janoschpreuss / wave-uc-dg-repro

wave-uc-dg-repro

This repository contains software and instructions to reproduce the numerical experiments in the paper

Unique continuation for the wave equation based on a discontinuous Galerkin time discretization

• Erik Burman and Janosch Preuss
• University College London

How to run / install

We describe two options to setup the software for running the experiments.

• downloading a docker image from Zenodo or Docker Hub which contains all dependencies and tools to run the application,
• or installing the required software in a conda environment.

The first option is very convenient on personal machines whereas the second option may be more suited for compute clusters on which one does not have admin rights. In fact, the 1D results shown in the article have obtained using the Docker image whereas the 3D results were obtained with the conda installation on a compute machine.

Docker image

Pulling the docker image from Docker Hub

• Please install the docker platform for your distribution as described here.
• After installation the Docker daemon has to be started. This can either be done on boot or manually. In most Linux distributions the command for the latter is either sudo systemctl start docker or sudo service docker start.
• Pull the docker image using the command docker pull janosch2888/wave-uc-dg-repro:v1.
• Run the image with sudo docker run -it janosch2888/wave-uc-dg-repro:v1 bash.
• Proceed further as described in How to reproduce.

Downloading the docker image from Zenodo

• For this option the first two steps are the same as above.
• The image can be downloaded from zenodo.
• Assuming that wave-uc-dg-repro.tar is the filename of the downloaded image, please load the image with sudo docker load < wave-uc-dg-repro.tar.
• Run the image with sudo docker run -it janosch2888/wave-uc-dg-repro:v1 bash.
• Proceed further as described in How to reproduce.

Installing in a conda environment

• Please install conda as described in detail here.
• Then download the file wave-uc-dg-repro.yml from Zenodo or from this github repository.

Open a bash shell in the folder that contains this file and execute (in the conda base shell)

conda env create -f wave-uc-dg-repro.yml
conda activate wave-uc-dg-repro 

Then clone the repository

git clone https://github.com/janoschpreuss/wave-uc-dg-repro.git

and change into the folder wave-uc-dg-repro. Now we can proceed further as described in How to reproduce.

How to reproduce

The python scripts for runnings the numerical experiments are located in the folder scripts. To run an experiment we change to this folder and run the corresponding file. After execution has finished the produced data will be available in the folder data. For the purpose of comparison, the folder data_ref contains a copy of the data which has been used for the plots in the paper.

To generate the plots as shown in the article from the data just produced we change to the folder plots and compile the corresponding latex file. Below we decribe the above process for each of the figures in the article in detail. For viewing the generated pdf file, say figure.pdf, the figure has to be copied to the host machine. This can be done by executing the following commands in a new terminal window (not the one in which docker is run):

CONTAINER_ID=$(sudo docker ps -alq)
sudo docker cp $CONTAINER_ID:/home/app/wave-uc-dg-repro/plots/figure.pdf \
/path/on/host/machine/figure.pdf

Here, /path/on/host/machine/ has to be adapted according to the file structure on the host machine. The file figure.pdf can then be found at the designated path on the host machine and inspected with a common pdf viewer. (The command above assumes that the reproduction image is the latest docker image to be started on the machine). Alternatively, if a recent latex distribution is available on the host machine it is also possible to copy data and tex files to the latter and compile the figures there.

Figure 1

Change to directory scripts. Run

python3 precond-1d-comp.py 1
python3 precond-1d-comp.py 2

The following data files will be produced:

• precond-1d-iters-order__j__.dat where j in [1,2] denotes the finite element order. This table contains the GMRes iteration numbers for the different methods. The columns DFB contains results for the decoupled forward-backward solve, the columns MTM-full and MTM-lo for the monolitic time-marching with full and minimal dual order, respectively. The column block is for the Block-Jacobi preconditioner and the column vanilla without any preconditioner.
• The files precond-1d-L2L2ut-order__j__.dat contain the L2L2-errors for the different methods, whereas the files precond-1d-LinftyL2u-order__j__.dat contain the L-infinity(time)-L2(space) errors. As above, j in [1,2] gives the order of the FEM.
• The files precond-1d-DFB-order2-residuals-ref-lvl__i__.dat contain the GMRes residuals for the DFB method for i in [2,3,4]. Here, i=3 corresponds to N=8 and i=4 to N=16. The files precond-1d-MTM-lo-order2-residuals-ref-lvl__i__.dat contain the corresponding residuals for the monolitic time-marching method with minimal dual stabilization.

To generate Figure 1, switch to the folder plots and run

latexmk -pdf Figure1.tex

Figure 2 and Table 1

Change to directory scripts. Run

python3 precond-3d-comp.py 1 1
python3 precond-3d-comp.py 2 1
python3 precond-3d-comp.py 3 1

The following data files will be produced:

• The files precond-3d-GCC-iters-order__j__.dat where j in [1,2,3] denotes the order of the FEM contain the iteration numbers and specific information about the degrees of freedom (dof) as given in Table 1 of the paper. The columns ndof-tot-DFB,ndof-inv-DFB and DFB-iter contain the total number of dofs, number of dofs in the linear system to be inverted and the iteration numbers for the DFB method. The columns ndof-tot-MTM-lo,ndof-inv-MTM-lo and MTM-lo-iter contain the same quantities for the monolitic time-marching.
• The files precond-3d-GCC-L2L2ut-order__j__.dat contain the L2L2-errors and the files precond-3d-GCC-LinftyL2u-order__j__.dat the L-infinity-L2 errors for both methods. The column deltat denotes the size of the time step.
• The vtk data of the absolute error shown in the center of Figure 2 is contained in the file abserr-cube-GCC-order1-MTM-lo-u.xdmf.

To generate Figure 2 and Table 1, switch to the folder plots and run

latexmk -pdf Figure2.tex
latexmk -pdf Table1.tex

Figure 3

Change to directory scripts. Run

python3 precond-3d-comp.py 1 0
python3 precond-3d-comp.py 2 0
python3 precond-3d-comp.py 3 0

The generated data files are called exactly the same way (and have the same structure) as the ones in Figure 2 except for GCCbeing replaced by noGCC. To generate Figure 3, switch to the folder plots and run

latexmk -pdf Figure3.tex

Figure 4

Change to directory scripts.

python3 noGCC-restricted-1d.py

Data files of the form noGCC-restricted-1d-order__j__.dat for j in [1,2,3] representing the order of the FEM will be created. The columns LinftyL2u-all and L2L2ut-all represent the L-infinity-L2 and L2-L2 errors in the whole space-time domain, whereas the columns LinftyL2u-restrict and L2L2ut-restrict contain the errors in the restricted set B_t (shown as dashed lines in the plot). The plot of the absolute error shown in the middle of Figure 4 is available in the file abs-err.png.

To generate Figure 4 switch to the folder plots and run

latexmk -pdf Figure4.tex