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 and paste the following code,

def integrand( sample ):
  x = sample["Parameters"][0]
  #sample["Evaluation"] = x**2
  y = sample["Parameters"][1]
  #sample["Evaluation"] = x**2+y**2
  z = sample["Parameters"][2]
  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 ‘’) and import the korali module

#!/usr/bin/env python3
import korali

Import the integrand,

import sys
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"][0]["Name"] = "x"
e["Variables"][0]["Number Of Gridpoints"] = 10
e["Variables"][0]["Lower Bound"] = 0.0
e["Variables"][0]["Upper Bound"] = 1.0
e["Variables"][0]["Sampling Distribution"] = "Uniform"

e["Variables"][1]["Name"] = "y"
e["Variables"][1]["Lower Bound"] = 0.0
e["Variables"][1]["Upper Bound"] = 1.0
e["Variables"][1]["Number Of Gridpoints"] = 10
e["Variables"][1]["Sampling Distribution"] = "Uniform"

e["Variables"][2]["Name"] = "z"
e["Variables"][2]["Lower Bound"] = 0.0
e["Variables"][2]["Upper Bound"] = 1.0
e["Variables"][2]["Number Of Gridpoints"] = 10
e["Variables"][2]["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.


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

Or, alternatively: python3 ./

The results are saved in the folder _korali_result/.