PDF Publication Title:
Text from PDF Page: 117
6.2 Proper Orthogonal Decomposition (POD) 6.2.3 POD and Singular Value Decomposition Computation of POD basis functions is closely linked with calculating the singular value decom- position (SVD) of the snapshot matrix Y ∈ RNx×Nt. Singular-value decomposition guarantees the existence of real numbers σ1 ≥ σ2 ≥ . . . ≥ σNx ≥ 0 and orthogonal matrices U ∈ RNx×Nx , V ∈ RNt×Nt, of YYT and YT Y, respectively, while σ12, . . . , σN2 as both have same eigenvalues). As a consequence, we obtain the singular value decomposition of the snapshot matrix as Y = UDV T (6.6) since U and V are orthogonal matrices. SVD can be used to obtain a complete set of POD basis functions. To obtain POD basis set, we solve Problem (6.2) for a given value of M. A necessary optimality condition for Problem (6.2) is given by the following eigenvalue problem YYT u = σ2u (6.7) From SVD, it is clear that the first M vectors {ui}Mi=1, M ≤ Nx of orthogonal matrix U solve (6.7), and thus solve Problem (6.2) as well. Hence {ui}Mi=1 represent the desired set of POD basis functions (or basis vectors in our case of finite spatial dimension). The amount of projection of all the snapshots {yj}Nt captured by each basis function {ui}M is given j=1 i=1 by corresponding singular values {σi2}Mi=1. Since σ1 ≥ σ2 ≥ ... ≥ σM, the first POD basis function captures maximum projection of Y on the new reduced basis set, second POD basis with columns {ui}Nx , i=1 with columns {vj}Nt , j=1 such that U T Y V = D, where D ∈ RNx ×Nt σ1 , . . . , σNx , called singular values of Y . Here, vectors {ui }Nx and {vj }Nt are the eigenvectors is a diagonal matrix with diagonal entries being x i=1 j=1 represent the eigenvalues of YYT (or YT Y, Chapter 6. Reduced-order Modeling for Optimization 103PDF Image | Design and Operation of Pressure Swing Adsorption Processes
PDF Search Title:
Design and Operation of Pressure Swing Adsorption ProcessesOriginal File Name Searched:
anshul_thesis.pdfDIY PDF Search: Google It | Yahoo | Bing
CO2 Organic Rankine Cycle Experimenter Platform The supercritical CO2 phase change system is both a heat pump and organic rankine cycle which can be used for those purposes and as a supercritical extractor for advanced subcritical and supercritical extraction technology. Uses include producing nanoparticles, precious metal CO2 extraction, lithium battery recycling, and other applications... More Info
Heat Pumps CO2 ORC Heat Pump System Platform More Info
CONTACT TEL: 608-238-6001 Email: greg@infinityturbine.com (Standard Web Page)