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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|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 include [[aerodynamics]], [[computational fluid dynamics]], [[image segmentation]], and [[Inverse problem|inverse problems]].<ref>{{Cite book|url=https://epubs.siam.org/doi/abs/10.1137/1.9780898718935|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|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> is the control variable and <math>\|\cdot\|_{L_{2}(\Omega)}^{2}</math> is the [[Euclidean norm]]. 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><br />
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