Abstract
We address adaptive multivariate polynomial approximation by means of the discrete least-squares method with random evaluations, to approximate in the L 2 probability sense a smooth function depending on a random variable distributed according to a given probability density. The polynomial least-squares approximation is computed using random noiseless pointwise evaluations of the target function. Here noiseless means that the pointwise evaluation of the function is not polluted by the presence of noise. Recent works Migliorati et al. (Found Comput Math 14:419–456, 2014), Cohen et al. (Found Comput Math 13:819–834, 2013), and Chkifa et al. (Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDEs, EPFL MATHICSE report 35/2013, submitted) have analyzed the univariate and multivariate cases, providing error estimates for (a priori) given sequences of polynomial spaces. In the present work, we apply the results developed in the aforementioned analyses to devise adaptive least-squares polynomial approximations. We build a sequence of quasi-optimal best n-term sets to approximate multivariate functions that feature strong anisotropy in moderately high dimensions. The adaptive approximation relies on a greedy selection of basis functions, which preserves the downward closedness property of the polynomial approximation space. Numerical results show that the adaptive approximation is able to catch effectively the anisotropy in the function.
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A. Chkifa, A. Cohen, G. Migliorati, F. Nobile, R. Tempone, Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDEs, EPFL MATHICSE report 35/2013 (submitted)
A. Cohen, M. Davenport, D. Leviatan, On the stability and accuracy of least squares approximations. Found. Comput. Math. 13, 819–834 (2013)
G.Migliorati, Polynomial approximation by means of the random discrete L 2 projection and application to inverse problems for PDEs with stochastic data. Ph.D. thesis, Dipartimento di Matematica “Francesco Brioschi”, Politecnico di Milano, Milano, Italy, and Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau, France, 2013
_________ , Multivariate Markov-type and Nikolskii-type inequalities for polynomials associated with downward closed multi-index sets, to appear in J. Approx. Theory
G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, Analysis of the discrete L 2 projection on polynomial spaces with random evaluations. Found. Comput. Math. 14, 419–456 (2014)
_________ Approximation of quantities of interest in stochastic PDEs by the random discrete L 2 projection on polynomial spaces. SIAM J. Sci. Comput. 35, A1440–A1460 (2013)
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Migliorati, G. (2015). Adaptive Polynomial Approximation by Means of Random Discrete Least Squares. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_54
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DOI: https://doi.org/10.1007/978-3-319-10705-9_54
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