Tiled Monte Carlo Algorithm : Control of the Statistical Error with Tiling and its Applications

While direct Monte Carlo (MC) algorithms are powerful tools for solving extremely large linear equations, the original MC algorithm is severely restricted in its applicability. We examine in detail the novel tiled MC linear algebra algorithm recently developed by us. The tiled algorithm, which samples a tile of the matrix instead of individual elements, was proven to converge for sufficiently large tile size. Random ma-trices and a matrix from chemical physics are used to explore and demonstrate the properties of the algorithm including the effect of tiling on both convergence and the statistical error, which in turn informs parameter selection. We also introduce and illustrate the use of practical numerical tools to assess convergence and the accuracy of computed statistics since the rigorous convergence test may not be readily computed for very large matrices.