Oberseminar Stochastik und Data Science : Pritpal Matharu : Adjoint-Based Enforcement of State Constraints in PDE Optimization Problems

Title: Adjoint-Based Enforcement of State Constraints in PDE Optimization Problems

Abstract: Adjoint-based
methods have become a workhorse in the solution of unconstrained PDE
optimization problems. They make it possible to conveniently determine
the gradient (sensitivity) of the objective functional with respect to a
control variable, which can then be used in various gradient descent
algorithms. Unlike most constraints imposed on the control variable,
constraints on the state variables are generally harder to satisfy since
they define, via solutions of the governing system, complicated
manifolds in the space of control variables. In this talk, we will
demonstrate how this traditional adjoint-based framework can be extended
to handle general constraints on the state variables. This is
accomplished by constructing a projection of the gradient of the
objective functional onto a subspace tangent to the manifold defined by
the constraint. We focus on the “optimize-then-discretize” paradigm in
the infinite-dimensional setting where the required regularity of both
the gradient and of the projection is ensured. This proposed approach
will be illustrated first with a simple test problem describing
optimization of heat transfer in one direction and then a more involved
problem where an optimal closure is found for a turbulent flow described
by the Navier-Stokes system in two dimensions.

Joint work with Bartosz Protas.

https://www.math.fau.de/stochastik/oberseminar-stochastik/