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Model Reduction Using Proper Orthogonal Decomposition
| Content Provider | Semantic Scholar |
|---|---|
| Author | Volkwein, Stefan |
| Copyright Year | 2011 |
| Abstract | In this lecture notes an introduction to model reduction utilizing proper orthogonal decomposition (POD) is given. The close connection between POD and singular value decomposition (SVD) of rectangular matrices is emphasized. As an application POD is used to derive a reduced-order model for non-linear initial value problems. The strategy is extended to linear-quadratic optimal control problems governed by ordinary differential equations. The relationship to classical model reduction techniques like balanced truncation is studied. 1. The POD method in R In this section we introduce the POD method in the Euclidean space R and study the close connection to the SVD of rectangular matrices; see [6]. We also refer to the monograph [3]. 1.1. POD and SVD. Let Y = [y1, . . . , yn] be a real-valued m× n matrix of rank d ≤ min{m,n} with columns yj ∈ R, 1 ≤ j ≤ n. Consequently, (1.1) ȳ = 1 n n ∑ j=1 yj can be viewed as the column-averaged mean of the matrix Y . SVD [10] guarantees the existence of real numbers σ1 ≥ σ2 ≥ . . . ≥ σd > 0 and orthogonal matrices U ∈ R with columns {ui}i=1 and V ∈ R with columns {vi}i=1 such that (1.2) U Y V = ( D 0 0 0 ) =: Σ ∈ R, where D = diag (σ1, . . . , σd) ∈ R and the zeros in (1.2) denote matrices of appropriate dimensions. Moreover the vectors {ui}i=1 and {vi}i=1 satisfy (1.3) Y vi = σiui and Y T ui = σivi for i = 1, . . . , d. They are eigenvectors of Y Y T and Y T Y , respectively, with eigenvalues λi = σ 2 i > 0, i = 1, . . . , d. The vectors {ui}i=d+1 and {vi}i=d+1 (if d < m respectively d < n) are eigenvectors of Y Y T and Y T Y with eigenvalue 0. From (1.2) we deduce that Y = UΣV T . Date: December 7, 2011. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.uni-konstanz.de/numerik/personen/volkwein/teaching/POD-Vorlesung.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
