Variables influence

In this example, we illustrate the use of the VariableInfluence plot on the Sobieski’s SSBJ problem.

from gemseo.api import configure_logger
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.problems.sobieski.core.problem import SobieskiProblem
from matplotlib import pyplot as plt

Import

The first step is to import some functions from the API and a method to get the design space.

configure_logger()

Out:

<RootLogger root (INFO)>

Description

The VariableInfluence post-processing performs first-order variable influence analysis.

The method computes \(\frac{d f}{d x_i} \cdot \left(x_{i_*} - x_{initial_design}\right)\), where \(x_{initial_design}\) is the initial value of the variable and \(x_{i_*}\) is the optimal value of the variable.

Create disciplines

At this point, we instantiate the disciplines of Sobieski’s SSBJ problem: Propulsion, Aerodynamics, Structure and Mission

disciplines = create_discipline(
    [
        "SobieskiPropulsion",
        "SobieskiAerodynamics",
        "SobieskiStructure",
        "SobieskiMission",
    ]
)

Create design space

We also read the design space from the SobieskiProblem.

design_space = SobieskiProblem().design_space

Create and execute scenario

The next step is to build an MDO scenario in order to maximize the range, encoded ‘y_4’, with respect to the design parameters, while satisfying the inequality constraints ‘g_1’, ‘g_2’ and ‘g_3’. We can use the MDF formulation, the SLSQP optimization algorithm and a maximum number of iterations equal to 100.

scenario = create_scenario(
    disciplines,
    formulation="MDF",
    objective_name="y_4",
    maximize_objective=True,
    design_space=design_space,
)
scenario.set_differentiation_method("user")
for constraint in ["g_1", "g_2", "g_3"]:
    scenario.add_constraint(constraint, "ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})

Out:

    INFO - 15:01:28:
    INFO - 15:01:28: *** Start MDOScenario execution ***
    INFO - 15:01:28: MDOScenario
    INFO - 15:01:28:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
    INFO - 15:01:28:    MDO formulation: MDF
    INFO - 15:01:29: Optimization problem:
    INFO - 15:01:29:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 15:01:29:    with respect to x_1, x_2, x_3, x_shared
    INFO - 15:01:29:    subject to constraints:
    INFO - 15:01:29:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:01:29:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:01:29:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:01:29:    over the design space:
    INFO - 15:01:29:    +----------+-------------+-------+-------------+-------+
    INFO - 15:01:29:    | name     | lower_bound | value | upper_bound | type  |
    INFO - 15:01:29:    +----------+-------------+-------+-------------+-------+
    INFO - 15:01:29:    | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 15:01:29:    | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 15:01:29:    | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 15:01:29:    | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 15:01:29:    | x_shared |      40     |   55  |      70     | float |
    INFO - 15:01:29:    | x_shared |     500     |  1000 |     1500    | float |
    INFO - 15:01:29:    | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 15:01:29:    | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 15:01:29:    | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 15:01:29:    | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 15:01:29:    +----------+-------------+-------+-------------+-------+
    INFO - 15:01:29: Solving optimization problem with algorithm SLSQP:
    INFO - 15:01:29: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 15:01:29: ...  20%|██        | 2/10 [00:00<00:00, 41.25 it/sec, obj=-2.12e+3]
 WARNING - 15:01:29: MDAJacobi has reached its maximum number of iterations but the normed residual 9.482098563532576e-06 is still above the tolerance 1e-06.
    INFO - 15:01:29: ...  30%|███       | 3/10 [00:00<00:00, 23.66 it/sec, obj=-3.76e+3]
    INFO - 15:01:29: ...  40%|████      | 4/10 [00:00<00:00, 17.11 it/sec, obj=-3.96e+3]
    INFO - 15:01:29: ...  50%|█████     | 5/10 [00:00<00:00, 13.35 it/sec, obj=-3.96e+3]
    INFO - 15:01:29: ...  70%|███████   | 7/10 [00:00<00:00, 10.91 it/sec, obj=-3.96e+3]
    INFO - 15:01:30: ...  90%|█████████ | 9/10 [00:01<00:00,  9.22 it/sec, obj=-3.96e+3]
 WARNING - 15:01:30: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 15:01:30: ... 100%|██████████| 10/10 [00:01<00:00,  8.53 it/sec, obj=-3.96e+3]
    INFO - 15:01:30: Optimization result:
    INFO - 15:01:30:    Optimizer info:
    INFO - 15:01:30:       Status: None
    INFO - 15:01:30:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
    INFO - 15:01:30:       Number of calls to the objective function by the optimizer: 12
    INFO - 15:01:30:    Solution:
 WARNING - 15:01:30:       The solution is not feasible.
    INFO - 15:01:30:       Objective: -3963.793570013662
    INFO - 15:01:30:       Standardized constraints:
    INFO - 15:01:30:          g_1 = [-0.01810854 -0.03338059 -0.04427602 -0.0518563  -0.05734441 -0.13720865
    INFO - 15:01:30:  -0.10279135]
    INFO - 15:01:30:          g_2 = 1.9000415800052295e-05
    INFO - 15:01:30:          g_3 = [-7.67474555e-01 -2.32525445e-01  4.34281075e-04 -1.83255000e-01]
    INFO - 15:01:30:       Design space:
    INFO - 15:01:30:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:01:30:       | name     | lower_bound |        value        | upper_bound | type  |
    INFO - 15:01:30:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:01:30:       | x_shared |     0.01    | 0.06000475010395003 |     0.09    | float |
    INFO - 15:01:30:       | x_shared |    30000    |        60000        |    60000    | float |
    INFO - 15:01:30:       | x_shared |     1.4     |         1.4         |     1.8     | float |
    INFO - 15:01:30:       | x_shared |     2.5     |         2.5         |     8.5     | float |
    INFO - 15:01:30:       | x_shared |      40     |          70         |      70     | float |
    INFO - 15:01:30:       | x_shared |     500     |         1500        |     1500    | float |
    INFO - 15:01:30:       | x_1      |     0.1     |         0.4         |     0.4     | float |
    INFO - 15:01:30:       | x_1      |     0.75    |         0.75        |     1.25    | float |
    INFO - 15:01:30:       | x_2      |     0.75    |         0.75        |     1.25    | float |
    INFO - 15:01:30:       | x_3      |     0.1     |  0.1563125997824079 |      1      | float |
    INFO - 15:01:30:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:01:30: *** End MDOScenario execution (time: 0:00:01.186009) ***

