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.{{Cite journal|date=2014|editor-last=Leugering|editor-first=Günter|editor2-last=Benner|editor2-first=Peter|editor3-last=Engell|editor3-first=Sebastian|editor4-last=Griewank|editor4-first=Andreas|editor5-last=Harbrecht|editor5-first=Helmut|editor6-last=Hinze|editor6-first=Michael|editor7-last=Rannacher|editor7-first=Rolf|editor8-last=Ulbrich|editor8-first=Stefan|title=Trends in PDE Constrained Optimization|journal=International Series of Numerical Mathematics|language=en-gb|publisher=Springer|volume=165|doi=10.1007/978-3-319-05083-6|isbn=978-3-319-05082-9|issn=0373-3149}} Typical domains where these problems arise include aerodynamics, computational fluid dynamics, image segmentation, and inverse problems.{{Cite book|title=Real-Time PDE-Constrained Optimization|date=2007-01-01|publisher=Society for Industrial and Applied Mathematics|isbn=978-0-89871-621-4|editor-last=Lorenz T. Biegler|series=Computational Science & Engineering|doi=10.1137/1.9780898718935|editor-last2=Omar Ghattas|editor-last3=Matthias Heinkenschloss|editor-last4=David Keyes|editor-last5=Bart van Bloemen Waanders}} A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:{{Cite web|url=https://www.maths.dundee.ac.uk/aathanassoulis/Pearson_May2018.pdf|title=PDE-Constrained Optimization in Physics, Chemistry & Biology: Modelling and Numerical Methods|last=Pearson|first=John|date=May 16, 2018|website=University of Edinburgh}} $\min_{y,u} \; \frac 1 2 \|y-\widehat{y}\|_{L_2(\Omega)}^2 + \frac\beta2 \|u\|_{L_2(\Omega)}^2, \quad \text{s.t.} \; \mathcal{D}y = u$ where $u$ is the control variable and $\|\cdot\|_{L_{2}(\Omega)}^{2}$ 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.{{Cite journal|last1=Biros|first1=George|last2=Ghattas|first2=Omar|date=2005-01-01|title=Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver|journal=SIAM Journal on Scientific Computing|volume=27|issue=2|pages=687–713|doi=10.1137/S106482750241565X|bibcode=2005SJSC...27..687B |issn=1064-8275}}{{Cite journal|last1=Antil|first1=Harbir|last2=Heinkenschloss|first2=Matthias|last3=Hoppe|first3=Ronald H. W.|last4=Sorensen|first4=Danny C.|date=2010-08-01|title=Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables|journal=Computing and Visualization in Science|language=en|volume=13|issue=6|pages=249–264|doi=10.1007/s00791-010-0142-4|s2cid=9412768|issn=1433-0369|url=https://nbn-resolving.org/urn:nbn:de:bvb:384-opus4-10652}}{{Cite journal|last1=Schöberl|first1=Joachim|last2=Zulehner|first2=Walter|date=2007-01-01|title=Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems|journal=SIAM Journal on Matrix Analysis and Applications|volume=29|issue=3|pages=752–773|doi=10.1137/060660977|issn=0895-4798}}

Applications

Aerodynamic shape optimization{{Cite web|url=http://aero-comlab.stanford.edu/Papers/jameson.vki03.pdf|title=Aerodynamic Shape Optimization Using the Adjoint Method|last=Jameson|first=Antony|date=2003|website=Stanford University}}{{Cite journal|last1=Hazra|first1=S. B.|last2=Schulz|first2=V.|last3=Brezillon|first3=J.|last4=Gauger|first4=N. R.|date=2005-03-20|title=Aerodynamic shape optimization using simultaneous pseudo-timestepping|url=http://www.sciencedirect.com/science/article/pii/S0021999104004061|journal=Journal of Computational Physics|language=en|volume=204|issue=1|pages=46–64|doi=10.1016/j.jcp.2004.10.007|bibcode=2005JCoPh.204...46H |issn=0021-9991|url-access=subscription}}
Drug delivery{{Cite journal|last1=Somayaji|first1=Mahadevabharath R.|last2=Xenos|first2=Michalis|last3=Zhang|first3=Libin|last4=Mekarski|first4=Megan|last5=Linninger|first5=Andreas A.|date=2008-01-01|title=Systematic design of drug delivery therapies|url=http://www.sciencedirect.com/science/article/pii/S0098135407001688|journal=Computers & Chemical Engineering|series=Process Systems Engineering: Contributions on the State-of-the-Art|language=en|volume=32|issue=1|pages=89–98|doi=10.1016/j.compchemeng.2007.06.014|issn=0098-1354|url-access=subscription}}{{Cite journal|last1=Antil|first1=Harbir|last2=Nochetto|first2=Ricardo H.|last3=Venegas|first3=Pablo|date=2017-10-19|title=Optimizing the Kelvin force in a moving target subdomain|journal=Mathematical Models and Methods in Applied Sciences|volume=28|issue=1|pages=95–130|doi=10.1142/S0218202518500033|issn=0218-2025|arxiv=1612.07763|s2cid=119604277}}
Mathematical finance{{Cite journal|last1=Egger|first1=Herbert|last2=Engl|first2=Heinz W.|date=2005|title=Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates|journal=Inverse Problems|volume=21|issue=3|pages=1027–1045|doi=10.1088/0266-5611/21/3/014|bibcode=2005InvPr..21.1027E |s2cid=11012681 }}
Epidemiology{{Cite journal |last1=Mehdaoui |first1=Mohamed |last2=Lacitignola |first2=Deborah |last3=Tilioua |first3=Mouhcine |date=2024 |title=Optimal social distancing through cross-diffusion control for a disease outbreak PDE model |url=https://www.sciencedirect.com/science/article/pii/S1007570424000418 |journal=Communications in Nonlinear Science and Numerical Simulation |volume=131 |pages=107855 |doi=10.1016/j.cnsns.2024.107855 |bibcode=2024CNSNS.13107855M |issn=1007-5704|url-access=subscription }}

= Optimal control of bacterial chemotaxis system =

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

References

External links

[https://www2.math.uconn.edu/~leykekhman/my_papers/HA_DL_PDECO-Jan_28_2018.pdf A Brief Introduction to PDE Constrained Optimization]
[https://archive.siam.org/meetings/op08/Heinkenschloss.pdf PDE Constrained Optimization]
[http://www.numerical.rl.ac.uk/people/rees/pdf/NA-08-10_new.pdf Optimal solvers for PDE-Constrained Optimization]
[https://www.mathcs.emory.edu/~haber/pubs/ParamC.pdf Model Problems in PDE-Constrained Optimization]

Category:Mathematical optimization

Category:Optimal control

Category:Partial differential equations

PDE-constrained optimization

Applications

= Optimal control of bacterial chemotaxis system =

See also

References

Further reading

External links