PDE-constrained optimization is a subset of mathematical optimization where at least one of the constraints may be expressed as a partial differential equation.
[1] Typical domains where these problems arise include aerodynamics, computational fluid dynamics, image segmentation, and inverse problems.
[2] A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:[3]
min
1 2
‖ y −
β 2
is the control variable and
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.
[4][5][6] The following example comes from p. 20-21 of Pearson.
[3] 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:
min
γ
γ
is the ideal cell density,
is the ideal concentration density, and
is the control variable.
This objective function is subject to the dynamics:
z − α ∇ ⋅
c + ρ c − w
+ ζ ( c − u )
{\displaystyle {\begin{aligned}{\partial z \over {\partial t}}-D_{z}\Delta z-\alpha \nabla \cdot \left[{\nabla c \over {(1+c)^{2}}}z\right]&=0\quad {\text{in}}\quad \Omega \\{\partial c \over {\partial t}}-\Delta c+\rho c-w{z^{2} \over {1+z^{2}}}&=0\quad {\text{in}}\quad \Omega \\{\partial z \over {\partial n}}&=0\quad {\text{on}}\quad \partial \Omega \\{\partial c \over {\partial n}}+\zeta (c-u)&=0\quad {\text{on}}\quad \partial \Omega \end{aligned}}}
is the Laplace operator.