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We define error tolerance λ as M σ2 Nx σ2 εPOD(M) i=M+1 i ε Nx POD 2 (6.9) (6.10) 6.2 Proper Orthogonal Decomposition (POD) is second best in capturing projection of Y, and so on. We note that Equation (6.6) can also be written as Y = UB, with U ∈ RNx×M being the reduced set of eigenvectors, and B = DVT ∈ RM×Nt. Consequently, each snapshot {yj}Nt of Y can be represented in terms j=1 of M linearly independent columns of U j M y = i=1 where bi,j are the elements of B. Hence, it is clear that the columns of U indeed represent a set of basis functions, and eventually turn out to be the POD basis functions based on the definition. We also note that Equation (6.6) can also be written as UD = YV , with D ∈ RM×Nt and U as defined above. It follows that the POD basis vectors are linear combinations of the snapshots, thus ensuring that such a basis set inherently describes the dynamics of the system by being closely linked to its actual numerical solution. 6.2.4 Subspace Selection Choice of the subspace dimension M controls the overall error in projection εPOD(M) in Prob- lem (6.3). Nevertheless, the choice of M is an important and critical task, since it determines the interrelation between accuracy and dimension of the POD based reduced order models. Since the singular values of the snapshot matrix Y convey the amount of projection captured, we utilize a criterion based on these singular values to choose M. Using SVD, the error in projection in Problem (6.3) is given as bi,jui, j = 1,...,Nt (6.8) (M) = σi POD λ=1−Nx σ2 = Nx σ2 = Nx σ2 =εnorm(M) i=1 i i=1 i i=1 i i=1 i i=M +1 Chapter 6. Reduced-order Modeling for Optimization 104PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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