Journal of Computational Physics
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RAℓSFA (Randomized Algorithm for Sparse Fourier Approximation) is a sublinear randomized algorithm. Given a discrete signal S of length N, the algorithm finds a near-optimal B-term Sparse Representation R in time and space poly(B, log(N), log(δ),&epsis; −1), such that ||S − R|| 2 ≤ (1 + &epsis;)||S − R Ropt|| 2. Its sublinear time cost outperforms the superlinear Ω( N log N) time requirement of the famous Fast Fourier Transform (FFT). However, a straightforward implementation of the RAℓSFA in [31] turns out to be very slow. Thus, it is important to improve the algorithm for practical purpose. This dissertation focuses on the improvement, implementation, and application of RAℓSFA. In order to make the theoretical sketch practical, we introduce several new ideas to speed up the algorithm. Both rigorous and heuristic arguments are presented. Theoretically, our RAℓSFA achieves the near-optimal approximation with high probability in time poly(B) log( N) log(1/δ)/&epsis;2. We also extend the algorithm to higher dimensional cases. Numerical examples show that our implementation is faster (by a factor of four thousand times) than original RAℓSFA. Furthermore, it already beats FFT for reasonably large N. The crossover point lies at N ≃ 70, 000 in one dimension, and at N ≃ 900 for data on a N x N grid in two dimensions for small B signals contaminated by noise. In some real applications, it is desirable to reconstruct the sparse representation from incomplete data. There are two main difficulties caused by irregularly spaced data. For one thing, a uniformly spaced sample may not always be available as a subset of the data; for another, the success probability of the RAℓSFA relies heavily on the percentage of available data. This thesis proposed a new adapted sublinear RAℓSFA algorithm. It introduced a greedy technique and Lagrange interpolation to overcome the above problems. One of the exciting applications of RAℓSFA is multiscale models. Typically, the solutions of these models have rapidly oscillating coefficients, with period proportional to a small parameter α. Chapter 4 presents the first study of a new sublinear spectral method with time cost O(|log α|poly(d)) per time step. It shows great potential in solving multiscale models fast, especially for small resolutions and high dimensions. This is a joint work with Dr. Ingrid Daubechies.