Multilevel Hierarchical Decomposition of Finite Element White Noise with Application to Multilevel Markov Chain Monte Carlo

In this work we develop a new hierarchical multilevel approach to generate Gaussian random field realizations in an algorithmically scalable manner that is well-suited to incorporate into multilevel Markov chain Monte Carlo (MCMC) algorithms. This approach builds off of other partial differential equation (PDE) approaches for generating Gaussian random field realizations; in particular, a single field realization may be formed by solving a reaction-diffusion PDE with a spatial white noise source function as the righthand side. While these approaches have been explored to accelerate forward uncertainty quantification tasks, e.g. multilevel Monte Carlo, the previous constructions are not directly applicable to multilevel MCMC frameworks which build fine scale random fields in a hierarchical fashion from coarse scale random fields. Our new hierarchical multilevel method relies on a hierarchical decomposition of the white noise source function in $L^2$ which allows us to form Gaussian random field realizations across multiple levels of discretization in a way that fits into multilevel MCMC algorithmic frameworks. After presenting our main theoretical results and numerical scaling results to showcase the utility of this new hierarchical PDE method for generating Gaussian random field realizations, this method is tested on a four-level MCMC algorithm to explore its feasibility

Parallel Element-Based Algebraic Multigrid for H (Curl) And H (Div) Problems Using the Parelag Library

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers (ParELAG) library, to produce multilevel preconditioners and solvers for H (curl) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and the Auxiliary-Space Maxwell Solver or the Auxiliary-Space Divergence Solver on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library