Monte Carlo for PDEs

Fall 2013

Review

• Last Class– Monte Carlo Linear Solver

• von Neumann and Ulam method• Randomize Stationary iterative methods• Variations of Monte Carlo solver

• This Class– Fredholm integral equations of the second kind– The Dirichlet Problem– Eigenvalue Problems

• Next Class– Green’s Function Monte Carlo

Fredholm integral equations of the second kind

• The integral equation

may be solved by Monte Carlo methods.

Since the integral can be approximated by a quadrature formula:

Fredholm integral equations of the second kind

• The integral equation can be transformed to be

evaluate it at the quadrature points:

Let be the vector , be the vector and be the matrix , the integral equation becomes

where is the unknown vector.

The Dirichlet Problem

• Dirichlet’s problem is to find a function , which is continuous and differentiable over a closed domain with boundary , satisfying

where is a prescribed function, and is the Laplacian operator.

Replacing by its finite-difference approximation,

The Dirichlet Problem

• Suppose the boundary lies on the mesh, the previous equations can be transformed into

– The order of is equal to the number of mesh points in .– has four elements equal to in each row corresponding to an interior

point of , all other elements being zero.– has boundary values corresponding to an boundary point of , all other

interior elements being zero.– The random walk starting from an interior point , terminates when it

hits a boundary point . The is an unbiased estimator of .

Eigenvalue Problems

• For a given symmetric matrix

assume that >, so that is the dominant eigenvalue and is the corresponding eigenvector.

For any nonzero vector , according to the power method,

We can obtain a good approximation of the dominant eigenvector of from the above.

Eigenvalue Problems

Similar to the idea behinds Monte Carlo solver that we can do sampling on to estimate its value, and then evaluate the dominant eigenvector by a proper scaling. From the Rayleigh quotient,the dominant eigenvalue be approximated based on the estimated vector of .

Summary

• This Class– Monte Carlo Linear Solver• von Neumann and Ulam method• Randomize Stationary iterative methods• Variations of Monte Carlo solver

– Fredholm integral equations of the second kind– The Dirichlet Problem– Eigenvalue Problems

What I want you to do?

• Review Slides• Work on Assignment 4