New Weak Error bounds and expansions for Optimal Quantization

Vincent Lemaire (1) , Thibaut Montes (1, 2) , Gilles Pagès (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 ICA - The Independent Calculation Agent

We propose new weak error bounds and expansion in dimension one for optimal quantization-based cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewise-defined locally Lipschitz or α-Hölder derivatives. This new results rest on the local behaviors of optimal quantizers, the L r-L s distribution mismatch problem and Zador's Theorem. This new expansion supports the definition of a Richardson-Romberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.

Mots clés: Optimal quantization; Numerical integration; Weak error; Romberg extrapolation; Variance reduction; Monte Carlo simulation; Product quantizer

Domaines: Analyse numérique [math.NA]; Probabilités [math.PR]; Finance quantitative [q-fin.CP]

Vincent Lemaire, Thibaut Montes, Gilles Pagès. New Weak Error bounds and expansions for Optimal Quantization. Journal of Computational and Applied Mathematics, In press, ⟨10.1016/j.cam.2019.112670⟩. ⟨hal-02361644v1⟩