2020/07/28 by Hillary Fairbanks, Umberto Villa, Fairbanks, Hillary R. +3
Decision Sciences · Engineering · Mathematics · #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Nuclear reactor physics and engineering #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2007.14440
openalex publication_date 2020/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we develop a new hierarchical multilevel approach to generate\nGaussian random field realizations in an algorithmically scalable manner that\nis well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC)\nalgorithms. This approach builds off of other partial differential equation\n(PDE) approaches for generating Gaussian random field realizations; in\nparticular, a single field realization may be formed by solving a\nreaction-diffusion PDE with a spatial white noise source function as the\nrighthand side. While these approaches have been explored to accelerate forward\nuncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous\nconstructions are not directly applicable to multilevel MCMC frameworks which\nbuild fine scale random fields in a hierarchical fashion from coarse scale\nrandom fields. Our new hierarchical multilevel method relies on a hierarchical\ndecomposition of the white noise source function in L2 which allows us to\nform Gaussian random field realizations across multiple levels of\ndiscretization in a way that fits into multilevel MCMC algorithmic frameworks.\nAfter presenting our main theoretical results and numerical scaling results to\nshowcase the utility of this new hierarchical PDE method for generating\nGaussian random field realizations, this method is tested on a four-level MCMC\nalgorithm to explore its feasibility.\n