[Paper Review] Algebraic-geometric matrix-free multigrid on dynamically adaptive Cartesian meshes.

Marion Weinzierl, Tobias Weinzierl|arXiv (Cornell University)|Jul 3, 2016

Advanced Numerical Methods in Computational Mathematics33 references3 citations

TL;DR

This paper proposes a matrix-free multigrid solver for dynamically adaptive Cartesian meshes that embeds algebraic multigrid operators as stencils within a spacetree hierarchy, enabling efficient in-situ matrix-vector operations without explicit matrix assembly. By representing operators hierarchically as differences from geometric counterparts, it achieves low memory overhead and supports parallel computation via multiscale domain decomposition.

ABSTRACT

We introduce a family of spacetree-based multigrid solvers that do not maintain a matrix data structure and rely on the spacetree's multiscale nature to derive coarse grids. The most sophisticated solvers from the family pick up the concept of BoxMG, which defines operator-dependent prolongation and restriction in combination with (Petrov-)Galerkin coarse grid operators. It yields robust solvers for nontrivial elliptic problems. We propose to embed the algebraic, problem- and grid-dependent multigrid operators as stencils into the grids, to evaluate all matrix-vector products in-situ throughout grid traversals, and, thus, to avoid assembly overhead that is notably cumbersome for dynamically adaptive grids. While such an approach is not literally matrix-free---the grid carries the matrix---we propose switch to a hierarchical two-scale representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be held and exchanged with small memory footprint. Our solvers support arbitrary dynamically adaptive grids and vertically integrate the multilevel operations through spacetree linearisation which yields good memory access characteristics, while standard colouring of mesh entities with a multiscale domain decomposition allow us to run the scheme on parallel computers.

Motivation & Objective

• To eliminate matrix assembly overhead in multigrid solvers for dynamically adaptive Cartesian grids.
• To enable robust multigrid convergence for complex elliptic problems on unstructured adaptive meshes.
• To support efficient parallel computation through multiscale domain decomposition and grid linearisation.
• To minimize memory usage by storing only algebraic operator differences from geometric ones.
• To integrate multilevel operations seamlessly within a spacetree data structure.

Proposed method

• The method uses a spacetree to represent the multigrid hierarchy, with coarse grids derived from the tree’s multiscale structure.
• Matrix-vector products are computed in-situ during grid traversals, avoiding explicit matrix storage.
• Algebraic operators are embedded as stencils and represented hierarchically as differences from geometric operators.
• The solver employs BoxMG-style prolongation and restriction operators tailored to problem and grid characteristics.
• A two-scale representation stores only the algebraic corrections to geometric operators, reducing memory footprint.
• Parallel execution is enabled via standard colouring of mesh entities and linearisation of the spacetree for coalesced memory access.

Experimental results

Research questions

• RQ1 How can matrix-free multigrid be efficiently implemented on dynamically adaptive Cartesian meshes without matrix assembly?
• RQ2 Can hierarchical representation of algebraic operators reduce memory usage while preserving solver robustness?
• RQ3 How does the spacetree-based multiscale structure support robust coarse grid transfer operators?
• RQ4 What is the performance impact of in-situ matrix-vector operations compared to traditional assembly?
• RQ5 Can the method scale efficiently on parallel architectures using multiscale domain decomposition?

Key findings

• The method eliminates explicit matrix assembly, significantly reducing overhead for dynamically adaptive grids.
• Hierarchical representation of algebraic operators reduces memory usage by storing only differences from geometric counterparts.
• The solver achieves robust convergence for nontrivial elliptic problems using problem- and grid-dependent prolongation and restriction.
• Spacetree linearisation enables good memory access patterns, enhancing cache efficiency.
• Vertical integration of multilevel operations through the spacetree supports efficient parallel execution.
• The approach enables scalable performance on parallel computers via multiscale domain decomposition and entity colouring.