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M Nt j2 6.2 Proper Orthogonal Decomposition (POD) that such a basis set inherently describes the dynamics in the best possible manner by being closely linked to the accurate numerical solution of the system. In particular, we use method of snaphots in this work to generate POD basis functions [175]. POD-based model reduction begins with the collection of snapshot sets which consist of solutions of the PDEs at several time instants during the evolution of the system. These snap- shot sets are obtained by solving a rigorous, large-dimensional system obtained after spatial discretization (and temporal also in some cases) of the PDEs. The determination of these sets is crucial to the effectiveness of POD-based reduced-order modeling. Hence, they must contain sufficient information to accurately represent the dynamics of the system. One then uses the set of snapshots to determine a POD basis set which can accurately capture the information contained in the snapshots using a much smaller set of basis functions. Let the snapshot set (solution of PDEs) be given as Y = {y1,...,yNt} (6.1) with the fields yj = y(x, tj ), where Nt is the number of snapshots and Nx is the number of spa- tial discretization nodes. Here columns {Y:,j}Nt , known as snapshots, are the spatial profiles j=1 of the state variable evaluated at time tj. Similary, rows {Yi,:}Nx are the time trajectories of i=1 the state variable evaluated at spatial location xi. Consequently, (1/Nt) Nt Yi,j is the time- j=1 averaged mean of the trajectory at location xi. POD procedure computes an orthonormal set of basis functions {φ1, . . . , φNx } which maximizes projection of each snapshot on to the first M ≤ Nx basis functions. In other words, it solves where (v,w) = (v,w)L2 denotes the L2-inner product with the corresponding norm ∥v∥ = ∥v∥L2 . Here (yj , φi) is the projection of jth snapshot on ith basis function φi. Instead of maximing a convex problem, it is reformulated such that the sum of the error between each max |(y ,φi)| s.t. ∥φi∥ = 1, (φi,φj)i̸=j = 0 i,j = 1,...,M (6.2) φ1 ,...,φM i=1 j =1 Chapter 6. Reduced-order Modeling for Optimization 101PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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