Model reduction Algorithm

A power bond with low relative activity can be conditioned or converted to a modulated source and signal. This leads to the partitioning and model reduction algorithm described in Section 2.3. 

The first model reduction algorithm based on the activity metric is shown in Fig. 2.2 and is called Model Order Reduction Algorithm (MORA). Given the full model, the goal of MORA is to order the importance of the energy elements in that model as given by their activity and reduce the size of the model based on a user-supplied threshold of the percent of the total activity to be retained in the reduced model. 

This section provides an example to illustrate the mechanics of the ECI-based model reduction algorithm and emphasize its advantages, namely its applicability to nonlinear systems, ability to achieve graph-level reduction, and ability to reduce the order and structure of the model, while taking into account the scenario of interest and preserving the realization of the model. 

Edwards, K. V. Manousiouthakis and T. F. Edgar. Kinetic Model Reduction Using Genetic Algorithms. Comput Chem Eng 22 239-246 (1998). 

Beliczynski et al., 1992] Beliczynski, B., Kale, I., and Cain, G. D. (1992). Approximation of FIR by HR digital filters An algorithm based on balanced model reduction. IEEE Trans. Acoustics, Speech, Signal Processing, 40(3) 532-542. 

K. Edwards, T.F. Edgar, and V.I. Manousiouthakis. Kinetic model reduction using genetic algorithms. Computers and Chemical Engineering, 22 239-246, 1998. 

The projection-based model order reduction algorithm begins with a spatial discretization of the governing PDEs to attain the dynamic system equations as Eq. 11. Specifically, here, X(t) is the state vector of unknowns (a function of time) on the discrete nodes, n is the total number of nodes A is formulated by the numerical discretization Z defines the functions of boundary conditions and source terms and B relates the input function to each state X. Equation 11 can be recast into the frequency domain in terms of transfer function T(s). T(s) then is expanded as a Taylor series at s = 0 yielding... 

Therefore, we conclude that d = 2 is a minimum dimension necessary to model this combustion process correctly. The success of our automatic dimension reduction algorithm can also nicely be seen by comparison with the quite sophisticated analytical singular perturbation treatment of Hoppensteadt ET AL. in [14]. 

An algorithm for model reduction using the ECI can be outlined as follows ... 

The use of complex models is hindered by two obstacles. First, the models contain large numbers of unknown kinetic parameters regression to determine the parameters of complex nonlinear models is both difficult and unreliable. Secondly, because of their sheer size and the presence of multiple time-scales, these models are difficult to solve. For these reasons, model simplification and order reduction are central problems for complex reaction systems. Ideally, a model order reduction algorithm would have broad applicability, permit analysis at several levels of detail, and provide an assessment of the modeling error. 

Kalachev, L.V., Field, R.J. Reduction of a model describing ozone oscillations in the troposphere example of an algorithmic approach to model reduction in atmospheric chemistry. J. Atm. Chem. 39, 65-93 (2001)... 

The hyper-focus on benchmark data is a problem that has been observed across the field of machine learning [2]. Typically, a paper on manifold learning or spectral dimensionality reduction will follow a standard evaluation model the algorithm is applied to synthetic data (typically the swiss role), and then applied to other trickier data sets such as Frey faces or MNIST digits. The use of standard benchmark data is not without its benefits it allows for a direct side-by-side comparison with other previously published methods and it also makes interpreting these results far easier. 

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