The proper orthogonal decomposition is a

numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...

that enables a reduction in the complexity of computer intensive simulations such as

computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...

and

structural analysis Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...

(like

crash simulation A crash simulation is a virtual recreation of a destructive crash test of a car or a highway guard rail system using a computer simulation in order to examine the level of safety of the car and its occupants. Crash simulations are used by autom ...

s). Typically in

fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

and turbulences analysis, it is used to replace the

Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...

by simpler models to solve. It belongs to a class of algorithms called ''

model order reduction Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical model ...

'' (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 Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

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 Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

from Pearson in the field of statistics, or the

singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...

in linear algebra because it refers to

eigenvalues and eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

of a physical field. In those domains, it is associated with the research of Karhunen and Loève, 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_k''(''x'') modulated by random time coefficients ''a_k''(''t'') so that: :

u(x,t)=\sum_^\infty a_k (t) \phi_k(x)

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 is averaging the samples over the space dimension ''n'', and correlating them with each other along the time samples ''p'': :

U = \begin 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

with n spatial elements, and p time samples The next step is to compute the

covariance matrix In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of ...

C :

C = \frac U^T U

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: :

\phi = \begin \phi_ & \cdots & \phi_ \\ \vdots &  & \vdots \\ \phi_ & \cdots & \phi_ \end{pmatrix}

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-CME345-Ch4.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 Continuum mechanics Numerical differential equations Partial differential equations Structural analysis Computational electromagnetics