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.^[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]$${\displaystyle \min _{y,u}\;{\frac {1}{2}}\|y-{\widehat {y}}\|_{L_{2}(\Omega )}^{2}+{\frac {\beta }{2}}\|u\|_{L_{2}(\Omega )}^{2},\quad {\text{s.t.}}\;{\mathcal {D}}y=u}$$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.^[4][5][6]

• Aerodynamic shape optimization^[7][8]
• Drug delivery^[9][10]
• Mathematical finance^[11]
• Epidemiology^[12]

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 ${\displaystyle z(t,{\bf {x}})}$ and concentration density ${\displaystyle c(t,{\bf {x}})}$ of a chemoattractant, it is possible to formulate a boundary control problem:$${\displaystyle \min _{z,c,u}\;{1 \over {2}}\int _{\Omega }\left[z(T,{\bf {x}})-{\widehat {z}}\right]^{2}+{\gamma _{c} \over {2}}\int _{\Omega }\left[c(T,{\bf {x}})-{\widehat {c}}\right]^{2}+{\gamma _{u} \over {2}}\int _{0}^{T}\int _{\partial \Omega }u^{2}}$$where ${\displaystyle {\widehat {z}}}$ is the ideal cell density, ${\displaystyle {\widehat {c}}}$ is the ideal concentration density, and ${\displaystyle u}$ is the control variable. This objective function is subject to the dynamics:$${\displaystyle {\begin{aligned}{\partial z \over {\partial t}}-D_{z}\Delta z-\alpha \nabla \cdot \left[{\nabla c \over {(1+c)^{2}}}z\right]&=0\quad {\text{in}}\quad \Omega \\{\partial c \over {\partial t}}-\Delta c+\rho c-w{z^{2} \over {1+z^{2}}}&=0\quad {\text{in}}\quad \Omega \\{\partial z \over {\partial n}}&=0\quad {\text{on}}\quad \partial \Omega \\{\partial c \over {\partial n}}+\zeta (c-u)&=0\quad {\text{on}}\quad \partial \Omega \end{aligned}}}$$where ${\displaystyle \Delta }$ is the Laplace operator.

• Multiphysics
• Shape optimization
• SU2 code

• Antil, Harbir; Kouri, Drew. P; Lacasse, Martin-D.; Ridzal, Denis (2018). Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, Springer. ISBN 978-1493986354.
• Tröltzsch, Fredi (2010). Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate Studies in Mathematics, American Mathematical Society. ISBN 978-0-8218-4904-0.

• A Brief Introduction to PDE Constrained Optimization
• PDE Constrained Optimization
• Optimal solvers for PDE-Constrained Optimization
• Model Problems in PDE-Constrained Optimization