program程序代写、代做Java编程设计

日期：2024-05-04 01:26

numerical-methods-montecarlointegrati

on-exercise

In this exercise, you should implement classes that provide a reasonably flexible

framework for Monte-Carlo integration.

The framework should be flexible enough to allow

• integration of different functions f : Rn → R

• flexible specification of the integration domains using a transformation from

[0,1]n to a subset of Rn

• flexible use for different random number generators

For convenience, we provide the interfaces that define the framework. You can find

these interfaces in the package

info.quantlab.numericalmethods.lecture.montecarlo.integration

Interfaces (provided)

• Integrand

• IntegrationDomain

• Integrator

• MonteCarloIntegratorFactory

The MonteCarloIntegratorFactory's method requires a class implementing

a RandomNumberGenerator. This interface and some classes implementing this interface can

be found in the package

info.quantlab.numericalmethods.lecture.randomnumbers

Integrand and IntegrationDomain

• Objects implementing Integrand provide a function f : A \rightarrow R defined on a domain

A.

• Objects implementing IntegrationDomain provide a bijective function g : [0,1]^{n} \rightarrow

A that transforms the integration domain and the determinant of the derivative (Jacobi matrix)

dg/dx.

• Objects implementing Integrator provide the integral \int_A f(z) dz using substitution z =

f(x).

Classes

You may use the classes providing random number generators that will be or were

developed during the lecture, e.g.,

• RandomNumberGeneratorFrom1D

• MersenneTwister

Task

The exercise consists of two separate tasks.

Task 1: A MonteCarloIntegrator

To complete your task:

•

i. Implement a class implementing the interface Integrator that performs a general

Monte-Carlo integration of arbitrary functions on general domains.

The function to integrate will be provided to the integrator's method integrate as an

object implementing the interface Integrand.

The integration domain will be provided to the integrator's method integrate as an

object implementing the interface IntegrationDomain.

•

ii. Implement a class implementing the interface MonteCarloIntegratorFactory that

allows creating an object of the class that you have implemented in 1). Note:

the MonteCarloIntegratorFactory simply calls the constructor of your class.

•

ii. To allow us to test you implementation, complete the implementation of the

method getMonteCarloIntegratorFactory of MonteCarloIntegrationAssignmentSolutio

n. This allows the creation of an object of your MonteCarloIntegratorFactory. Our unit

tests will use this to test your code.

•

ii. Feel free to create your own UnitTests and JavaDoc documentation.

Suggestion: you may test your integrator with different random number generators,

e.g. MersenneTwister via

final long seed = 3141;

RandomNumberGenerator randomNumberGenerator = new RandomNumberGeneratorFrom1D(new

MersenneTwister(seed), domain.getDimension());

or a HaltonSequence.

Task 2: Using your MonteCarloIntegrator to calculate the integral

of a DoubleBinaryFunction

•

v. Complete the method getIntegral of MonteCarloIntegrationAssignmentSolution. Use

your Monte-Carlo integrator with approximately 1 million sample points to calculate

the integral.

Tasks 3: Implement a SimpsonsIntegrator for the general

Simpson's rule in d dimension

To complete your task:

•

vi. Implement a class implementing the interface Integrator that performs a general

(composite) Simpson's rule integration in d dimension of arbitrary functions on

general domains.

The function to integrate will be provided to the integrator's method integrate as an

object implementing the interface Integrand.

The integration domain will be provided to the integrator's method integrate as an

object implementing the interface IntegrationDomain.

•

vii. Implement a class implementing the interface IntegratorFactory that allows creating

an object of the class that you have implemented in 1). Note:

the IntegratorFactory simply calls the constructor of your class.

•

vii. To allow us to test you implementation, complete the implementation of the

method getSimpsonsIntegratorFactory of MonteCarloIntegrationAssignment. This

allows the creation of an object of your IntegratorFactory. Our unit tests will use this

to test your code.

•

vii. Feel free to create your own UnitTests and JavaDoc documentation.

Hints

• Note that your Simpsons integral and your Monte-Carlo integral only operator

on [0,1]^d (the object implementing the Domain will provide you with the

transformation).

• Your Simpsons integrator should accept the numberOfValuationPoints as an

argument. This should be the minimum total number of valuation points. Since

the Simpsons rule uses an odd number of points in every dimension, you may

use the following code to round this number appropriately

to numberOfSamplePointsEffective, using numberOfSamplePointsPerDimension per

dimension.

int dimension = integrationDomain.getDimension();

int numberOfValuationPointsPerDimension = 2 * (int)

(Math.ceil(Math.pow(numberOfValuationPoints, 1.0/dimension))/2) + 1;

int numberOfValuationPointsEffective = (int)

Math.pow(numberOfValuationPointsPerDimension, dimension);

• You might realise that you need to think a bit to find a short algorithm to implement the

Simpsons integration in arbitrary dimensions. It is possible to create a fairly short

implementation if you implement a multi-index index - an array of length dimension where

each entry runs from 0 to numberOfSamplePointsPerDimension-1.

Unit Tests

We encourage you to write your own unit tests.

Further Research

This project offers the opportunity to explore Monte-Carlo integration in more detail for

those interested. Here are a few suggestions:

• Explore the dependency on the dimension: Consider the integration of x →

product(i=0,...,d-1) sin(xi) for 0 < xi < π. The value of the integral is 2^d. This is an

d-dimensional integral. For this function, compare the accuracy of Monte-Carlo

integration and Simpsons integration with d = 1, 2, 4, 8 using for example n =

5^8 = 390625 sample points.

• Explore the dependency on the smoothness of the function: Consider the

integration of (x0,x1) → x0

2 + x1

2 < 1.0 ? 1.0 : 0.0 for 0 < xi < 1. The analytic value of

this integral π. For this function, compare the accuracy of Monte-Carlo

integration and Simpsons integration using n = 101^2 = 10201 sample points.