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 is the control variable and is the Euclidean norm. Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of numerical methods.

Applications

The following example comes from p. 20-21 of Pearson. Chemotaxis is the movement of an organism in response to an external chemical stimulus. One problem of particular interest is in managing the spatial dynamics of bacteria that are subject to chemotaxis to achieve some desired result. For a cell density and concentration density of a chemoattractant, it is possible to formulate a boundary control problem:where is the ideal cell density, is the ideal concentration density, and is the control variable. This objective function is subject to the dynamics:where is the Laplace operator.