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ode15s to obtain snapshots. With 100 time snapshots in the snapshot matrix, POD basis set was computed using SVD. Finally, ROM was constructed as in Equation (6.21) and analyzed for different values of subspace dimension M. Figure 6.1 compares the original solution profile of the Burgers equation with that obtained from ROM for different values of M. It also reports the normalized error in projection εPOD(M) for corresponding M. We observe that norm for just 7 POD basis functions, error in projection is as low as 0.7%, and we also obtain a substantial model reduction (almost (1/7)th of the model obtained after spatial discretization with 50 nodes). Moreover, the solution of ROM with 7-rank approximation is almost identical to the original solution. Also, the solutions obtained with 3-rank and 5-rank approximation are fairly accurate with a little high error in projection. Hence, this illustrates the power of reduced-order modeling to predict dynamic behavior of the system with significant accuracy and with the help of a model quite small in order. 6.4 ROM-based Optimization As mentioned before, reduced-order modeling offers computational advantages for PDE-constrained optimization problems. Since the number of DAEs in a ROM is much smaller compared to the number of DAEs obtained after spatial discretization of PDEs, replacing the latter with the former yields a much smaller and computationally-efficient optimization problem which can be solved cheaply. Mathematically, if the original optimization problem is given as min s.t. Φ(y(tf ), z(tf ), p) dyi = f(y(t),z(t),p) i = 1,...,Nx (6.22a) (6.22b) (6.22c) (6.22d) (6.22e) dt s(y(t), z(t), p) = 0 g(y(t), z(t), p) ≤ 0 bL ≤ p ≤ bU with differential variables y(t), algebraic variables z(t), decision variables p with lower and upper bounds bL and bU , respectively, and Nx ODEs obtained after spatial discretization, the 6.4 ROM-based Optimization Chapter 6. Reduced-order Modeling for Optimization 110PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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