivatives, coderivatives and directional differentiability, to derive stationarity conditions for such problems in function spaces and to design solution algorithms.

A related strand concerns generalized Nash equilibrium problems with PDE constraints, including risk-neutral and mean-field formulations.

Optimization for trustworthy AI

At SURE-AI, the Norwegian Centre for Sustainable, Risk-Averse and Ethical AI, I lead the work package on optimization in AI. The aim is to bring the mathematics of risk-averse optimization to bear on learning systems: risk-aware and explainable inference, the stability of statistical estimators, and energy-efficient algorithms with guarantees.