The purpose of this thesis is twofold. Firstly, we introduce a novel method for estimating the state of a system governed by a linear evolution equation. The method utilizes the adjoint of the partial differential equation (PDE) and a basis for the Hilbert space to accurately reconstruct the initial condition. The method also provides a filter bank which can be utilized for the purpose of reconstructing initial conditions based on given data. We then extend the method to include source identification and simultaneous state/parameter estimation for a certain class of problems.;Secondly, we develop and analyze the multi-parameter regularization necessary for the accurate approximation of inverse problem solutions. The regularization is essential for both the state estimation method developed in this thesis, as well as for the general inverse problem theory. The multi-parameter regularization allows for solutions which may have a multi-scale profile. Specifically, we address problems involving sparsely distributed measurements. In addition, solutions which are, themselves, locally supported are treated, such as collections of point sources. The method developed is widely applicable and accurate, as demonstrated in this thesis.
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