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Getting started

Trying it out

While we consider pyMOR mainly as a library for building MOR applications, we ship a few example scripts. These can be found in the src/pymordemos directory of the source repository (some are available as Jupyter notebooks in the notebooks directory). Try launching one of them using the pymor-demo script:

pymor-demo thermalblock --plot-err --plot-solutions 3 2 3 32

The demo scripts can also be launched directly from the source tree:

./thermalblock.py --plot-err --plot-solutions 3 2 3 32

This will reduce the so called thermal block problem using the reduced basis method with a greedy basis generation algorithm. The thermal block problem consists in solving the stationary heat equation

- ∇ ⋅ [ d(x, μ) ∇ u(x, μ) ] = 1     for x in Ω
                  u(x, μ)   = 0     for x in ∂Ω

on the domain Ω = [0,1]^2 for the unknown u. The domain is partitioned into XBLOCKS x YBLOCKS blocks (XBLOCKS and YBLOCKS are the first two arguments to thermalblock.py). The thermal conductivity d(x, μ) is constant on each block (i,j) with value μ_ij:

(0,1)------------------(1,1)
|        |        |        |
|  μ_11  |  μ_12  |  μ_13  |
|        |        |        |
|---------------------------
|        |        |        |
|  μ_21  |  μ_22  |  μ_23  |
|        |        |        |
(0,0)------------------(1,0)

The real numbers μ_ij form the XBLOCKS x YBLOCKS - dimensional parameter on which the solution depends.

Running thermalblock.py will first produce plots of two detailed solutions of the problem for different randomly chosen parameters using linear finite elements. (The size of the grid can be controlled via the --grid parameter. The randomly chosen parameters will actually be the same for each run, since a the random generator is initialized with a fixed default seed in default_random_state.)

After closing the window, the reduced basis for model order reduction is generated using a greedy search algorithm with error estimator. The third parameter SNAPSHOTS of thermalblock.py determines how many different values per parameter component μ_ij should be considered. I.e. the parameter training set for basis generation will have the size SNAPSHOTS^(XBLOCKS x YBLOCKS). After the basis of size 32 (the last parameter) has been computed, the quality of the obtained reduced model (on the 32-dimensional reduced basis space) is evaluated by comparing the solutions of the reduced and detailed models for new, randomly chosen parameter values. Finally, plots of the detailed and reduced solutions, as well as the difference between the two, are displayed for the random parameter values which maximises reduction error.

The thermalblock demo explained

In the following we will walk through the thermal block demo step by step in an interactive Python shell. We assume that you are familiar with the reduced basis method and that you know the basics of Python programming as well as working with NumPy. (Note that our code will differ a bit from thermalblock.py as we will hardcode the various options the script offers and leave out some features.)

First, start a Python shell. We recommend using IPython

ipython

You can paste the following input lines starting with >>> by copying them to the system clipboard and then executing

%paste

inside the IPython shell.

First, we will import the most commonly used methods and classes of pyMOR by executing:

from pymor.basic import *
from pymor.core.logger import set_log_levels
set_log_levels({'pymor.algorithms.greedy': 'ERROR', 'pymor.algorithms.gram_schmidt.gram_schmidt': 'ERROR', 'pymor.algorithms.image.estimate_image_hierarchical': 'ERROR'})

Next we will instantiate a class describing the analytical problem we want so solve. In this case, a thermal_block_problem:

p = thermal_block_problem(num_blocks=(3, 2))

We want to discretize this problem using the finite element method. We could do this by hand, creating a Grid, instatiating DiffusionOperatorP1 finite element diffusion operators for each subblock of the domain, forming a LincombOperator to represent the affine decomposition, instantiating a L2ProductFunctionalP1 as right hand side, and putting it all together into a StationaryModel. However, since thermal_block_problem returns a StationaryProblem, we can use a predifined discretizer to do the work for us. In this case, we use discretize_stationary_cg:

fom, fom_data = discretize_stationary_cg(p, diameter=1./50.)

fom is the StationaryModel which has been created for us, whereas fom_data contains some additional data, in particular the Grid and the BoundaryInfo which have been created during discretization. We can have a look at the grid,

print(fom_data['grid'])

Tria-Grid on domain [0,1] x [0,1]
x0-intervals: 50, x1-intervals: 50
elements: 10000, edges: 15100, vertices: 5101

and, as always, we can display its class documentation using help(fom_data['grid']).

