×

Compressive sampling for energy spectrum estimation of turbulent flows. (English) Zbl 1328.76032

Summary: Recent results from compressive sampling (CS) have demonstrated that accurate reconstruction of sparse signals often requires far fewer samples than suggested by the classical Nyquist-Shannon sampling theorem. Typically, signal reconstruction errors are measured in the \(\ell^2\) norm and the signal is assumed to be sparse or compressible. Our spectrum estimation by sparse optimization (SpESO) method uses a priori information about isotropic homogeneous turbulent flows with power law energy spectra and applies the methods of CS to one- and two-dimensional turbulence signals to estimate their energy spectra with small logarithmic errors. SpESO is distinct from existing energy spectrum estimation methods which are based on sparse support of the signal in Fourier space. SpESO approximates energy spectra with an order of magnitude fewer samples than needed with Shannon sampling. Our results demonstrate that SpESO performs much better than lumped orthogonal matching pursuit, and as well as or better than wavelet-based best \(M\)-term or \(M/2\)-term methods, even though these methods require complete sampling of the signal before compression.

MSC:

76F05 Isotropic turbulence; homogeneous turbulence
65F22 Ill-posedness and regularization problems in numerical linear algebra
65T60 Numerical methods for wavelets

Software:

CoSaMP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Alam, N. K.-R. Kevlahan, and O. V. Vasilyev, {\it Scaling of space-time modes with Reynolds number in two-dimensional turbulence}, J. Fluid Mech., 570 (2007), pp. 217-226. · Zbl 1120.76032
[2] J. Almeida, J. Prior, and M. B. Plenio, {\it Computation of 2-D Spectra Assisted by Compressed Sampling}, preprint, http://arxiv.org/abs/1207.2404arXiv:1207.2404, 2012.
[3] F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia, {\it High-order velocity structure functions in turbulent shear flows}, J. Fluid Mech., 140 (1984), pp. 63-89.
[4] D. Dony Ariananda, G. Leus, and Z. Tian, {\it Multi-coset sampling for power spectrum blind sensing}, in Proceedings of the 17th International Conference on Digital Signal Processing (DSP), 2011.
[5] R. Benzi, L. Biferale, A. Crisanti, G. Paladin, M. Vergassola, and A. Vulpiani, {\it A random process for the construction of multiaffine fields}, Phys. D, 65 (1993), pp. 352-358. · Zbl 0772.60093
[6] T. Blumensath, {\it Compressed Sensing with Nonlinear Observations and Related Nonlinear Optimisation Problems}, preprint, http://arxiv.org/abs/1205.1650arXiv:1205.1650, 2012.
[7] E. Candès, {\it The restricted isometry property and its implications for compressed sensing}, C. R. Math. Acad. Sci. Paris, 346 (2008), pp. 589-592. · Zbl 1153.94002
[8] E. Candès and J. Romberg, {\it Sparsity and incoherence in compressive sampling}, Inverse Problems, 23 (2007), pp. 969-985. · Zbl 1120.94005
[9] E. Candès, J. Romberg, and T. Tao, {\it Robust uncertainty principles: Exact signal reconstruction from highly incomplete Fourier information}, IEEE Trans. Inform. Theory, 52 (2006), pp. 489-509. · Zbl 1231.94017
[10] E. J. Candès and J. Romberg, {\it Practical signal recovery from random projections}, in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5914, SPIE, Bellingham, WA, 2005.
[11] E. J. Candès and J. Romberg, {\it Quantitative robust uncertainty principles and optimally sparse decompositions}, Found. Comput. Math., 6 (2006), pp. 227-254. · Zbl 1102.94020
[12] E. J. Candès, J. K. Romberg, and T. Tao, {\it Stable signal recovery from incomplete and inaccurate measurements}, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223. · Zbl 1098.94009
[13] E. J. Candès and T. Tao, {\it Decoding by linear programming}, IEEE Trans. Inform. Theory, 51 (2005), pp. 4203-4215. · Zbl 1264.94121
[14] I. Daubechies, {\it Orthonormal bases of compactly supported wavelets}, Comm. Pure Appl. Math., 41 (1988), pp. 909-996. · Zbl 0644.42026
[15] M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, {\it Signal processing with compressive measurements}, IEEE J. Selected Topics Signal Process., 4 (2010), pp. 445-460.
[16] G. De Stefano, D. E. Goldstein, O. V. Vasilyev, and N. K. R. Kevlahan, {\it Towards Lagrangian dynamic SGS model for SCALES of isotropic turbulence}, in Direct and Large-Eddy Simulation VI, E. Lamballais, R. Friedrich, B. J. Geurts, and O. Metais, O, eds., Springer, 2006, pp. 175-182.
