280 R. Deshmukh et al. Thus, POD based Galerkin ROMs do not account for sufficient energy dissipation, resulting in over-prediction of kinetic energy [1, 11, 13]. Moreover, the energy accumulation over a period of time may also cause the ROM to become unstable [11, 13, 15]. Related to this, it is now well-established that POD modes are ineffective in capturing the local dynamics (or transience) of full-order systems [11, 12]. This is because the POD modes are always active, or have nonzero coefficients for all time windows. For these reasons it is clear that the optimality of POD modes, in terms of energy capture, is non-ideal for model reduction of highly unsteady, nonlinear flow fields. Techniques such as Balanced Truncation [16], Balanced POD (BPOD) [2], and Eigensystem Realisation Algorithm (ERA) [17] have addressed some of the limitations identified earlier. However, balanced truncation is intractable for large data (for more than 10,000 degrees of freedom) [3], BPOD is only applicable to response data of linear systems as it requires adjoint system information [2, 17], and modes generated by ERA cannot be used for projection of non-linear dynamics [17]. In a recent study, a technique is developed to generate a stable Galerkin projection based ROM [11]. However, building the ROM is an iterative process, and requires multiple time-integrations until an energy-balance is achieved. These issues highlight the need to explore alternative basis identification techniques that not only generalize well to changing flow conditions, but also accurately capture essential multi-scale features. Olshausen and Field [18] argue that most naturally occurring phenomena are conveniently represented using non-Gaussian distributions, whereas the PCA approach is suitable when the structure of the data can be represented using Gaussian distributions. In Gaussian distributions, the linear correlation between statistical structures is the most important relation. Observations from a naturally occurring phenomena, such as natural images, contain higher order statistics. To this end, a technique based on sparse coding was proposed to extract the higher order features from natural image data [18]. This approach, which is also referred to as sparse dictionary learning [19], generates a finite dictionary of modes in which only a subset is active—i.e. has nonzero coefficients—for a given time window. Furthermore, sparse coding describes a nonlinear system in a locally linear manner by tailoring the modes to local behavior of the system [20]. Thus, compared to the POD approach—where the principal components of the observed data are identified, ordered and then truncated to a compact set—sparse coding is formulated as a procedure to identify a compact representation that best spans the entire observed data. The sparse coding approach has been successfully applied in a number of topics, such as in image processing [21], audio analysis [22], neuroscience [18, 23, 24], and electrical power disaggregation [25]. In a previous study conducted by the authors, sparse coding approach was examined in the context of reduced order modeling of dynamical systems [26]. The ROMs generated using sparse bases were compared against the standard POD ROMs in terms of stability and accuracy. The sparse ROMs were found to perform better than the POD ROMs when the same number of modes were used. This paper is a continuation of that study, where the performance of the sparse and POD bases are assessed in the context of different energy components of the dynamical system—in this case turbulent kinetic energy of a lid driven cavity. The remainder of this extended abstract is organized as follows. The POD and the sparse coding approaches are presented in Sect. 26.2. Results describing the application of POD and sparse modes to model the unsteady flow fields are presented in Sect. 26.3. Concluding remarks are presented in Sect. 26.4. 26.2 Method of Solution High resolution data are computed by solving the 2-D incompressible Navier Stokes (NS) equations using a Direct Numerical Simulation (DNS) CFD solver. The POD approach is based on the method of snapshots developed by Sirovich [10]. Sparse modes are evaluated using the algorithm developed by Friedman et al. [27]. These approaches are detailed next. 26.2.1 Full Order Models The DNS solution to the non-dimensionalized NS equations given by (26.1) is generated using the PICar3D code [28]. r uD0; @u @t C.u:r/uD rpC 1 Rer 2u (26.1) where r is the gradient operator, t is non-dimensional time, Re is Reynolds number, pis pressure, anduis velocity.
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