janoschpreuss / linearized-NSE-data-assimilation

linearized-NSE-data-assimilation

README

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

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

• authors: Erik Burman, Deepika Garg, Janosch Preuss
• University College London

How to run / install

• Please download the docker image from zenodo.
• Assuming that linearized-nse-repro.tar is the filename of the downloaded image, please load the image with docker load < linearized-nse-repro.tar. Usually this command has to be executed with root privileges, which means e.g. on linux systems that sudo has to be added in front of this command.
• Run the image with docker run --init -ti -p 8888:8888 linearized-nse-repro:v1. Open the url shown in the terminal in your browser.
• Proceed further as described in How to reproduce

How to reproduce

Parameters to be changed by the user are located in the fourth block titled as "setting parameters":

• order describes the polynomial degree of the finite elements
• domain_case which describes the geometrical setup (convex, non-convex ..)
• theta describes the perturbation order (as defined in the paper)
• give_pressure is a flag which describes whether global pressure data is added tot the Lagrangian (see Section 5.2)

Figure 2 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = None
• domain_case = subdomain_cases[2] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 2a can be found in the folder plots and is called stokes-convex-k123.tex.

Figure 2 b) Run the same file as Figure 2 a) the only changes are:

• domain_case = subdomain_cases[3]

• The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 2b can be found in the folder plots and is called stokes-nonconvex-k123.tex.

Figure 3 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 0
• domain_case = subdomain_cases[2] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 3a can be found in the folder plots and is called stokes-convex-k123-data-perturbation0.tex.

Figure 3 b)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 1
• domain_case = subdomain_cases[2] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 3b can be found in the folder plots and is called stokes-convex-k123-data-perturbation1.tex.

Figure 3 c)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 2
• domain_case = subdomain_cases[2] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 3c can be found in the folder plots and is called stokes-convex-k123-data-perturbation2.tex.

Figure 4 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 0
• domain_case = subdomain_cases[3] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 4a can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation0.tex.

Figure 4 b)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 1
• domain_case = subdomain_cases[3] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 4b can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation1.tex.

Figure 4 c)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 2
• domain_case = subdomain_cases[3] For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 4a can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation2.tex.

Figure 5 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = None
• domain_case = subdomain_cases[2]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 5a can be found in the folder plots and is called stokes-convex-k123.tex.

Figure 5 b)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = None
• domain_case = subdomain_cases[3]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 5b can be found in the folder plots and is called stokes-nonconvex-k123.tex.

Figure 6 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 0
• domain_case = subdomain_cases[2]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 6a can be found in the folder plots and is called stokes-convex-k123-data-perturbation0.tex.

Figure 6 b)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 1
• domain_case = subdomain_cases[2]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 6b can be found in the folder plots and is called stokes-convex-k123-data-perturbation1.tex.

Figure 6 c)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 0
• domain_case = subdomain_cases[2]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-convex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 6c can be found in the folder plots and is called stokes-convex-k123-data-perturbation2.tex.

Figure 7 a)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 0
• domain_case = subdomain_cases[3]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 7a can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation0.tex.

Figure 7 b)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 1
• domain_case = subdomain_cases[3]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 7b can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation1.tex.

Figure 7 c)

Run the file Ill_posed_Stokes.ipynb using the following parameters:

• theta = 2
• domain_case = subdomain_cases[3]
• dual_finite_element_u = ufl.VectorElement("CG", mesh.ufl_cell(), 1)
• primal_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), order-1)
• dual_finite_element_p = ufl.FiniteElement("CG", mesh.ufl_cell(), 1) For each polynomial degree order in [2,3] a separate run is required.

The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named stokes-ill-posed-nonconvex-Helmholtz-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 7c can be found in the folder plots and is called stokes-nonconvex-k123-data-perturbation2.tex.

Figure 9 a)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = None
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 1 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 9a can be found in the folder plots and is called Navier-stokes-downstream-k123.tex.

Figure 9 b) Run the same file as Figure 9 a) only change is:

• give_pressure = True

Figure 9 c) Run the same file as Figure 9 a) only change is:

• diffusion_coff= 0.01

Figure 9 d) Run the same file as Figure 9 c) only change is:

• give_pressure = True

Figure 9 e) Run the same file as Figure 9 a) only change is:

• diffusion_coff= 0.0001

Figure 9 f) Run the same file as Figure 9 e) only change is:

• give_pressure = True

Figure 9 g) Run the same file as Figure 9 a) only change is:

• diffusion_coff= 0

Figure 9 h) Run the same file as Figure 9 g) only change is:

• give_pressure = True

Figure 10 a)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 0
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 1 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 10a can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation0.tex.

Figure 10 b) Run the same file as Figure 10 a) only change is:

• give_pressure = True

Figure 10 c)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 0
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 0.01 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 10c can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation0.tex.

Figure 10 d) Run the same file as Figure 10 c) only change is:

• give_pressure = True

Figure 10 e)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 0
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 0.0001 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 10e can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation0.tex.

Figure 10 f) Run the same file as Figure 10 e) only change is:

• give_pressure = True

Figure 10 g)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 0
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 0 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 10g can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation0.tex.

Figure 10 h) Run the same file as Figure 10 g) only change is:

• give_pressure = True

Figure 11 a)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 1
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 1 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 11a can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation1.tex.

Figure 11 b) Run the same file as Figure 11 a) only change is:

• give_pressure = True

Figure 11 c)

Run the file Ill_posed_new_Navier_Stokes using the following parameters:

• theta = 2
• give_pressure = False
• domain_case = subdomain_cases[5]
• diffusion_coff= 1 For each polynomial degree order in [1,2,3] a separate run is required. The convergence can be seen in the plot generated as the end of the jupyter notebook. The data for generating the plot included in the paper will be saved to the folder data in .dat files named Navier-Stokes-ill-posed-flow_downstream-order__j__-theta__i__.dat where j and i represent the corresponding values of theta and order respectively. The .tex file for generating the Fig. 11c can be found in the folder plots and is called Navier-stokes-downstream-k123-data-perturbation2.tex.

Figure 11 d) Run the same file as Figure 11 c) only change is:

• give_pressure = True