Automatic basis function construction and global-local approximation of dynamic system response
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Finding a proper mathematical model is the first step in solving many science and engineering problems. Due to the complexity and limited prior knowledge of real-life systems, most system models are found by using the system input and output data. Numerous methods for input-output data modeling exist in the literature. Methods such as polynomials and artificial neural networks use a single global model to approximate the given input-output map based on the assumption that all the model parameters can be optimized simultaneously. However, this may not be true for problems with highly irregular localized behaviors. For problems like these, multiresolution methods such as wavelets, splines and the finite element methods often generate better results. Traditional multiresolution methods apply certain constraints on the local basis functions to avoid introducing discontinuities to the final global model. This requirement often prevents the modeler from choosing the optimal local basis functions based on the specific problems. Another challenge in multiresolution modeling is that the number of available training data for certain local models is often relatively small compared to the number of basis functions. In some extreme cases, the number of available training data can be considerably less than the number of basis functions. For problems like these, traditional parameter estimation methods such as least squares perform poorly. In this dissertation, multiresolution input-output data modeling algorithms are developed with particular interest in addressing the above-mentioned problems in the literature. In the proposed multiresolution modeling algorithms, a weighted averaging approach known as the Global-Local Orthogonal MAPping is used to merge local models to a continuous global model. Two local model construction methods – a basis function selection method based on the l 1 norm regularization and a basis function extraction methods based on the partial least squares are also developed. The proposed methods are then applied to study some challenging problems in the dynamic systems area. In the area of dynamic system modeling, a hierarchical nonlinear system identification model is proposed for modeling the respiration-induced tumor motion to address some challenging problems in the image-guided radiation therapy. In the area of uncertainty propagation in dynamic systems, a multiresolution polynomial chaos approach is developed to study the parametric and initial condition uncertainty problems in both linear and nonlinear dynamic systems.