improved docstrings and markdown for cleaniness and consistency in re…

…adthedocs

docs/code/reliability/inverse_form/README.rst

@@ -1,4 +1,2 @@
 Inverse FORM Examples
-^^^^^
-
-The following example is Example 7.2 from Chapter 7 of :cite:`FORM_XDu`.
+^^^^^^^^^^^^^^^^^^^^^^^^^

docs/code/reliability/inverse_form/inverse_form_cantilever.py

@@ -1,25 +1,25 @@
 """
 Inverse FORM - Cantilever Beam
------
+-----------------------------------
+The following example is Example 7.2 from Chapter 7 of :cite:`FORM_XDu`.
 
-A cantilever beam (example 7.2 in :cite:`FORM_XDu`) is considered to fail if the displacement at the tip exceeds the
-threshold :math:`D_0`. The performance function :math:`G(\textbf{U})` of this problem is given by
+A cantilever beam is considered to fail if the displacement at the tip exceeds the threshold :math:`D_0`.
+The performance function :math:`G(\\textbf{U})` of this problem is given by
 
-.. math:: G = D_0 - \frac{4L^3}{Ewt} \sqrt{ \left(\frac{P_x}{w^2}\right)^2 + \left(\frac{P_y}{t^2}\right)^2}
+.. math:: G = D_0 - \\frac{4L^3}{Ewt} \\sqrt{ \\left(\\frac{P_x}{w^2}\\right)^2 + \\left(\\frac{P_y}{t^2}\\right)^2}
 
 Where the external forces are modeled as random variables :math:`P_x \sim N(500, 100)` and :math:`P_y \sim N(1000,100)`.
-The constants in the problem are length (:math:`L=100`), elastic modulus (:math:E=30\times 10^6), cross section width
-(:math:`w=2`) and cross section height (:math:`t=4`).
+The constants in the problem are length (:math:`L=100`), elastic modulus (:math:`E=30\\times 10^6`), cross section width
+(:math:`w=2`), cross section height (:math:`t=4`), and :math:`D_0=3`.
 
 """
 # %% md
 #
-# Import the necessary modules.
+# First, we import the necessary modules.
 
 # %%
+
 import numpy as np
-import matplotlib.pyplot as plt
-plt.style.use('ggplot')
 from scipy import stats
 from UQpy.distributions import Normal
 from UQpy.reliability.taylor_series import InverseFORM
@@ -34,7 +34,7 @@
 
 # %%
 
-model = PythonModel(model_script='performance_function.py', model_object_name="cantilever_beam")
+model = PythonModel(model_script='local_pfn.py', model_object_name="cantilever_beam")
 runmodel_object = RunModel(model=model)
 
 # %% md
@@ -70,4 +70,4 @@
 print('Design point in original space:', inverse_form.design_point_x[0])
 print('Hasofer-Lind reliability index:', beta)
 print('Probability of failure at design point:', stats.norm.cdf(-beta))
-print('Number of iterations:', inverse_form.iteration_record[0])
+print('Number of iterations:', inverse_form.iteration_record[0])

docs/code/reliability/inverse_form/performance_function.py → docs/code/reliability/inverse_form/local_pfn.py

@@ -1,3 +1,9 @@
+"""
+
+Auxiliary file
+==============================================
+
+"""
 import numpy as np
 
 

docs/source/reliability/inverse_form.rst

@@ -9,7 +9,7 @@ where :math:`\textbf{U}^\star` is the expansion point, :math:`G(\textbf{U})` is
 the standard normal space and :math:`\nabla G(\textbf{U}^\star)` is the gradient of :math:`G(\textbf{U})` evaluated at
 :math:`\textbf{U}^\star`. The probability failure is approximated as
 
-.. math:: P_{fail} = \Phi(-\beta_{HL})
+.. math:: p_{fail} = \Phi(-\beta_{HL})
 
 where :math:`\Phi(\cdot)` is the standard normal cumulative distribution function and :math:`\beta_{HL}=||\textbf{U}^*||`
 is the norm of the design point known as the Hasofer-Lind reliability index.
@@ -24,14 +24,14 @@ are:
 
 
 Problem Statement
------
+-----------------
 Compute :math:`u^* = \text{argmin}\ G(\textbf{U})` such that :math:`||\textbf{U}||=\beta`.
 
 The feasibility criteria :math:`||\textbf{U}||=\beta` may be equivalently defined as
 :math:`\beta = -\Phi^{-1}(p_{fail})`, where :math:`\Phi^{-1}(\cdot)` is the inverse standard normal CDF.
 
 Algorithm
------
+-----------------
 This method implements a gradient descent algorithm to solve the optimization problem within the tolerances specified by
 :math:`\text{tolerance}_{\textbf{U}}` (:code:`tolerance_u`) and :math:`\text{tolerance}_{\nabla G(\textbf{U})}` (:code:`tolerance_gradient`).
 
@@ -50,12 +50,12 @@ The :class:`.InverseFORM` class is imported using the following command:
 >>> from UQpy.reliability.taylor_series import InverseFORM
 
 Methods
-"""""""
+-----------------
 .. autoclass:: UQpy.reliability.taylor_series.InverseFORM
     :members: run
 
 Attributes
-""""""""""
+-----------------
 .. autoattribute:: UQpy.reliability.taylor_series.InverseFORM.alpha
 .. autoattribute:: UQpy.reliability.taylor_series.InverseFORM.alpha_record
 .. autoattribute:: UQpy.reliability.taylor_series.InverseFORM.beta
@@ -71,4 +71,4 @@ Examples
 """"""""""
 .. toctree::
 
-   FORM Examples <../auto_examples/reliability/inverse_form/index>
+   InverseFORM Examples <../auto_examples/reliability/inverse_form/index>

src/UQpy/reliability/taylor_series/InverseFORM.py

@@ -111,16 +111,17 @@ def __init__(
         self.alpha_record: list = []
         """Record of alpha :math:`\\alpha=\\frac{\\nabla G(u)}{||\\nabla G(u)||}`"""
         self.beta: float = self.beta
-        """Feasibility criteria for the optimization :math:`||\\textbf{U}|| = \\beta`"""
+        """Feasibility criteria for the optimization :math:`||\\textbf{U}|| = \\beta_{HL}`"""
         self.beta_record: list = []
-        """Record of Hasofer-Lind reliability index that defines the feasibility criteria :math:`||u||=\\beta_{HL}`"""
+        """Record of Hasofer-Lind reliability index that defines the feasibility criteria 
+         :math:`||\\textbf{U}||=\\beta_{HL}`"""
         self.design_point_u: list = []
         """Design point in the standard normal space :math:`\\textbf{U}`"""
         self.design_point_x: list = []
         """Design point in the parameter space :math:`\\textbf{X}`"""
         self.error_record: list = []
         """Record of the final error defined by 
-         :math:`error_u = ||u_new - u||` and :math:`error_{gradient} = || \\nabla G(u_new) - \\nabla G(u)||`"""
+         :math:`error_u = ||u_{new} - u||` and :math:`error_{\\nabla G(u)} = || \\nabla G(u_{new}) - \\nabla G(u)||`"""
         self.iteration_record: list = []
         """Record of the number of iterations before algorithm termination"""
         self.failure_probability_record: list = []