Optimal weighted least-squares methods

Cohen, Albert; Migliorati, Giovanni

The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 181-203.

We consider the problem of reconstructing an unknown bounded function $u$ defined on a domain $X\subset {ℝ}^d$ from noiseless or noisy samples of $u$ at $n$ points ${\left(x^i\right)}_{i=1,\cdots ,n}$. We measure the reconstruction error in a norm $L^2(X,d\rho )$ for some given probability measure $d\rho$. Given a linear space $V_m$ with $\dim \left(V_m\right)=m\le n$, we study in general terms the weighted least-squares approximations from the spaces $V_m$ based on independent random samples. It is well known that least-squares approximations can be inaccurate and unstable when $m$ is too close to $n$, even in the noiseless case. Recent results from [6, 7] have shown the interest of using weighted least squares for reducing the number $n$ of samples that is needed to achieve an accuracy comparable to that of best approximation in $V_m$, compared to standard least squares as studied in [4]. The contribution of the present paper is twofold. From the theoretical perspective, we establish results in expectation and in probability for weighted least squares in general approximation spaces $V_m$. These results show that for an optimal choice of sampling measure $d\mu$ and weight $w$, which depends on the space $V_m$ and on the measure $d\rho$, stability and optimal accuracy are achieved under the mild condition that $n$ scales linearly with $m$ up to an additional logarithmic factor. In contrast to [4], the present analysis covers cases where the function $u$ and its approximants from $V_m$ are unbounded, which might occur for instance in the relevant case where $X={ℝ}^d$ and $d\rho$ is the Gaussian measure. From the numerical perspective, we propose a sampling method which allows one to generate independent and identically distributed samples from the optimal measure $d\mu$. This method becomes of interest in the multivariate setting where $d\mu$ is generally not of tensor product type. We illustrate this for particular examples of approximation spaces $V_m$ of polynomial type, where the domain $X$ is allowed to be unbounded and high or even infinite dimensional, motivated by certain applications to parametric and stochastic PDEs.

Classification : 41A10, 41A25, 41A65, 62E17, 93E24
Mots clés : multivariate approximation, weighted least squares, error analysis, convergence rates, random matrices, conditional sampling, polynomial approximation.

[1] Adcock, B.; Hansen, A. C. A generalized sampling theorem for stable reconstructions in arbitrary bases, J. Fourier Anal. Appl., Volume 18 (2012), pp. 685-716 | DOI | MR | Zbl

[2] Chardon, G.; Cohen, A.; Daudet, L. Sampling and reconstruction of solutions to the Helmholtz equation, Sampl. Theory Signal Image Process., Volume 13 (2014), pp. 67-89 | MR | Zbl

[3] Chkifa, A.; Cohen, A.; Migliorati, G.; Nobile, F.; Tempone, R. Discrete least squares polynomial approximation with random evaluations - application to parametric and stochastic elliptic PDEs, M2AN, Volume 49 (2015), pp. 815-837 | DOI | MR | Zbl

[4] Cohen, A.; Davenport, M. A.; Leviatan, D. On the stability and accuracy of least squares approximations, Found. Comput. Math., Volume 13 (2013), pp. 819-834 | DOI | MR | Zbl

[5] Devroye, L. Non-Uniform Random Variate Generation, Springer, 1985

[6] Doostan, A.; Hampton, J. Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression, Comput. Methods Appl. Mech. Engrg., Volume 290 (2015), pp. 73-97 | DOI | MR | Zbl

[7] Jakeman, J. D.; Narayan, A.; Zhou, T. A Christoffel function weighted least squares algorithm for collocation approximations, Math. Comp., Volume 86 (2017), pp. 1913-1947 | MR | Zbl

[8] Máté, A.; Nevai, P.; Totik, V. Szegö’s extremum problem on the unit circle, Annals of Mathematics,, Volume 134 (1991), pp. 433-453 | DOI | Zbl

[9] Migliorati, G. Multivariate Markov-type and Nikolskii-type inequalities for polynomials associated with downward closed multi-index sets, J. Approx. Theory, Volume 189 (2015), pp. 137-159 | DOI | MR | Zbl

[10] Migliorati, G.; Nobile, F.; Tempone, R. Convergence estimates in probability and in expectation for discrete least squares with noisy evaluations at random points, J. Multivar. Analysis, Volume 142 (2015), pp. 167-182 | DOI | MR | Zbl

[11] Migliorati, G.; Nobile, F.; von Schwerin, E.; Tempone, R. Analysis of discrete $L^2$ projection on polynomial spaces with random evaluations, Found. Comput. Math., Volume 14 (2014), pp. 419-456 | MR | Zbl

[12] Nevai, P. Géza Freud, orthogonal polynomials and Christoffel Functions. A case study, J. Approx. theory, Volume 48 (1986), pp. 3-167 | DOI | Zbl

[13] Saff, E. B.; Totik, V. Logarithmic Potentials with External Fields, Springer, 1997 | Zbl

[14] Tropp, J. User friendly tail bounds for sums of random matrices, Found. Comput. Math., Volume 12 (2012), pp. 389-434 | DOI | MR | Zbl