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where εP OD (M ) is the normalized error in projection. A value of M is chosen such that norm εPOD(M) ≤ λ∗ for a desired threshold error tolerance λ∗. In other words, the normalized norm projection error should be less than the tolerance level λ∗ [175, 65, 27]. It is also known as an M -rank approximation since the rank of the matrix U after truncation, and that of the solution matrix obtained after solving the POD-based reduced-order model, is always M ≤ Nx. It is commonly observed that the first few singular values are significantly larger than the subsequent ones, thus representing most of the captured projection of the system. Therefore, based on the aforementioned criterion, basis functions corresponding to those smaller singular values are dropped, which eventually leads to a much smaller subspace spanned by a very few basis functions. Hence, a significant model reduction is achieved since generally the value of M is much smaller compared to the value of Nx. For instance, M can be less than 10, whereas N can be on the order of 100s, and the error εPOD(M) can still be only 1-3%. x norm 6.3 Reduced-order Modeling 6.3.1 Methodology After computing POD basis functions, a reduced-order model (ROM) is derived by projecting the underlying PDEs of the system onto the corresponding POD subspace. We use a Galerkin- type projection scheme to project our set of PDEs in this work. Let the set of hyperbolic/parabolic PDEs be given as ∂y ∂y∂2y ∂t = f y, ∂x, ∂x2 In terms of the new set of POD basis functions, the state variable y(x, t) is written as M (6.11) (6.12) y(x, t) = ai(t)φi(x) i=1 where {ai}Mi=1 are the unknown temporal coefficients in the expansion. We solve this system of Chapter 6. Reduced-order Modeling for Optimization 105 6.3 Reduced-order ModelingPDF Image | Design and Operation of Pressure Swing Adsorption Processes
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