Numerical comparison of some dimension reduction techniques for time series data from PDEs
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There has been increasing use of dimensionality reduction (DR) techniques to deal with high dimensional data in order to minimize the number of dimensions, referred to as the number of variables or degrees of freedom. These techniques are applied to the data prior to modeling and give insight into the data, help reduce storage space required, and improve the computational cost of the models, due to reducing the number of variables. In extreme cases, some partial differential equation (PDE) solutions may be characterized by a finite number of degrees of freedom (or variables). In this thesis, four DR techniques, namely empirical orthogonal function, diffusion maps, extended dynamic mode decomposition, and approximated Lax pairs, will be used for data sets formed from the approximating solution sequence of the two different PDEs. The idea of using the spectral theory for dimensionality reduction will be deployed to form the basis set presented as the set of the coordinates for the DR methods. The main motivation is to analyze and compare the four DR techniques. Consequently, it will be presented which DR method indicates the best approximation of the original data set for both types of data under the mean squared error criterion, in contrast to other methods.