On Sparse Interpolation and the Design of Deterministic Interpolation Points

In this paper, we build up a framework for sparse interpolation. We first investigate the theoretical limit of the number of unisolvent points for sparse interpolation under a general setting and try to answer some basic questions of this topic. We also explore the relation between classical interpolation and sparse interpolation. We second consider the design of the interpolation points for the $s$-sparse functions in high dimensional Chebyshev bases, for which the possible applications include uncertainty quantification, numerically solving stochastic or parametric PDEs and compressed sensing. Unlike the traditional random sampling method, we present in this paper a deterministic method to produce the interpolation points, and show its performance with $\ell_1$ minimization by analyzing the mutual incoherence of the interpolation matrix. Numerical experiments show that the deterministic points have a similar performance with that of the random points.

[1]  Xiu Yang,et al.  Reweighted ℓ1ℓ1 minimization method for stochastic elliptic differential equations , 2013, J. Comput. Phys..

[2]  Yuan Xu,et al.  Lagrange Interpolation on Chebyshev Points of Two Variables , 1996 .

[3]  Yang Wang,et al.  Adaptive sub-linear Fourier algorithms , 2012, ArXiv.

[4]  Stephen J. Dilworth,et al.  Explicit constructions of RIP matrices and related problems , 2010, ArXiv.

[5]  Gerlind Plonka,et al.  A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators , 2013 .

[6]  Laura Rebollo-Neira,et al.  Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval , 2009, Signal Process..

[7]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[8]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[9]  R. DeVore,et al.  Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs , 2010 .

[10]  Dongbin Xiu,et al.  Stochastic Collocation Methods on Unstructured Grids in High Dimensions via Interpolation , 2012, SIAM J. Sci. Comput..

[11]  Mark A. Iwen,et al.  Combinatorial Sublinear-Time Fourier Algorithms , 2010, Found. Comput. Math..

[12]  Ronald A. DeVore,et al.  Deterministic constructions of compressed sensing matrices , 2007, J. Complex..

[13]  Rémi Gribonval,et al.  Sparse representations in unions of bases , 2003, IEEE Trans. Inf. Theory.

[14]  Jean-Jacques Fuchs,et al.  On sparse representations in arbitrary redundant bases , 2004, IEEE Transactions on Information Theory.

[15]  Holger Rauhut,et al.  Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit , 2008, Found. Comput. Math..

[16]  Houman Owhadi,et al.  A non-adapted sparse approximation of PDEs with stochastic inputs , 2010, J. Comput. Phys..

[17]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[18]  Lie Wang,et al.  Stable Recovery of Sparse Signals and an Oracle Inequality , 2010, IEEE Transactions on Information Theory.

[19]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[20]  A. Weil.  On Some Exponential Sums. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Zhiqiang Xu,et al.  Deterministic sampling of sparse trigonometric polynomials , 2010, J. Complex..

[22]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[23]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[24]  H. Rauhut.  Random Sampling of Sparse Trigonometric Polynomials , 2005, math/0512642.

[25]  Anru Zhang,et al.  Sharp RIP bound for sparse signal and low-rank matrix recovery , 2013 .

[26]  D. Xiu,et al.  STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁 1 -MINIMIZATION , 2012 .

[27]  R. Tempone,et al.  ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS , 2012 .

[28]  Holger Rauhut,et al.  Sparse Legendre expansions via l1-minimization , 2012, J. Approx. Theory.

[29]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[30]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

[31]  Daniel Potts,et al.  Sparse polynomial interpolation in Chebyshev bases , 2014 .

[32]  Albert Cohen,et al.  Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs , 2015 .

[33]  Albert Cohen,et al.  High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs , 2013, Foundations of Computational Mathematics.

[34]  Yang Wang,et al.  Adaptive Sub-Linear Time Fourier Algorithms , 2013, Adv. Data Sci. Adapt. Anal..

[35]  Ward Cheney,et al.  A course in approximation theory , 1999 .

[36]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .