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 Euclidean norm. Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of numerical methods.[4][5][6]

Applications

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

Optimal control of bacterial chemotaxis system

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.

See also

• Multiphysics
• Shape optimization
• SU2 code

References

1. Leugering, Günter; Benner, Peter; Engell, Sebastian; Griewank, Andreas; Harbrecht, Helmut; Hinze, Michael; Rannacher, Rolf; Ulbrich, Stefan, eds. (2014). "Trends in PDE Constrained Optimization". International Series of Numerical Mathematics. Springer. doi:10.1007/978-3-319-05083-6. ISSN 0373-3149.
2. Lorenz T. Biegler; Omar Ghattas; Matthias Heinkenschloss; David Keyes; Bart van Bloemen Waanders, eds. (2007-01-01). Real-Time PDE-Constrained Optimization. Computational Science & Engineering. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898718935. ISBN 978-0-89871-621-4.
3. Pearson, John (May 16, 2018). "PDE-Constrained Optimization in Physics, Chemistry & Biology: Modelling and Numerical Methods" (PDF). University of Edinburgh.
4. Biros, George; Ghattas, Omar (2005-01-01). "Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver". SIAM Journal on Scientific Computing. 27 (2): 687–713. doi:10.1137/S106482750241565X. ISSN 1064-8275.
5. Antil, Harbir; Heinkenschloss, Matthias; Hoppe, Ronald H. W.; Sorensen, Danny C. (2010-08-01). "Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables". Computing and Visualization in Science. 13 (6): 249–264. doi:10.1007/s00791-010-0142-4. ISSN 1433-0369.
6. Schöberl, Joachim; Zulehner, Walter (2007-01-01). "Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems". SIAM Journal on Matrix Analysis and Applications. 29 (3): 752–773. doi:10.1137/060660977. ISSN 0895-4798.
7. Jameson, Antony (2003). "Aerodynamic Shape Optimization Using the Adjoint Method" (PDF). Stanford University.
8. Hazra, S. B.; Schulz, V.; Brezillon, J.; Gauger, N. R. (2005-03-20). "Aerodynamic shape optimization using simultaneous pseudo-timestepping". Journal of Computational Physics. 204 (1): 46–64. doi:10.1016/j.jcp.2004.10.007. ISSN 0021-9991.
9. Somayaji, Mahadevabharath R.; Xenos, Michalis; Zhang, Libin; Mekarski, Megan; Linninger, Andreas A. (2008-01-01). "Systematic design of drug delivery therapies". Computers & Chemical Engineering. Process Systems Engineering: Contributions on the State-of-the-Art. 32 (1): 89–98. doi:10.1016/j.compchemeng.2007.06.014. ISSN 0098-1354.
10. Antil, Harbir; Nochetto, Ricardo H.; Venegas, Pablo (2017-10-19). "Optimizing the Kelvin force in a moving target subdomain". Mathematical Models and Methods in Applied Sciences. 28 (01): 95–130. arXiv:1612.07763. doi:10.1142/S0218202518500033. ISSN 0218-2025.
11. Egger, Herbert; Engl, Heinz W. (2005). "Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates". Inverse Problems. 21 (3): 1027–1045.

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.
• 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