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 are in optimal design, optimal control, and inverse problems.

Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of numerical methods. A standard formulation of PDE-constrained optimization encountered in a number of problems takes the form:$$\min _{y,u}\;{1 \over {2}}\|y-{\widehat {y}}\|_{L_{2}(\Omega )}^{2}+{\beta \over {2}}\|u\|_{L_{2}(\Omega )}^{2},\quad {\text{s.t.}}\;{\mathcal {D}}y=u$$where $u$ in the control variable and $\|\cdot \|_{L_{2}(\Omega )}^{2}$ is the Euclidean norm.

Applications

• Aerodynamic shape optimization
• Drug delivery
• Mathematical finance

Example: 2D Navier-Stokes problem

This example problem aims to minimize the norm of the vorticity $\omega =\nabla \times {\bf {v}}$ over a given 2D region $D$ with some multivariate control ${\bf {u}}$:$$\min _{{\bf {u}}\in {\mathcal {U}}}\;{1 \over {2}}\int _{D}\|\omega \|^{2}d{\bf {x}}+{\alpha \over {2}}\int _{\Gamma _{c}}\|\nabla _{s}{\bf {u}}\|^{2}d{\bf {x}}$$where the dynamics are governed by the steady, incompressible Navier-Stokes equations:$${\begin{aligned}-\nu \Delta {\bf {v}}+{\bf {v}}\cdot \nabla {\bf {v}}+\nabla p&={\bf {f}}\quad {\text{in}}\quad D\\\nabla \cdot {\bf {v}}&=0\quad {\text{in}}\quad D\\(\nu \nabla {\bf {v}}-pI){\widehat {\bf {n}}}&={\bf {0}}\quad {\text{on}}\quad \Gamma _{out}\\{\bf {v}}&={\bf {u}}\quad {\text{on}}\quad \Gamma _{c}\\{\bf {v}}&={\bf {b}}\quad {\text{on}}\quad \partial D\backslash (\Gamma _{c}\cup \Gamma _{out})\end{aligned}}$$

See also

• Multiphysics
• Shape optimization
• SU2 code

Further reading

• 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.
• Biegler, Lorenz T.; Ghattas, Omar; Heinkenschloss, Matthias; Keyes, David; van Bloemen Waanders, Bart (2007). Real-Time PDE-Constrained Optimization. Computational Science and Engineering, Society for Industrial and Applied Mathematics. ISBN 978-0898716214.
• 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.

External links

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