(Super-)fast convergence rates for Chernoff approximations via centered differences and Richardson extrapolation
Abstract
We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth m, the rates $ O(n^{-2m}) $ on the Sobolev scales H2m+δ, H4m, and H5m, respectively, with arbitrary δ>0. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates e−cn2 for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.
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BibTeX
@misc{koudjinan2026superfast,
title = {{(Super-)fast convergence rates for Chernoff approximations via centered differences and Richardson extrapolation}},
author = {Comlan Edmond Koudjinan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16094},
eprint = {2610.16094},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16094},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - (Super-)fast convergence rates for Chernoff approximations via centered differences and Richardson extrapolation
AU - Comlan Edmond Koudjinan
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16094
UR - https://arxiv.org/abs/2610.16094
AB - We use centered differences to construct Chernoff families and derive even asymptotic expansions for their iterated products. Richardson extrapolation is then used to cancel polynomial part and thereby raise the convergence order. As illustrations, we apply the construction to uniformly parabolic, Schrödinger-type second-order operators and to a third-order dispersive operator, obtaining, at extrapolation depth m, the rates $ O(n^{-2m}) $ on the Sobolev scales H^2m+δ, H^4m, and H^5m, respectively, with arbitrary δ>0. In the analytic category, quantitative all-order estimates allow the extrapolation depth to grow with the base resolution and yield the super-fast rates e^−cn^2 for the two second-order models and $e^{-cn} $ for the third-order model. The constructions are explicit in terms of the coefficients of the underlying differential operators and do not use the target semigroup in their definition. This provides a constructive positive answer to the challenge of super-fast Chernoff approximation formulated by Remizov in Problem 30 of arXiv:2410.00416.
ER -