Let’s solve the thermal block problem and visualize the solution:

U = fom.solve([1.0, 0.1, 0.3, 0.1, 0.2, 1.0])
fom.visualize(U, title='Solution')

Each class in pyMOR that describes a Parameter-dependent mathematical object, like the StationaryModel in our case, derives from ParametricObject. ParametricObjects automatically determine the Parameters they depend on from ParametricObjects that have been passed as __init__ arguments and from the parameters_own and parameters_internal attributes that have been set in __init__. Let’s have a look:

print(fom.parameters)

{diffusion: 6}

This tells us, that the Parameters which solve expects should be a dictionary with one key 'diffusion' whose value is a one-dimensional NumPy array of size 6, corresponding to the block structure of the problem. However, as an exception to this rule, the interface methods of Models allow simply passing the list [1.0, 0.1, 0.3, 0.1, 0.2, 1.0] by internally calling parse.

Next we want to use the greedy algorithm to reduce the problem. For this we need to choose a reductor which will keep track of the reduced basis and perform the actual RB-projection. We will use CoerciveRBReductor, which will also assemble an error estimator to estimate the reduction error. This will significantly speed up the basis generation, as we will only need to solve the high-dimensional problem for those parameters in the training set which are actually selected for basis extension. To control the condition of the reduced system matrix, we must ensure that the generated basis is orthonormal w.r.t. the H1_0-product on the solution space. For this we pass the h1_0_semi_product attribute of the model as inner product to the reductor, which will also use it for computing the Riesz representatives required for error estimation. Moreover, we have to provide the reductor with a ParameterFunctional which computes a lower bound for the coercivity of the problem for given parameter values.

reductor = CoerciveRBReductor(
   fom,
   product=fom.h1_0_semi_product,
   coercivity_estimator=ExpressionParameterFunctional('min(diffusion)', fom.parameters)
)

Moreover, we need to select a training set of parameter values. The problem p already comes with a ParameterSpace, from which we can easily sample these values. E.g.:

training_set = p.parameter_space.sample_uniformly(4)
print(training_set[0])

{diffusion: [0.1, 0.1, 0.1, 0.1, 0.1, 0.1]}

Now we start the basis generation:

greedy_data = rb_greedy(fom, reductor, training_set, max_extensions=32)

The max_extensions parameter defines how many basis vectors we want to obtain. greedy_data is a dictionary containing various data that has been generated during the run of the algorithm:

print(greedy_data.keys())

dict_keys(['max_errs', 'max_err_mus', 'extensions', 'time', 'rom'])

The most important items is 'rom' which holds the reduced Model obtained from applying our reductor with the final reduced basis.

rom = greedy_data['rom']

All vectors in pyMOR are stored in so called VectorArrays. For example the solution U computed above is given as a VectorArray of length 1. For the reduced basis we have:

RB = reductor.bases['RB']
print(type(RB))
print(len(RB))
print(RB.dim)

<class 'pymor.vectorarrays.numpy.NumpyVectorArray'>
32
5101

Let us check if the reduced basis really is orthonormal with respect to the H1-product. For this we use the gramian method:

import numpy as np
gram_matrix = RB.gramian(fom.h1_0_semi_product)
print(np.max(np.abs(gram_matrix - np.eye(32))))

1.5144135945277526e-15

Looks good! We can now solve the reduced model for the same parameter values as above. The result is a vector of coefficients w.r.t. the reduced basis, which is currently stored in RB. To form the linear combination, we can use the reconstruct method of the reductor:

u = rom.solve([1.0, 0.1, 0.3, 0.1, 0.2, 1.0])
print(u)
U_red = reductor.reconstruct(u)
print(U_red.dim)

[[ 0.56008169  0.19410562  0.00463453  0.01675562  0.04900982  0.07760119
  -0.08043888 -0.02889488 -0.03121126  0.26467367  0.13991773 -0.09805741
   0.16005484  0.04417512 -0.05245844 -0.00132281 -0.01304436  0.01330245
  -0.02185056  0.00625769 -0.00868736 -0.00115484  0.01895446 -0.00153234
   0.02306277  0.0035532   0.01061803 -0.00225572  0.00199581 -0.0068476
   0.00186538  0.0047302 ]]
5101

Finally we compute the reduction error and display the reduced solution along with the detailed solution and the error:

ERR = U - U_red
print(ERR.norm(fom.h1_0_semi_product))
fom.visualize((U, U_red, ERR),
             legend=('Detailed', 'Reduced', 'Error'),
             separate_colorbars=True)

[0.00480011]

We can nicely observe that, as expected, the error is maximized along the jumps of the diffusion coefficient.

Download the code: getting_started.md getting_started.ipynb

Learning more

As a next step, you should read our Technical Overview which discusses the most important concepts and design decisions behind pyMOR. You can also follow our growing set of pyMOR Tutorials, which focus on specific aspects of pyMOR.

Should you have any problems regarding pyMOR, questions or feature requests, do not hesitate to contact us via GitHub discussions!