Michalewicz function¶

Introduction¶

Examples can be found in BATMAN’s installer subrepository test-cases. To create a new study, use the same structure as this example on the Michalewicz function:

Michalewicz
|
|__ data
|   |__ script.sh
|   |__ function.py
|
|__ settings.json

The working directory consists in two parts:

data: contains all the simulation files necessary to perform a new simulation. It can be a simple python script to a complex code. The content of this directory will be copied for each snapshot. In all cases, script.sh launches the simulation.
settings.json: contains the case setup.

Note

The following section is a step-by-step tutorial that can be applied to any case.

Michalewicz Function¶

Step 1: Simulation directory¶

For this tutorial, the Michalewicz function was choosen. It is a multimodal d-dimensional function which has $d!$ local minima - for this test-case:

$f(x)=-\sum_{i=1}^d \sin(x_i)\sin^{2m}\left(\frac{ix_i^2}{\pi}\right),$

where m defines the steepness of the valleys and ridges.

Note

It is to difficult to search a global minimum when $m$ reaches large value. Therefore, it is recommended to have $m < 10$ .
In this case we used the two-dimensional form, i.e. $d = 2$ .

To summarize, we have the Michalewicz 2*D* function as follows:

$f(x)=-\sin(x_1)\sin^{20}\left(\frac{x_1^2}{\pi}\right)-\sin(x_2)\sin^{20}\left(\frac{2x_2^2}{\pi}\right).$

../_images/response_Michalewicz_true_2D.png

../_images/response_Michalewicz_true_3D.png

Step 2: Setting up the case¶

BATMAN’s settings are managed via a python file located in scripts. An example template can be found within all examples directory. This file consists in five blocks with different functions:

Block 1 - Space of Parameters

The space of parameters is created using the two extrem points of the domain here we have $x_1, x_2 \in [1, \pi]^2$ . Also we want to make 50 snapshots using a halton sequence.

"space": {
    "corners": [
        [1.0, 1.0],
        [3.1415, 3.1415]
    ],
    "sampling": {
        "init_size": 50,
        "method": "halton"
    },
}

Block 2 - Snapshot provider

Then, we configure the snapshot itself. We define the name of the header and output file as well as the dimension of the output. Here BATMAN will look at the variable F, which is a scalar value, within the file point.dat.

"snapshot": {
    "max_workers": 10,
    "parameters": ["x1", "x2"],
    "variables": ["F"],
    "provider": {
        "type": "file"
        "command": "bash script.sh",
        "context_directory": "data",
        "coupling_directory": "batman-coupling",
        "clean": false
    },
    "io": {
        "point_filename": "point.json",
        "data_filename": "point.dat",
        "data_format": "fmt_tp_fortran"
    }
}

Note

For a simple python function, you can pass it directly in the settings file:

"provider": {
    "type": "plugin",
    "module": "python_module",
    "function": "func_name"
}

with module the name of the python module containing the function function. For an example, see test_cases/Ishigami.

Block 3 - POD

In this example, a POD is not necessary as it will result in only one mode. However, its use is presented. We can control the quality of the POD, chose a re-sampling strategy, etc.

"pod": {
    "dim_max": 100,
    "quality": 0.8,
    "tolerance": 0.99,
    "type": "static"
}

Block 4 - Surrogate

A model is build on the snapshot matrix to approximate a new snapshot. The Kriging method is selected. To construct a response surface, we need to make predictions.

surrogate = {'method' : 'kriging',
             'predictions' : [[1., 2.], [2., 2.]],
             }

To fill in easily predictions, use the script prediction.py.

Block 5 - UQ

Once the model has been created, it can be used to perform a statistical analysis. Here, Sobol’ indices are computed using Sobol’s method using 50000 samples.

"uq": {
    "sample": 50000,
    "pdf": ["Uniform(1., 3.1415)", "Uniform(1., 3.1415)"],
    "type": "aggregated",
    "method": "sobol"
}

Step 3: Running BATMAN¶

To launch BATMAN, simply call it with:

batman settings.json -qsu

BATMAN’s log are found within BATMAN.log. Here is an extract:

BATMAN main ::
    POD summary:
    modes filtering tolerance    : 0.99
    dimension of parameter space : 2
    number of snapshots          : 50
    number of data per snapshot  : 1
    maximum number of modes      : 100
    number of modes              : 1
    modes                        : [ 2.69091785]
batman.pod.pod ::
    pod quality = 0.45977, max error location = (3.0263943749999997, 1.5448927777777777)

----- Sobol' indices -----
batman.uq ::
    Second order: [array([[ 0.        ,  0.06490131],
       [ 0.06490131,  0.        ]])]
batman.uq ::
    First order: [array([ 0.43424729,  0.49512012])]
batman.uq ::
    Total: [array([ 0.51371718,  0.56966205])]

In this example, the quality of the model is estimated around $Q_2\sim 0.46$ which means that the model is able to represents around 46% of the variability of the quantity of interest. Also, from Sobol’ indices, both parameters appears to be as important.

Post-treatment¶

Result files are separated in 4 directories under output:

Case
|
|__ data
|
|__ settings.json
|
|__ output
    |
    |__ surrogate
    |
    |__ predictions
    |
    |__ snapshots
    |
    |__ uq

snapshots contains all snapshots computations, predictions contains all predictions, surrogate contains the model and uq contains the statistical analysis. Using predictions we can plot the response surface of the function as calculated using the model:

../_images/response_Michalewicz_model_2D.png

It can be noted that using 50 snapshots on this case is not enought to capture all the non-linearities of the function.

Note

Usually, physical phenomena are smoother. Thus, less points are needed for a 2 parameters problem when dealing with real physics.

Michalewicz function¶

Introduction¶

Michalewicz Function¶

Step 1: Simulation directory¶

Step 2: Setting up the case¶

Step 3: Running BATMAN¶

Post-treatment¶

Refinement strategies¶