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@article{Fantuzzi:2021:10.1098/rsta.2021.0038,
author = {Fantuzzi, G and Arslan, A and Wynn, A},
doi = {10.1098/rsta.2021.0038},
journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences},
title = {The background method: Theory and computations},
url = {http://dx.doi.org/10.1098/rsta.2021.0038},
volume = {380},
year = {2021}
}

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TY  - JOUR
AB  - The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader "auxiliary function" framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined.
AU  - Fantuzzi,G
AU  - Arslan,A
AU  - Wynn,A
DO  - 10.1098/rsta.2021.0038
PY  - 2021///
SN  - 1364-503X
TI  - The background method: Theory and computations
T2  - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
UR  - http://dx.doi.org/10.1098/rsta.2021.0038
UR  - https://arxiv.org/abs/2107.11206
UR  - https://doi.org/10.1098/rsta.2021.0038
UR  - http://hdl.handle.net/10044/1/92696
VL  - 380
ER  -

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