**Papers Published**

- Daubechies, I; Runborg, O; Zou, J,
*A sparse spectral method for homogenization multiscale problems*, Multiscale Modeling & Simulation, vol. 6 no. 3 (August, 2007), pp. 711-740, Society for Industrial & Applied Mathematics (SIAM), ISSN 1540-3459

(last updated on 2019/07/19)**Abstract:**

We develop a new sparse spectral method, in which the fast Fourier transform (FFT) is replaced by RAℓSFA (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( logε ) per time step, whereas standard methods cost at least O(ε-1). 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. © 2007 Society for Industrial and Applied Mathematics.