[17] D. L. Donoho, M. Vetterli, R. A. Devore, and I. Daubechies, {\it Data compression and harmonic analysis}, IEEE Trans. Inform. Theory, 44 (1998), pp. 2435-2476. · Zbl 1125.94311
[18] D. L. Donoho, {\it Compressed sensing}, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289-1306. · Zbl 1288.94016
[19] M. Farge, N. Kevlahan, V. Perrier, and E. Goirand, {\it Wavelets and turbulence}, Proc. IEEE, 84 (1996), pp. 639-669.
[20] M. Farge, K. Schneider, and N. K.-R. Kevlahan, {\it Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis}, Phys. Fluids, 11 (1999), pp. 2187-2201. · Zbl 1147.76386
[21] S. Foucart and H. Rauhut, {\it A Mathematical Introduction to Compressive Sensing}, Birkhäuser, 2013. · Zbl 1315.94002
[22] U. Frisch, {\it Turbulence}, Cambridge University Press, Cambridge, UK, 1996.
[23] J. Fröhlich and K. Schneider, {\it Numerical simulation of decaying turbulence in an adaptive wavelet basis}, Appl. Comput. Harmon. Anal., 3 (1996), pp. 393-397. · Zbl 0865.76061
[24] C. La and M. N. Do, {\it Signal reconstruction using sparse tree representations}, in Wavelets XI, M. Papadakis, A. F. Laine, and M. A. Unser, eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5914, SPIE, Bellingham, WA, 2005, pp. 273-283.
[25] G. Leus and D. D. Ariananda, {\it Power spectrum blind sampling}, IEEE Signal Process. Lett., 18 (2011), pp. 443-446.
[26] M. A. Lexa, M. E. Davies, and J. S. Thompson, {\it Compressive and Noncompressive Power Spectral Density Estimation from Periodic Nonuniform Samples}, preprint, http://arxiv.org/abs/1110.2722arXiv:1110.2722, 2011.
[27] Y. Li, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eyink, {\it A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence}, J. Turbulence, 9 (2008). · Zbl 1273.76210
[28] S. G. Mallat, {\it Multiresolution approximations and wavelet orthonormal bases of \(\mathbf{L}^2(\mathbf{R})\)}, Trans. Amer. Math. Soc., 315 (1989). · Zbl 0686.42018
[29] S. G. Mallat, {\it A Wavelet Tour of Signal Processing: The Sparse Way}, 3rd ed., Elsevier Computer Science Library, Academic Press, Amsterdam, 2009. · Zbl 1170.94003
[30] D. Needell and J. A. Tropp, {\it CoSaMP: Iterative signal recovery from incomplete and inaccurate samples}, Appl. Comput. Harmon. Anal., 26 (2008), pp. 301-321. · Zbl 1163.94003
[31] H. Nobach, E. Müller, and C. Tropea, {\it Refined reconstruction techniques for LDA data analysis}, in Proceedings of the 8th International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.
[32] E. Perlman, R. Burns, Y. Li, and C. Meneveau, {\it Data exploration of turbulence simulations using a database cluster}, in Proceedings of the 2007 ACM/IEEE Conference on Supercomputing, SC ’07, ACM, New York, 2007, pp. 23:1-23:11.
[33] V. Perrier, T. Philipovitch, and C. Basdevant, {\it Wavelet spectra compared to Fourier spectra}, J. Math. Phys., 36 (1995), pp. 1506-1519. · Zbl 0833.42020
[34] J. Romberg, {\it Compressive sensing by random convolution}, SIAM J. Imaging Sci., 2 (2009), pp. 1098-1128. · Zbl 1176.94017
[35] K. Schneider and O. V. Vasilyev, {\it Wavelet methods in computational fluid dynamics}, Ann. Rev. Fluid Mech., 42 (2010), pp. 473-503. · Zbl 1345.76085
[36] J. A. Tropp and A. C. Gilbert, {\it Signal recovery from random measurements via orthogonal matching pursuit}, IEEE Trans. Inform. Theory, 53 (2007), pp. 4655-4666. · Zbl 1288.94022
[37] J. A. Tropp, M. B. Wakin, M. F. Duarte, D. Baron, and R. G. Baraniuk, {\it Random filters for compressive sampling and reconstruction}, in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2006, pp. III-872-III-875.
[38] R. Venkataramani and Y. Bresler, {\it Further results on spectrum blind sampling of \(2\) D signals}, in Proceedings of the 1998 International Conference on Image Processing, Vol. 2, IEEE, Piscataway, NJ, 1998, pp. 752-756.
[39] J. Wang, {\it Generalized orthogonal matching pursuit}, IEEE Trans. Signal Process., 60 (2012), pp. 6202-6216. · Zbl 1393.94479
[40] J. Wang and B. Shim, {\it On the recovery limit of sparse signals using orthogonal matching pursuit}, IEEE Trans. Signal Process., 60 (2012), pp. 4973-4976. · Zbl 1393.94719
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.