Integration

The integration problem is considering the numerical approximation of integrals

\[I = \int f(x)\mathrm{d}x\]

using

\[I \approx \sum\limits_{i=1}^N w_i f(x_i)\]

The supported methods are Monte Carlo Integration and Quadrature. For Quadrature the weights for the Rectangle rule, the Trapezoidal rule and the Simpson rule are given, and there is the possibility to provide own weights and evaluation points.

Usage

e["Problem"]["Type"] = "Integration"

Compatible Solvers

This problem can be solved using the following modules:

Variable-Specific Settings

These are settings required by this module that are added to each of the experiment’s variables when this module is selected.

Lower Bound

  • Usage: e[“Variables”][index][“Lower Bound”] = real number

  • Description: Lower bound for integration.

Upper Bound

  • Usage: e[“Variables”][index][“Upper Bound”] = real number

  • Description: Upper bound for integration.

Number Of Gridpoints

  • Usage: e[“Variables”][index][“Number Of Gridpoints”] = unsigned integer

  • Description: Number of Gridpoints along given axis.

Sampling Distribution

  • Usage: e[“Variables”][index][“Sampling Distribution”] = string

  • Description: Indicates the name of the distribution to use for Monte Carlo integration.

Distribution Index

  • Usage: e[“Variables”][index][“Distribution Index”] = unsigned integer

  • Description: Stores the the index number of the selected prior distribution.

Sample Points

  • Usage: e[“Variables”][index][“Sample Points”] = List of real number

  • Description: Contains values at which to evaluate the function.

Quadrature Weights

  • Usage: e[“Variables”][index][“Quadrature Weights”] = List of real number

  • Description: Contains quadrature weights to evaluate the integral.

Name

  • Usage: e[“Variables”][index][“Name”] = string

  • Description: Defines the name of the variable.

Configuration

These are settings required by this module.

Integrand

  • Usage: e[“Problem”][“Integrand”] = Computational Model

  • Description: Stores the function to integrate.

Integration Method

  • Usage: e[“Problem”][“Integration Method”] = string

  • Description: Indicates the name of the integration method to use.

  • Options:

    • “Rectangle”: Uses the Rectangle Rule to perform the integral.

    • “Trapezoidal”: Uses the Trapezoidal Rule to perform the integral.

    • “Simpson”: Uses the Simpson Rule to perform the integral.

    • “Monte Carlo”: Uses Monte Carlo Integration to perform the integral.

    • “Custom”: Uses a Rule based on provided weights to perform the integral.

Variable Defaults

These following configuration will be assigned to each of the experiment variables by default. Any settings defined by the user will override the given settings specified in these defaults.

{
"Distribution Index": -1,
"Quadrature Weights": [],
"Sample Points": [],
"Sampling Distribution": " "
}