Today I was privileged to listen to Kim Levere present her Ph.D. (Mathematics) dissertation defence. The talk was titled A collage-based approach to inverse problems for nonlinear systems of partial differential equations. I’ve included the abstract below.
I’ve known Kim for several years now – first as her TA, later as a friend – and she has always been an exceptional student. I remember marking her papers and being amazed at what she was able to do. And today, after many years of education, hard work, dedication, research, and perseverance, she finished.
I’m so proud of her. In fact, during her presentation I couldn’t help but grin from ear to ear because she presented some very complicated (and impressive) mathematics in a way that made everyone – even those who weren’t necessarily familiar with Hilbert Spaces and Partial Differential Equations, for example – comfortable. And that is no small feat.
Truly, I’d say the most exceptional thing about Kim – beyond her ability to comprehend and process high-end mathematical theory – is her ability to present that theory in a way that is attainable to those that haven’t necessarily studied it. I already know based on the feedback of students that we’ve both taught, that she is absolutely loved. She pushes the students to places they never thought they’d be able to go, but also does so by making mathematics accessible. I would love to be a student in one of her classes.
Wherever Kim ends up, she’s going to make an incredible difference; in terms of her research, in terms of her teaching, and in terms of what she’ll bring to a department. I can’t wait to see what she does.
Well done Kim. Congratulations indeed!
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  with broad application, in particular to ordinary differential equations.
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.
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