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min ε φ1,...,φM s.t. y − j=1 i=1 y,φi φi = j=1 i=M+1 i,j = 1,...,M y,φi φi (6.3a) (6.3b) ε (M)= y − j=1 y ,ψi ψi (6.5) i=1 102 (M)= ∥φi∥ = 1, (φi,φj)i̸=j = 0 for any other orthonormal basis {ψi}Nx i=1 y = i=1 6.2 Proper Orthogonal Decomposition (POD) snapshot and its projection represented with a truncated first M ≤ Nx basis functions is minimized. POD Nt M 2 Nt Nx 2 j j j where each snapshot yj can be represented in terms of the new basis set using its projection (yj,φi) in the direction of φi Nx jj y,φi φi, j∈{1,...,Nt} (6.4) We choose M based on a subspace selection strategy (discussed in section 6.2.4). Usually, a significant model reduction is achieved since the value of M is much smaller compared to the value of Nx. Moreover, even with such less M, generally the POD error in projection εPOD(M) is significantly small. Thus, eventually we obtain a much smaller subspace spanned by a very few basis functions. 6.2.2 Optimality Property It can be shown that POD basis functions provide an optimal basis set for the representation of the dynamics under consideration [65, 115]. Here, from a physical point of view, these first M basis functions capture more “energy” of the snapshot field than any other set of M orthonormal spatial basis functions. In other words, if we desire to represent each snapshot with exactly M orthonormal basis functions, M POD basis functions will provide the best representation and least error in projection, i.e., Nt M 2 Nt M 2 POD j j j j y ,φi φi ≤ y − j=1 and for all M ≤ Nx. Chapter 6. Reduced-order Modeling for Optimization i=1PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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