Parallel Algorithms for PDE-Constrained Optimization

PDE-constrained optimization refers to the optimization of systems governed by partial differential equations (PDEs). The simulation problem is to solve the PDEs for the state variables (e.g. displacement, velocity, temperature, electric field, magnetic field, species concentration), given appropriate data (e.g. geometry, coefficients, boundary conditions, initial conditions, source functions). The optimization problem seeks to determine some of these data—the decision variables—given performance goals in the form of an objective function and possibly inequality or equality constraints on the behavior of the system. Since the behavior of the system is modeled by the PDEs, they appear as (usually equality) constraints in the optimization problem. We will refer to these PDE constraints as the state equations. Let u represent the state variables, d the decision variables, J the objective function, c the residual of the state equations, and h the residual of the inequality constraints. We