It is well known that model reduction methods borrow techniques typically found in data compression, and current state-of-the-art techniques for data compression are based on the wavelet transform. Given these facts, it is surprising that model reduction using wavelets has not received much attention and has not been adequately addressed in the literature. This research seeks to determine if wavelets can be used for model reduction and if wavelet model reduction is a viable alternative to existing model reduction methods.;In this work we propose a novel method for model reduction using wavelets. Specifically, we introduce techniques for deriving wavelet reduced-order models for solving inverse problems. Algorithms are developed for both orthogonal and bi-orthogonal wavelets, and two methods are proposed using Galerkin and Petrov-Galerkin wavelet reduced-order models. Also, we propose a computationally efficient method for wavelet model reduction using a reduced filter bank structure that has O (n) complexity and further reduces the time required for online computations.;To evaluate the performance of the wavelet reduced-order models, we apply our methods to the nonlinear Burgers' partial differential equation. The numerical results are then compared to model reduction based on the proper orthogonal decomposition (POD) and the full-order model.;We conclude that wavelet model reduction is an alternative to the POD and can enable the researcher to begin and execute online, real-time simulations faster while being memory efficient and simple to design. Further, wavelet model reduction does not require costly offline computations or snapshots. This gives model reduction with wavelets the additional advantage of data independence, meaning that the wavelet basis does not need to be re-computed as the full-order model changes.
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