Stochastic optimization is the study of optimization problems under uncertainty, with applications across economics, physics, ecology, and medicine. A model of the uncertainty is assumed to be available: for example, future energy demand cannot be predicted exactly, but it can be approximated by probability distributions that depend on season or weather. In manufacturing, irregularities may affect the physical properties of products, yet repeated production yields statistical models of these variations.
A central focus of our research group is the optimization of systems governed by physical laws. These laws are typically described by systems of ordinary or partial differential equations, with uncertainty appearing in physical parameters or external inputs. The resulting problems are generally infinite-dimensional.
The theoretical challenges are manifold. Existence of solutions and optimality theory must be developed in function spaces, requiring tools from functional analysis and stochastic analysis. Many physically relevant problems also involve nonsmoothness, since design or control variables are frequently constrained to feasible sets. Such nonsmooth structures create additional difficulties for both analysis and computation.
On the computational side, closed-form solutions rarely exist for the physical states of interest, so numerical approximation techniques such as finite differences or finite elements are required. When uncertainty influences the physical laws themselves, these problems become significantly more computationally expensive, as multiple scenarios must be considered simultaneously. Efficiently balancing the numerical error from discretization with the stochastic error from sampling is a central concern.
Our group designs algorithms that address these challenges by extending classical stochastic approximation and regularization techniques to infinite-dimensional settings, while also drawing on advances from data science. By combining rigorous analysis with efficient computation, we aim to develop robust tools for uncertainty-aware design and decision-making in physics-based applications, from engineering and materials science to economics and beyond.
