We explore the performance of several algorithms for the solution of stochastic partial differential equations including the stochastic Galerkin method and the stochastic sparse grid collocation method. We also introduce a new method called the adaptive kernel density estimation (KDE) collocation method, which addresses some of the deficiencies present in other stochastic PDE solution methods. This method combines an adaptive sparse grid collocation method with KDE to optimally allocate stochastic degrees of freedom.

Several components of this method can be computationally expensive, such as automatic bandwidth selection for the kernel density estimate, evaluation of the kernel density estimate, and computation of the coefficients of the approximate solution. Fortunately all of these operations are easily parallelizable. We present an implementation of adaptive KDE collocation that makes use of NVIDIA's complete unified device architecture (CUDA) to perform the computations in parallel on graphics processing units (GPUs).