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Multilevel Monte Carlo for the Feynman-Kac Formula for the Laplace Equation

Since its formulation in the late 1940s, the Feynman-Kac formula has proven to be an effective tool for both theoretical reformulations and practical simulations of differential equations. The link it establishes between such equations and stochastic processes can be exploited to develop Monte Carlo sampling methods that are effective, especially in high dimensions. There exist many techniques of improving standard Monte Carlo sampling methods, a relatively new development being the so-called Multilevel Monte Carlo method. This paper investigates the applicability of multilevel ideas to the stochastic representation of partial differential equations by the Feynman- Kac formula, using the Walk on Sphere algorithm to generate the required random paths. We focus on the Laplace equation, the simplest elliptic PDE, while mentioning some extension possibilities.