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Monte Carlo for PDEs Fall 2013

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

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Monte Carlo for PDEs Fall 2013

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  1. Monte Carlo for PDEsFall 2013

  2. Review • Last Class • Monte Carlo Linear Solver • von Neumann and Ulam method • Randomize Stationary iterative methods • Variations of Monte Carlo solver • This Class • Fredholmintegral equations of the second kind • The Dirichlet Problem • Eigenvalue Problems • Next Class • Green’s Function Monte Carlo

  3. 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:

  4. 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.

  5. 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,

  6. 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 .

  7. 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.

  8. 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 .

  9. 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

  10. What I want you to do? • Review Slides • Work on Assignment 4

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