{'max_iter': 10, 'algo': 'SLSQP'}

Post-process scenario

Lastly, we post-process the scenario by means of the BasicHistory plot.

Tip

Each post-processing method requires different inputs and offers a variety of customization options. Use the API function get_post_processing_options_schema() to print a table with the options for any post-processing algorithm. Or refer to our dedicated page: Post-processing algorithms.

scenario.post_process("VariableInfluence", save=False, show=False, fig_size=(15, 12))
# Workaround for HTML rendering, instead of ``show=True``
plt.show()
Partial variation of the functions wrt design variables, 9 variables required to explain 99% of -y_4 variations, 5 variables required to explain 99% of g_1_0 variations, 5 variables required to explain 99% of g_1_1 variations, 5 variables required to explain 99% of g_1_2 variations, 5 variables required to explain 99% of g_1_3 variations, 5 variables required to explain 99% of g_1_4 variations, 4 variables required to explain 99% of g_1_5 variations, 4 variables required to explain 99% of g_1_6 variations, 1 variables required to explain 99% of g_2 variations, 7 variables required to explain 99% of g_3_0 variations, 7 variables required to explain 99% of g_3_1 variations, 3 variables required to explain 99% of g_3_2 variations, 3 variables required to explain 99% of g_3_3 variations

Out:

 WARNING - 15:01:30: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 15:01:30: VariableInfluence for function -y_4
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [1 4 3 2 5 9 7 8 0]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/develop/lib/python3.9/site-packages/gemseo/post/variable_influence.py:234: UserWarning: FixedFormatter should only be used together with FixedLocator
  axe.set_xticklabels(x_labels, fontsize=font_size, rotation=rotation)
    INFO - 15:01:30: VariableInfluence for function g_1_0
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [0 7 3 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_1
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [7 0 3 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_2
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [7 0 3 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_3
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [7 0 3 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_4
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [7 0 3 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_5
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [3 7 5 6]
    INFO - 15:01:30: VariableInfluence for function g_1_6
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [3 7 5 6]
    INFO - 15:01:30: VariableInfluence for function g_2
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [0]
    INFO - 15:01:30: VariableInfluence for function g_3_0
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [1 9 5 2 4 0 8]
    INFO - 15:01:30: VariableInfluence for function g_3_1
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [1 9 5 2 4 0 8]
    INFO - 15:01:30: VariableInfluence for function g_3_2
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [1 9 2]
    INFO - 15:01:30: VariableInfluence for function g_3_3
    INFO - 15:01:30: Most influential variables indices to explain % of the function variation: 99
    INFO - 15:01:30: [9 1 2]

Total running time of the script: ( 0 minutes 2.288 seconds)

Gallery generated by Sphinx-Gallery