Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An iteration count estimate for a mesh-dependent steepest descent method
  based on finite elements and Riesz inner product representation   Existing implementations of gradient-based optimisation methods typically
assume that the problem is posed in Euclidean space. When solving optimality
problems on function spaces, the functional derivative is then inaccurately
represented with respect to $\ell^2$ instead of the inner product induced by
the function space. This error manifests as a mesh dependence in the number of
iterations required to solve the optimisation problem. In this paper, an
analytic estimate is derived for this iteration count in the case of a simple
and generic discretised optimisation problem. The system analysed is the
steepest descent method applied to a finite element problem. The estimate is
based on Kantorovich's inequality and on an upper bound for the condition
number of Galerkin mass matrices. Computer simulations validate the iteration
number estimate. Similar numerical results are found for a more complex
optimisation problem constrained by a partial differential equation.
Representing the functional derivative with respect to the inner product
induced by the continuous control space leads to mesh independent convergence.

6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc