PDE-constrained optimization

PDE-constrained optimization is a subset of mathematical optimization where at least one of the constraints may be expressed as a partial differential equation. Typical domains where these problems arise include aerodynamics, computational fluid dynamics, image segmentation, and inverse problems. A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:

where ${\displaystyle u}$ is the control variable and ${\displaystyle \|\cdot \|_{L_{2}(\Omega )}^{2}}$ is the squared Euclidean norm and is not a norm itself. Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of numerical methods.