The repository contains the source code for the numerical experiments considered in Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics by Sebastian Blauth.

To run the code, you have to install cashocs first, which includes all necessary prerequisites. The results presented in this repository have been obtained with version 1.2.1 of cashocs (which uses FEniCS 2019.1).

The repository consists of the following four test cases:

• A shape optimization problem with a Poisson equation (named poisson) which is considered in Section 4.2 of the manuscript.

• A shape identification problem in electrical impedance tomography (named eit) which is considered in Section 4.3 of the manuscript.

• A shape optimization problem in Stokes flow (named stokes) which is considered in Section 4.4 of the manuscript.

• A shape optimization problem for a pipe with Navier-Stokes flow (named pipe) which is considered in Section 4.5 of the manuscript.

For each case, there exist two files: case_benchmark.py file, containing the actual benchmark, and case_config.ini, the config file for the respective setting. Here, case is one of poisson, eit, stokes, or pipe. In particular, the benchmark for the Poisson problem can be run with the command

python3 poisson_benchmark.py

and the other benchmarks can be run analogously.

Note, that each problem is solved by the gradient descent method (abbreviated gd), limited memory BFGS methods with a memory size of 1, 3, and 5 (abbreviated lbfgs_1, lbfgs_3, and lbfgs_5), and the Fletcher-Reeves (cg_FR), Polak-Ribiere (cg_PR), Hestenes-Stiefel (cg_HS), Dai-Yuan (cg_DY), and Hager-Zhang (cg_HZ) nonlinear conjugate gradient methods.

Finally, there are two post processing functionalities available in the "visualization" folder: optimization_history.py creates plots showing the evolution of the cost functional and shape gradient norm over the optimization and saves them as .pdf files, and performance_analysis.py computes how many iterations the methods need to reach a certain tolerance, and generates the LaTeX tables used in the manuscript.

This software is citeable under the following DOI: 10.5281/zenodo.4432327.

If you use these nonlinear CG methods for your work, please cite the paper

Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics
Sebastian Blauth
SIAM Journal on Optimization, Volume 31, Issue 3
https://doi.org/10.1137/20M1367738

If you are using BibTeX, you can use the following entry:

@Article{Blauth2020Nonlinear,
    author   = {Sebastian Blauth},
    journal  = {SIAM J. Optim.},
    title    = {{N}onlinear {C}onjugate {G}radient {M}ethods for {PDE} {C}onstrained {S}hape {O}ptimization {B}ased on {S}teklov-{P}oincaré-{T}ype {M}etrics},
    year     = {2021},
    number   = {3},
    pages    = {1658--1689},
    volume   = {31},
    doi      = {10.1137/20M1367738},
    fjournal = {SIAM Journal on Optimization},
}