| Popis: |
The first part of this dissertation combines continuum limits of nonlocally interacting particles with stability analysis of nonlinear PDE to analyze the steady states of systems of pairwise-interacting particles. Models employing these assumptions cover a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. In this joint work with Theodore Kolokolnikov and Andrea Bertozzi, we study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in two-dimensional Euclidean space, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Part II continues the work on the compressive spectral method, which proposes the sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. In this joint work with Hayden Schaeffer and Stanley Osher, we show that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDE and a modified equation resulting from the sparse approximation. We also extend the method to solve elliptic equations and introduce sparse approximation of differential operators in the Fourier domain. The effectiveness of the method is demonstrated on homogenization examples, where its complexity is dependent only on the sparsity of the problem and constant in many cases. |
