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# Integration

In this tutorial we show how to integrate a given function.

## Problem Description

We are given the function $$f(x,y,z)=x^2+y^2+z^2$$ for $$x,y,z\in [0,1]^3$$. We want to find the integral of this function over its domain.

## The Objective Function

Create a folder named model. Inside, create a file with name integrand.py and paste the following code,

def integrand( sample ):
x = sample["Parameters"]
#sample["Evaluation"] = x**2
y = sample["Parameters"]
#sample["Evaluation"] = x**2+y**2
z = sample["Parameters"]
sample["Evaluation"] = x**2+y**2+z**2


This is the function we want to integrate.

## Optimization with CMAES

First, open a file (you could name it ‘run-integration.py’) and import the korali module

#!/usr/bin/env python3
import korali


Import the integrand,

import sys
sys.path.append('./model')
from integrand import *


## The Korali Engine and Experiment Objects

Next we construct a korali.Engine and a korali.Experiment object and set the computational model,

k = korali.Engine()
e = korali.Experiment()


## The Problem Type

Then, we set the type of the problem to Integration, set the function to integrate and chose the integration method

e["Problem"]["Type"] = "Integration"
e["Problem"]["Integrand"] = lambda modelData: integrand(modelData)
e["Problem"]["Integration Method"] = "Rectangle"#"Rectangle" #"Trapezoidal", "Simpson", "Monte Carlo", "Custom"


## The Variables

In this problem there is three variables, X, Y and Z, whose domain we set to [0,1] and in case of Monte Carlo Integration assume an uniform distribution. Furthermore we assume 10 samples per dimension

e["Variables"]["Name"] = "x"
e["Variables"]["Number Of Gridpoints"] = 10
e["Variables"]["Lower Bound"] = 0.0
e["Variables"]["Upper Bound"] = 1.0
e["Variables"]["Sampling Distribution"] = "Uniform"

e["Variables"]["Name"] = "y"
e["Variables"]["Lower Bound"] = 0.0
e["Variables"]["Upper Bound"] = 1.0
e["Variables"]["Number Of Gridpoints"] = 10
e["Variables"]["Sampling Distribution"] = "Uniform"

e["Variables"]["Name"] = "z"
e["Variables"]["Lower Bound"] = 0.0
e["Variables"]["Upper Bound"] = 1.0
e["Variables"]["Number Of Gridpoints"] = 10
e["Variables"]["Sampling Distribution"] = "Uniform"


## The Solver

We choose the solver Integrator, don’t set the execution per generation, to have the summation be performed in one generation,

e["Solver"]["Type"] = "Integrator"
#e["Solver"]["Executions Per Generation"] = 100


For a detailed description of Integrator settings see Integrator.

Finally, we need to add a call to the run() routine to start the Korali engine.

k.run(e)


## Running

We are now ready to run our example: ./run-integration

Or, alternatively: python3 ./run-integration.py

The results are saved in the folder _korali_result/.