Abstract principal component analysis and applications to model reduction

The work in the present thesis is to develop a general complex system reduction framework for analyzing complex mixing wave signals generated from some special dynamic systems. This general framework is named Abstract Principal Component Analysis (APCA) which is an extension of the classical Principal Component Analysis to abstract operator spaces. This work is motivated by the growing need to facilitate both theoretical modeling and numerical computation with high dimensional complex systems. This model reduction framework could be utilized as a data driven approach to study mixing dynamics data sets arising from physical problems and informational problems.;The general APCA framework is developed with a basic setting of Banach algebra including operator elements. Different specializations of APCA solve model reduction problems such as segmenting sample points to linear subspaces and decomposing single wave modes from mixing signals. Mode extraction is implemented in the local single stage pattern when wave motions preserve their characteristic features such as traveling speed and scaling parameters. Basic wave motion types include scaling in the dependent variable (standing wave), scaling in the independent variable (scaling wave) and moving with fixed profile (traveling wave). Complex wave motions include compositions of three basic motion types. Processing schemes for these composition wave modes are consistent with compositions of processing schemes for single wave modes. There is an alternative optimization approach for certain procedures in mode extraction, which are equivalent as procedures in APCA. Synthetic numerical examples are presented to demonstrate the performance of mode decomposition algorithms. .;Applications of the APCA model reduction framework are illustrated primarily with those partial differential equations describing wave signal propagation processes, such as the one dimensional Burgers' equation and the two dimensional Kadomtsev-Petviashvili (KP) equation. These equations permit solutions which are approximately superpositions of single independent wave modes. The most discussed wave modes are tanh shape modes for the Burgers' equation and soliton modes for the KP equation. We illustrate decomposition results with comparisons between mixing signals and corresponding decomposed single modes. With mode characteristic parameters and reconstructed mode functions in a short time, long time solution behaviors can be predicted without solving partial differential equations in a long time period.;In order to investigate independent signal mode information from the global complex mode evolution, we have developed several techniques and some additional processing procedures. The global rigid motion reduction is introduced with some earlier work designed for the purpose of facilitating molecular dynamics simulations. This global rigid motion reduction serves as a preprocessing procedure on the training data. In the presence of noise, mode extraction results involve moderate level of error. Attentions are given to signal processing issues such as independent signal mode number overestimation, mode characteristic parameter estimation error and mode reconstruction error. Mode number selection schemes are used to choose the independent mode number. An optimization approach would provide a more robust processing scheme for solving mode parameters. We use different snapshots as multi-stage mode evolutions and use the mode function alignment as a post-processing procedure to reduce noise effects on mode profiles. Global multi-stage complex motions are approximated with local single stage motions from different subintervals in time. Multi-stage motion parameters are approximated by piecewise constant local single stage motion parameters. Multi-stage complex wave motions are approximated by piecewise single stage motions and the approximation accuracy is determined by comparing the reconstructed snapshots with original given snapshots. (Abstract shortened by UMI.)...