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M d a j M j=1 M j=1 d φ j M d 2 φ j aj(t)dx, aj(t)dx2 j=1 φj(x)dt −f j=1 aj(t)φj(x), φidx=0, ∂y ∂y ∂2y ∂t −f y,∂x,∂x2 φi dx=0, i=1,...,M (6.14) i=1,...,M (6.15) Substituting Equation (6.12) in the above expression, we obtain Since the POD basis functions are orthonormal, we finally obtain our reduced-order model d a i M M j=1 d φ j M d 2 φ j dt = f j=1 aj(t)φj(x), aj(t) dx , aj(t) dx2 φi dx, i = 1,...,M (6.16) j=1 In case of a finite dimensional problem, integral inner product is replaced by an L2 inner product. It should be noted that in the final reduced-order model we obtain only M ordinary differential equations (ODEs) compared to Nx ODEs that we usually obtain after applying spa- tial discretization techniques such as finite difference, finite element, or finite volume. Since M is significantly less compared to Nx, we obtain a significantly low-order model compared to the one obtained after conventional spatial discretization. More importantly, in a PDE-constrained optimization problem, replacing the set of ODEs obtained after spatial discretization with the smaller set of ODEs of the reduced-order model yields a much smaller and computationally efficient optimization problem which can be solved cheaply. 6.3 Reduced-order Modeling PDEs using method of weighted residuals in which the inner product of the residual of PDEs with an orthonormal set of basis functions {ωi}Pi=1 is set to zero, i.e., ∂y ∂y ∂2y ∂t −f y,∂x,∂x2 ωi dx=0, i=1,...,P (6.13) In particular, for Galerkin projection, we choose such a basis set {ωi}Pi=1 same as the set of basis functions in terms of which the state variable is defined, i.e., POD basis functions in this case, with P = M. Thus, Equation (6.13) becomes Chapter 6. Reduced-order Modeling for Optimization 106PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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