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Abstract
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PDEs appear in many areas of science and engineering including finance, image processing and economics.The analytic solutions, although preferred, is difficult or impossible to achieve and therefore, we need to approximate the solution. The deterministic approximation methods like FD or FEM are highly sensitive w.r.t. dimension of the space so that they result non-efficient algorithms in dimensions $d>3$. However, the Monte Carlo scheme is less sensitive to dimension and could be used to develop numerical schemes. The Monte Carlo methods for PDEs is based on providing the PDE with a probabilistic representation for the solution. The first such a representation is based on the well-known Feynman-Kac formula for the solution of the linear parabolic and elliptic PDEs. In this talk, we will review some of the main methods which provide us with a probabilistic representation for the solution of the non-linear PDEs. Using this representation, one can implement Monte Carlo methods to approximate the solution of the PDEs. |
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