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'''PDE-constrained optimization''' is a subset of [[mathematical optimization]] where at least one of the [[Constrained optimization|constraints]] may be expressed as a [[partial differential equation]].<ref>{{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}}</ref> Typical domains where these problems arise include [[aerodynamics]], [[computational fluid dynamics]], [[image segmentation]], and [[Inverse problem|inverse problems]].<ref>{{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}}</ref> A standard formulation of PDE-constrained optimization encountered in a number of disciplines is given by:<ref name=":0">{{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}}</ref><math display="block">\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</math>where <math>u</math> is the control variable and <math>\|\cdot\|_{L_{2}(\Omega)}^{2}</math> 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 for partial differential equations|numerical methods]].<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><br />
{{essay-like|date=March 2020}}
'''PDE-constrained optimization''' is a subset of [[mathematical optimization]] where at least one of the [[Constrained optimization|constraints]] may be expressed as a [[partial differential equation]].<ref>{{Cite journal|last=|first=|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|url=https://link.springer.com/book/10.1007/978-3-319-05083-6|journal=International Series of Numerical Mathematics|language=en-gb|publisher=Springer|volume=|pages=|doi=10.1007/978-3-319-05083-6|issn=0373-3149|via=}}</ref> Typical domains where these problems arise are in [[optimal design]], [[optimal control]], and [[Inverse problem|inverse problems]].<ref>{{Cite web|url=https://web.stanford.edu/class/cme334/docs/2011-11-08-choi_pdeopt.pdf|title=PDE-constrained Optimization and Beyond|last=Choi|first=Youngsoo|date=2011|website=Stanford University|url-status=live|archive-url=|archive-date=|access-date=}}</ref>
Closed-form solutions are generally unavailable for PDE-constrained optimization problems, necessitating the development of [[Numerical methods for partial differential equations|numerical methods]].<ref>{{Cite journal|last=Biros|first=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|url=https://epubs.siam.org/doi/abs/10.1137/S106482750241565X|journal=SIAM Journal on Scientific Computing|volume=27|issue=2|pages=687–713|doi=10.1137/S106482750241565X|issn=1064-8275}}</ref><ref>{{Cite journal|last=Antil|first=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|url=https://doi.org/10.1007/s00791-010-0142-4|journal=Computing and Visualization in Science|language=en|volume=13|issue=6|pages=249–264|doi=10.1007/s00791-010-0142-4|issn=1433-0369}}</ref><ref>{{Cite journal|last=Schöberl|first=Joachim|last2=Zulehner|first2=Walter|date=2007-01-01|title=Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems|url=https://epubs.siam.org/doi/abs/10.1137/060660977|journal=SIAM Journal on Matrix Analysis and Applications|volume=29|issue=3|pages=752–773|doi=10.1137/060660977|issn=0895-4798}}</ref> A standard formulation of PDE-constrained optimization encountered in a number of problems takes the form:<ref>{{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|url-status=live|archive-url=|archive-date=|access-date=}}</ref><math display="block">\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</math>where <math>u</math> in the control variable and <math>\|\cdot\|_{L_{2}(\Omega)}^{2}</math> is the [[Euclidean norm]].<br />
== Applications ==
* Aerodynamic shape optimization<ref>{{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|url-status=live|archive-url=|archive-date=|access-date=}}</ref><ref>{{Cite journal|lastlast1=Hazra|firstfirst1=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}}</ref>
* [[Drug delivery]]<ref>{{Cite journal|lastlast1=Somayaji|firstfirst1=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}}</ref><ref>{{Cite journal|lastlast1=Antil|firstfirst1=Harbir|last2=Nochetto|first2=Ricardo H.|last3=Venegas|first3=Pablo|date=2017-10-19|title=Optimizing the Kelvin force in a moving target subdomain|url=https://www.worldscientific.com/doi/abs/10.1142/S0218202518500033|journal=Mathematical Models and Methods in Applied Sciences|volume=28|issue=011|pages=95–130|doi=10.1142/S0218202518500033|issn=0218-2025|arxiv=1612.07763|s2cid=119604277}}</ref>
* [[Mathematical finance]]<ref>{{Cite journal|lastlast1=Egger|firstfirst1=Herbert|last2=Engl|first2=Heinz W.|date=2005|title=Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates|urljournal=Inverse Problems|volume=21|issue=3|pages=1027–1045|doi=https://iopscience.iop.org/article/10.1088/0266-5611/21/3/014/pdf|journalbibcode=Inverse2005InvPr..21.1027E Problems|volume=21|issue=3|pages=1027-1045|vias2cid=11012681 }}</ref>
* [[Epidemiology]]<ref>{{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 }}</ref>
=== Optimal control of bacterial chemotaxis system ===
=== Example: 2D Navier-Stokes problem ===
The following example comes from p. 20-21 of Pearson.<ref name=":0" /> [[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 <math>z(t,{\bf x})</math> and concentration density <math>c(t,{\bf x})</math> of a [[Chemotaxis#Chemoattractants and chemorepellents|chemoattractant]], it is possible to formulate a boundary control problem:<math display="block">\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}</math>where <math>\widehat{z}</math> is the ideal cell density, <math>\widehat{c}</math> is the ideal concentration density, and <math>u</math> is the control variable. This objective function is subject to the dynamics:<math display="block">\begin{aligned}
This example problem aims to minimize the norm of the [[vorticity]] <math>\omega = \nabla \times {\bf v}</math> over a given 2D region <math>D</math> with some multivariate control <math>{\bf u}</math>:<ref>{{Cite web|url=https://scholarship.rice.edu/handle/1911/102048|title=Trust Region SQP Methods With Inexact Linear System Solves For Large-Scale Optimization|last=Ridzal|first=Denis|date=2006|website=Rice University|url-status=live|archive-url=|archive-date=|access-date=}}</ref><math display="block">\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}</math>where the dynamics are governed by the [[Steady flow|steady]], [[Incompressible flow|incompressible]] [[Navier–Stokes equations|Navier-Stokes equations]]:<math display="block">\begin{aligned}
-{\nupartial z\Delta over{\bfpartial vt}} +- D_{z}\bfDelta z - v}\cdotalpha \nabla {\bfcdot v} +\left[ {\nabla pc\over{(1+c)^{2}}}z \right] &= {\bf f}0 \quad \text{in} \quad D\Omega \\
{\nablapartial c\cdot over{\bfpartial vt}} - \Delta c + \rho c - w{z^{2}\over{1+z^{2}}} &= 0 \quad \text{in} \quad D\Omega \\
(\nu\nabla {\bf v} -partial pI)z\widehatover{\bfpartial n}} &= {\bf 0} \quad \text{on} \quad \Gamma_{out}partial\Omega \\
{\bfpartial vc\over{\partial n}} &=+ {\bfzeta (c-u}) &= 0 \quad \text{on} \quad \Gamma_{c} \partial\Omega
\end{aligned}</math>where <math>\Delta</math> is the [[Laplace operator]].
{\bf v} &= {\bf b} \quad \text{on} \quad \partial D\backslash(\Gamma_{c}\cup\Gamma_{out})
\end{aligned}</math><br />
== See also ==
* Antil, Harbir; Kouri, Drew. P; Lacasse, Martin-D.; Ridzal, Denis (2018). ''[https://www.springer.com/gp/book/9781493986354 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). ''[https://bookstore.ams.org/gsm-112 Optimal Control of Partial Differential Equations: Theory, Methods, and Applications]''. Graduate Studies in Mathematics, American Mathematical Society. {{ISBN|978-0-8218-4904-0|}}.
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