Multilevel Method for Turbulence Energy Spectrum Estimation by Compressive Sampling

Abstract

<p>Recent developments in signal processing called Compressive Sampling (CS) show that the measurement and reconstruction of sparse signals often requires fewer samples than is estimated by the sampling theorem. CS is a combination of a linear sampling scheme and a reconstruction method and, typically, the signal is assumed to be sparse, compressible, or having a prior distribution, with the reconstruction error measured in the \ell^2 norm. This thesis investigates the application of CS to turbulence signals, particularly for estimating some statistics or nonlinear functions of the signals. The main original research result of the thesis is a proposed method, Spectrum Estimation by Sparse Optimization (SpESO), which uses a priori information about isotropic homogeneous turbulent flows and the multilevel structure of wavelet transforms to reconstruct energy spectra from compressive measurements, with errors measured on a logarithmic scale. The method is tested numerically on a variety of 1D and 2D turbulence signals, and is able to approximate energy spectra with an order of magnitude fewer samples than with traditional fixed rate sampling. The results demonstrate that SpESO performs much better than Lumped Orthogonal Matching Pursuit (LOMP), and as well or better than wavelet-based best M-term methods in many cases, even though these methods require complete sampling of the signal before compression.</p>