Stability Results for Scattered Data Interpolation by Trigonometric Polynomials

A fast and reliable algorithm for the optimal interpolation of scattered data on the torus $\mathbb{T}^d$ by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at $M$ arbitrary nodes is of order ${\cal O}(M\log M)$. This result is obtained by the use of localized trigonometric kernels where the localization is chosen in accordance with the spatial dimension $d$. Numerical examples show the efficiency of the new algorithm.

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

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Hrushikesh Narhar Mhaskar,et al.  On the detection of singularities of a periodic function , 2000, Adv. Comput. Math..

[4]  Ññøøøññøø Blockin.  Random Sampling of Multivariate Trigonometric Polynomials , 2004 .

[5]  Lothar Reichel,et al.  Discrete least squares approximation by trigonometric polynomials , 1991 .

[6]  Thomas Strohmer,et al.  Smooth approximation of potential fields from noisy scattered data , 1998 .

[7]  T. Strohmer,et al.  Fast multi-dimensional scattered data approximation with Neumann boundary conditions , 2003, math/0301152.

[8]  K. Gröchenig.  RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

[9]  Gabriele Steidl,et al.  A note on fast Fourier transforms for nonequispaced grids , 1998, Adv. Comput. Math..

[10]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[11]  N. Sivakumar,et al.  Stability results for scattered‐data interpolation on Euclidean spheres , 1998, Adv. Comput. Math..

[12]  Gabriele Steidl,et al.  Fast Fourier Transforms for Nonequispaced Data: A Tutorial , 2001 .

[13]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[14]  Heike Faßbender.  On numerical methods for discrete least-squares approximation by trigonometric polynomials , 1997, Math. Comput..

[15]  J. Navarro-Pedreño.  Numerical Methods for Least Squares Problems , 1996 .

[16]  C. Micchelli,et al.  On multivariate -splines , 1989 .