Abstract. The authors of [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 6634-6639] proposed sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. We show here that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDEs 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.

Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on codimension one curves for two-species particle interaction in R 2 , 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. Some intriguing steady state patterns are shown through numerical examples.