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Fast Numerical Methods for Stochastic Partial Differential Equations

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Technical Report,01 Jul 2012,30 Dec 2015

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Florida A and M University Tallahassee United States

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Uncertainty quantification has been an active research area in the past 15 years because of its potential of significant applications ranging from signal processing to aircraft wing designs. It is well understood that effective numerical methods for stochastic partial differential equations SPDES are essential for uncertainty quantification. In the last decade much progress has been made in the construction of numerical algorithms to efficiently solve SPDES with random coefficients and white noise perturbations. However, high dimensionality and low regularity are still the bottleneck in solving real world applicable SPDES with efficient numerical methods. This research is intended to address the numerical analysis as well as algorithm aspects of SPDES. Three major contributions are made in this project i Construction and convergence analysis of Quasi Monte Carlo based Particle Swarm Optimization PSO method ii Efficient adaptive domain sparse grid method for SPDES iii High order methods of SPDES via systems of forward backward stochastic differential equations. Our work contains algorithm constructions, rigorous error analysis, and extensive numerical experiments to demonstrate our algorithm efficiency and validity of our theoretical analysis.

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