In this thesis, the image segmentation methods based on the Mumford&amp / #8211 / Shah variational approach have been studied. By obtaining an optimum point of the Mumford-Shah functional which is a piecewise smooth approximate image and a set of edge curves, an image can be decomposed into regions. This piecewise smooth approximate image is smooth inside of regions, but it is allowed to be discontinuous region wise. Unfortunately, because of the irregularity of the Mumford Shah functional, it cannot be directly used for image segmentation. On the other hand, there are several approaches to approximate the Mumford-Shah functional. In the first approach, suggested by Ambrosio-Tortorelli, it is regularized in a special way. The regularized functional (Ambrosio-Tortorelli functional) is supposed to be gamma-convergent to the Mumford-Shah functional. In the second approach, the Mumford-Shah functional is minimized in two steps. In the first minimization step, the edge set is held constant and the resultant functional is minimized. The second minimization step is about updating the edge set by using level set methods. The second approximation to the Mumford-Shah functional is known as the Chan-Vese method. In both approaches, resultant PDE equations (Euler-Lagrange equations of associated functionals) are solved by finite difference methods. In this study, both approaches are implemented in a MATLAB environment. The overall performance of the algorithms has been investigated based on computer simulations over a series of images from simple to complicated.

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Inverse problems occur in a wide variety of applications and are an active area of research in many disciplines. We consider inverse problems for a broad class of nonlinear systems of partial differential equations (PDEs). We develop collage-based approaches for solving inverse problems for nonlinear PDEs of elliptic, parabolic and hyperbolic type. The original collage method for solving inverse problems was developed in [29] with broad application, in particular to ordinary differential equations (ODEs). Using a consequence of Banach’s fixed point theorem, the collage theorem, one can bound the approximation error above by the so-called collage distance, which is more readily minimizable. By minimizing the collage distance the approximation error can be controlled. In the case of nonlinear PDEs we consider the weak formulation of the PDE and make use of the nonlinear Lax-Milgram representation theorem and Galerkin approximation theory in order to develop a similar upper-bound on the approximation error. Supporting background theory, including weak solution theory,is presented and example problems are solved for each type of PDE to showcase the methods in practice. Numerical techniques and considerations are discussed and results are presented. To demonstrate the practical applicability of this work, we study two real-world applications. First, we investigate a model for the migration of three fish species through floodplain waters. A development of the mathematical model is presented and a collage-based method is applied to this model to recover the diffusion parameters. Theoretical and numerical particulars are discussed and results are presented. Finally, we investigate a model for the “Gao beam”, a nonlinear beam model that incorporates the possibility of buckling. The mathematical model is developed and the weak formulation is discussed. An inverse problem that seeks the flexural rigidity of the beam is solved and results are presented. Finally, we discuss avenues of future research arising from this work. / Natural Sciences and Engineering Research Council of Canada, Department of Mathematics & Statistics