Celebratio Mathematica — Daubechies

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I. Daubechies, O. Runborg, and W. Sweldens: “Normal multiresolution approximation of curves,” Constr. Approx. 20 : 3 (2004), pp. 399–463. MR 2057535 Zbl 1051.42025 article

Wim Sweldens	Related
Olof Runborg	Related

I. Daubechies, O. Runborg, and J. Zou: “A sparse spectral method for homogenization multiscale problems,” Multiscale Model. Simul. 6 : 3 (2007), pp. 711–740. MR 2368964 Zbl 1152.65099 article

We develop a new sparse spectral method, in which the fast Fourier transform (FFT) is replaced by RA\( \mathcal{\ell} \)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 \( B \ll N \). To illustrate its potential, we consider the parabolic homogenization problem with a characteristic fine scale size \( \varepsilon \). For fixed tolerance the sparse method has a computational cost of \( O(|\log\varepsilon|) \) per time step, whereas standard methods cost at least \( O(\varepsilon^{-1}) \). 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 \( \varepsilon \) is very small. We also show some ways to extend the methods to hyperbolic and elliptic problems.

Olof Runborg	Related
Jing Zou	Related