Title: Iterative numerical methods for finding solitary waves, and techniques for their acceleration
Abstract: This talk will consist of two parts. First, I will review the concept behind iterative methods for obtaining solutions of linear and nonlinear (systems of) equations. One prominent application is finding stationary solitary wave solutions of wave equations. I will review methods that can find both dynamically stable and unstable stationary waves. In the second part of the talk, I will consider three techniques to accelerate these methods. The first of these techniques eliminates the numerical stiffness of the system of equations caused by the need to include high Fourier harmonics. In numerical linear algebra, this technique is known as preconditioning. The second technique eliminates the slowest-decaying mode in the error of the iterative solution. This is analogous to the Richardson interpolation procedure for solving linear systems of equations $Ax=b$. The third technique is known as the Momentum method in Machine learning and consists of modifying the iterated matrix so as to greatly decrease its condition number. 

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