A Sparse Spectral Method for the Homogenization of Multiscale Problems

We develop a new sparse spectral method, in which the Fast Fourier Transform (FFT) is replaced by RA`SFA (Randomized Algorithm of Sparse Fourier Analysis); this is a sublinear randomized algorithm that takes time O(B log N) to recover a B-term Fourier representation for a signal of length N , where we assume B N . To illustrate its potential, we consider the parabolic homogenization problem with a characteristic fine scale size ε. For fixed tolerance the sparse method has a computational cost of O(| log ε|) per time step, whereas standard methods cost at least O(ε−d). We present a theoretical analysis as well as numerical results; they show the advantage of the new method in speed over the traditional spectral methods when ε is very small. We also show some ways to extend the methods to hyperbolic and elliptic problems.