41 Reduced Order Models for Systems with Disparate Spatial and Temporal Scales 449 In SOD, we are looking for a basis vector k 2 Rn such that a projection of the data matrix onto this vector has minimal roughness and maximal variance. The solution to the SOD problem, is achieved by solving the generalized eigenvalue problem of the matrix pair †yy and †PyPy in Eq. (41.3): †yy k D k†PyPy k (41.5) where k are smooth orthogonal values (SOVs), k 2 Rn are smooth projection modes (SPMs), smooth orthogonal modes (SOMs) are ˆ D ‰ T, and smooth orthogonal coordinates (SOCs) are given by Q D Y‰, where ‰DŒ 1; 2; : : : ; n 2 R2n 2n. The degree of smoothness of the coordinates is described by the magnitude of the corresponding SOV. Thus, the greater in magnitude the SOV, the smoother in time is the corresponding coordinate. It should be noted that if we were to replace †PyPy with the identity matrix, the formulation will yield the proper orthogonal decomposition. 41.2.2 Separated Multivariate Analysis for Reduced Order Models The full data matrices in many dynamical systems are ill-conditioned. Therefore, the extracted modes using the aforementioned multivariate analysis methods may be noisy. These modes may have some unnecessary sign changes and they may not account for the true modal structure of the system. Here we propose to do multivariate analysis on position and velocity data matrices separately, since they usually have a lower condition number. The separated multivariate analysis as its name suggests will be done on Xand Vseparately. The solution of separated POD analysis on position data matrix is given by: †xx x k D x k x k (41.6) Likewise, the solution of separated POD analysis on velocity data matrix is given by: †vv v k D v k v k (41.7) In the above equations, †xx and †vv are auto-covariance matrices of position and velocity data. Also, the superscript x and v denote that the corresponding scalar or vector is related to the position and velocity data matrices, respectively. The obtained x k’s and v k’s are arranged in a position modal matrix ˆ x D x 1; x 2; : : : ; x n 2 Rn n and a velocity modal matrix ˆv D v 1; v 2; : : : ; v n 2 Rn n. After choosingl dominant position modes given in the matrix formˆ x l aswell as pdominant velocity modes given in the matrix formˆv p, they can be put together to, at last, yield the modal matrix: ˆk D" ˆ x l 0n p 0n l ˆv p # 2n k (41.8) where k Dl Cpis the dimension of the modal matrix. For separated SOD analysis, we need to solve the following generalized eigenvalue problems on position and velocity data and their time derivatives. For position data, one has: †xx x k D x k†vv x k (41.9) similarly for velocity data, one obtain: †vv v k D v k†aa v k (41.10) inwhich†vv and†aa are autocovariance matrices of velocity and acceleration data, respectively. Similar to POD, the modal matrix‰will be composed of l-dimensional ‰ x l and p-dimensional ‰v p. Thus, one gets: ‰k D" ‰ x l 0n p 0n l ‰v p # 2n k : (41.11)
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