Editing Proper orthogonal decomposition

{{Short description|Numerical method that reduces the complexity of computationally intensive simulations}}{{Machine learning bar}}

The '''proper orthogonal decomposition''' is a [[Numerical analysis|numerical method]] that enables a reduction in the complexity of computer intensive simulations such as [[computational fluid dynamics]] and [[structural analysis]] (like [[crash simulation]]s). Typically in [[fluid dynamics]] and [[Turbulence modeling|turbulences analysis]], it is used to replace the [[Navier–Stokes equations]] by simpler models to solve.<ref>{{Cite journal|last1=Berkooz|first1=G|last2=Holmes|first2=P|last3=Lumley|first3=J L|date=January 1993|title=The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows|url=http://dx.doi.org/10.1146/annurev.fl.25.010193.002543|journal=Annual Review of Fluid Mechanics|volume=25|issue=1|pages=539–575|doi=10.1146/annurev.fl.25.010193.002543|bibcode=1993AnRFM..25..539B|issn=0066-4189}}</ref> 

It belongs to a class of algorithms called ''[[model order reduction]]'' (or in short ''model reduction''). What it essentially does is to train a model based on simulation data. To this extent, it can be associated with the field of [[machine learning]].

== POD and PCA ==
The main use of POD is to '''decompose''' a physical field (like pressure, temperature in fluid dynamics or stress and deformation in structural analysis), depending on the different variables that influence its physical behaviors. As its name hints, it's operating an Orthogonal Decomposition along with the Principal Components of the field. As such it is assimilated with the [[principal component analysis]] from [[Karl Pearson|Pearson]] in the field of statistics, or the [[singular value decomposition]] in linear algebra because it refers to [[eigenvalues and eigenvectors]] of a physical field. In those domains, it is associated with the research of Karhunen<ref>{{Cite book|last=Karhunen|first=Kari|title=Zur spektral theorie stochasticher prozesse|year=1946}}</ref> and Loève,<ref>{{Cite journal|last1=David|first1=F. N.|last2=Loeve|first2=M.|date=December 1955|title=Probability Theory.|url=http://dx.doi.org/10.2307/2333409|journal=Biometrika|volume=42|issue=3/4|pages=540|doi=10.2307/2333409|jstor=2333409|issn=0006-3444|url-access=subscription}}</ref> and their [[Karhunen–Loève theorem]].

== Mathematical expression ==
The first idea behind the Proper Orthogonal Decomposition (POD), as it was originally formulated in the domain of fluid dynamics to analyze turbulences, is to decompose a random vector field '''''u(x, t)''''' into a set of deterministic spatial functions ''Φ<sub>k</sub>''(''x'') modulated by random time coefficients ''a<sub>k</sub>''(''t'') so that:

:<math>u(x,t)=\sum_{k=1}^\infty a_k (t) \phi_k(x)</math>
[[File:POD snapshots.png|thumb|POD snapshots]]
The first step is to sample the vector field over a period of time in what we call snapshots (as display in the image of the POD snapshots). This snapshot method<ref>{{Cite journal|last=Sirovich|first=Lawrence|date=1987-10-01|title=Turbulence and the dynamics of coherent structures. I. Coherent structures|journal=Quarterly of Applied Mathematics|volume=45|issue=3|pages=561–571|doi=10.1090/qam/910462|issn=0033-569X|doi-access=free}}</ref> is averaging the samples over the space dimension '''''n''''', and correlating them with each other along the time samples '''''p''''':

:<math>U = \begin{pmatrix} u(x_1,t_1) & \cdots &  u(x_n,t_1)\\ \vdots &  & \vdots \\ u(x_1,t_p) & \cdots &  u(x_n,t_p) \end{pmatrix}</math> with '''n''' spatial elements, and '''p''' time samples

The next step is to compute the [[covariance matrix]] C

:<math>C = \frac{1}{(p-1)} U^T U</math>[[File:Mor-diagram.png|thumb]]

We then compute the eigenvalues and eigenvectors of C and we order them from the largest eigenvalue to the smallest.

We obtain n eigenvalues λ1,...,λn and a set of n eigenvectors arranged as columns in an n × n matrix Φ:

: <math>\phi = \begin{pmatrix} \phi_{1,1} & \cdots & \phi_{1,n} \\ \vdots &  & \vdots \\ \phi_{n,1} & \cdots & \phi_{n,n} \end{pmatrix}</math>

== References ==
<references />

== External links ==
* MIT: http://web.mit.edu/6.242/www/images/lec6_6242_2004.pdf
* Stanford University - Charbel Farhat & David Amsallem https://web.stanford.edu/group/frg/course_work/CME345/CA-AA216-CME345-Ch4.pdf
* [https://depositonce.tu-berlin.de/bitstream/11303/9456/5/podnotes_aiaa2019.pdf Weiss, Julien: A Tutorial on the Proper Orthogonal Decomposition. In: 2019 AIAA Aviation Forum. 17–21 June 2019, Dallas, Texas, United States.]
*French course from CNRS https://www.math.u-bordeaux.fr/~mbergman/PDF/OuvrageSynthese/OCET06.pdf
*Applications of the Proper Orthogonal Decomposition Method http://www.cerfacs.fr/~cfdbib/repository/WN_CFD_07_97.pdf

[[Category:Finite element method| ]]
[[Category:Continuum mechanics]]
[[Category:Numerical differential equations]]
[[Category:Partial differential equations]]
[[Category:Structural analysis]]
[[Category:Computational electromagnetics]]