Sequential Approximation of Functions in Sobolev Spaces Using Random Samples

We present an iterative algorithm for approximating an unknown function sequentially using random samples of the function values and gradients. This is an extension of the recently developed sequential approximation (SA) method, which approximates a target function using samples of function values only. The current paper extends the development of the SA methods to the Sobolev space and allows the use of gradient information naturally. The algorithm is easy to implement, as it requires only vector operations and does not involve any matrices. We present tight error bound of the algorithm, and derive an optimal sampling probability measure that results in fastest error convergence. Numerical examples are provided to verify the theoretical error analysis and the effectiveness of the proposed SA algorithm.

[1]  Xuemei Chen,et al.  Almost Sure Convergence of the Kaczmarz Algorithm with Random Measurements , 2012 .

[2]  Kailiang Wu,et al.  Numerical Aspects for Approximating Governing Equations Using Data , 2018, J. Comput. Phys..

[3]  P. Davis.  Interpolation and approximation , 1965 .

[4]  Kailiang Wu,et al.  A Randomized Tensor Quadrature Method for High Dimensional Polynomial Approximation , 2017, SIAM J. Sci. Comput..

[5]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[6]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[7]  Kailiang Wu,et al.  Sequential function approximation on arbitrarily distributed point sets , 2018, J. Comput. Phys..

[8]  M. Powell,et al.  Approximation theory and methods , 1984 .

[9]  Stephen J. Wright,et al.  An accelerated randomized Kaczmarz algorithm , 2013, Math. Comput..

[10]  Yonina C. Eldar,et al.  Acceleration of randomized Kaczmarz method via the Johnson–Lindenstrauss Lemma , 2010, Numerical Algorithms.

[11]  Kailiang Wu,et al.  Sequential function approximation with noisy data , 2018, J. Comput. Phys..

[12]  Nikolaos M. Freris,et al.  Randomized Extended Kaczmarz for Solving Least Squares , 2012, SIAM J. Matrix Anal. Appl..

[13]  Alan Genz,et al.  Testing multidimensional integration routines , 1984 .

[14]  Claude Brezinski,et al.  Convergence acceleration of Kaczmarz’s method , 2013, 1302.0196.

[15]  Dongbin Xiu,et al.  A Randomized Algorithm for Multivariate Function Approximation , 2017, SIAM J. Sci. Comput..

[16]  R. Vershynin,et al.  A Randomized Kaczmarz Algorithm with Exponential Convergence , 2007, math/0702226.

[17]  D. Needell.  Randomized Kaczmarz solver for noisy linear systems , 2009, 0902.0958.

[18]  E. Cheney.  Introduction to approximation theory , 1966 .

[19]  T. Wallace,et al.  Acceleration of kaczmarz using orthogonal subspace projections , 2013, 2013 Biomedical Sciences and Engineering Conference (BSEC).