Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The background method: Theory and computations   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-B\'enard 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.

9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an