On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients

In this paper, we propose a novel non-standard Local Fourier Analysis (LFA) variant for accurately predicting the multigrid convergence of problems with random and jumping coefficients. This LFA method is based on a specific basis of the Fourier space rather than the commonly used Fourier modes. To show the utility of this analysis, we consider, as an example, a simple cell-centered multigrid method for solving a steady-state single phase flow problem in a random porous medium. We successfully demonstrate the prediction capability of the proposed LFA using a number of challenging benchmark problems. The information provided by this analysis helps us to estimate a-priori the time needed for solving certain uncertainty quantification problems by means of a multigrid multilevel Monte Carlo method

Reducing communication in sparse solvers

Sparse matrix operations dominate the cost of many scientific applications. In parallel, the performance and scalability of these operations is limited by irregular point-to-point communication. Multiple methods are investigated throughout this dissertation for reducing the cost associated with communication throughout sparse matrix operations. Algorithmic changes reduce communication requirements, but also affect accuracy of the operation, leading to reduced convergence of scientific codes. We investigate a method of systematically removing relatively small non-zeros throughout an algebraic multigrid hierarchy, yielding significant reductions to the cost of sparse matrix-vector multiplication that outweigh affects of reduced accuracy of the multiplication. Therefore, the reduction in per-iteration communication costs outweigh the cost of extra solver iterations. As a result, sparsification yields improvement of both the performance and scalability of algebraic multigrid. Alterations to the parallel implementation of MPI communication also yield reduced costs with no effect on accuracy. We investigate methods of agglomerating messages on-node before injecting into the network, reducing the amount of costly inter-node communication. This node-aware communication yields improvements to both performance and scalability of matrix operations, particularly in strong scaling studies. Furthermore, we show an improvement in the cost of algebraic multigrid as a result of reduced communication costs in sparse matrix operations. Finally, performance models can be used to analyze the costs of matrix operations, indicating the source of dominant communication costs, such as initializing messages or transporting bytes of data. We investigate methods of improving traditional performance models of irregular point-to-point communication through the addition of node-awareness, queue search costs, and network contention penalties

Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies

The application of multilevel methods to solving large algebraic systems obtained by discretization of PDEs has seen great success. However, these methods often perform sub-optimally when treating problems with anisotropies. For problems posed over unstructured meshes, optimal automatic multigrid coarsening is not a fully solved problem for the smoothed aggregation multigrid. The focus of this thesis is on enhancing robustness of the coarsening in the Smoothed Aggregation (SA) multigrid. We focus on improving the standard detection of coupling, on which the coarsening decisions in SA are based. Our approach takes the form of a two-pass test which allows for a more robust local control over the coupling detection, as well as added exibility permitting utilization of new coupling detection measures in a more systematic way. For isotropic problems, smoothed aggregation coarsening is known to offer very favorable operator complexity, but achieving similar behavior in the presence of anisotropy is more challenging. Special attention is paid to addressing the issue of controlling the complexity of the method. We discuss several existing approaches to curbing coarse-level operator fill-in, and offer generalizations and improvements. Numerical experiments are provided to demonstrate the performance of the improved coarsening on model examples of anisotropic problems featuring both cases where anisotropies are aligned with the grid, as well as cases where they are not

Shifted Laplacian multigrid for the elastic Helmholtz equation

The shifted Laplacian multigrid method is a well known approach for preconditioning the indefinite linear system arising from the discretization of the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth's subsurface is the elastic medium. The elastic Helmholtz equation is much harder to solve than its acoustic counterpart, partially because it is three times larger, and partially because it models more complicated physics. Despite this, there are very few solvers available for the elastic equation compared to the array of solvers that are available for the acoustic one. In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We demonstrate the efficiency and properties of our solver using numerical experiments for problems with heterogeneous media in two and three dimensions

The transport of nanoparticles in subsurface with fractured, anisotropic porous media: Numerical simulations and parallelization

The flow of fluids through fractured porous media has been an important topic in the research of subsurface flow. The several orders of magnitude in size between the fractures and the rock matrix causes difficulties for simulating such flow scenario. The fluid velocities in fractures are also several orders of magnitude higher than that in the rock matrix due to high permeability and porosity. If there exists pollutant such as nanoparticles in the fluids, the pollutant may be transported rapidly and the rock matrix’s properties near the fractures are hence changed. In this research, we simulate the transport phenomena of nanoparticles in the fluid flow through fractured porous media. The permeability fields which contain different anisotropy angles are considered in the simulation. Fractures are represented explicitly by volumetric grid cells and the numerical algorithm is parallelized in order to reduce the simulation time. We investigate the effect of the appearance of fractures and rotated anisotropy on the transport of nanoparticles, particles deposition, entrapment and detachment. The results show that flow directions are affected by the direction of anisotropy and the transport of nanoparticles in the fractures is significantly faster than that in rock matrix due to high fluid velocities. The direction of anisotropy distorted the pressure field and changed the fluid flow directions, which determined the time needed for the pollutant front to reach the fractures. The parallel efficiency of the overall algorithm is also discussed and the experimental results show that it is deeply affected by the performance of the multigrid solver