MCGS - Multicolor Gauss-Seidel Smoother

MCGS is a lightweight library for performing parallelized Gauss-Sidel smoothing, focusing on sparse systems with imbalanced topologies. Implementations for related tasks such as graph coloring and reordering are included as well.

Features

• Gauss-Seidel iterations in parallel (OpenMP) or serial
• SOR (successive over-relaxation) in parallel (OpenMP) or serial
• Graph coloring in parallel (loose implementation of doi:10.1006/jagm.2000.1097)
• Matrix/vector reordering and reverse reordering

Usage

Typical workflow

• construct your linear system $A x = b$
• construct and adaptor for $A$ (see mcgs::CSRAdaptor)
• compute a coloring of $A$ (see mcgs::color)
• construct a partition of $A$ with respect to the coloring (see mcgs::makePartition)
• reorder the system with respect to the coloring (see mcgs::reorder)
• perform Gauss-Seidel iterations using the reordered partition (see mcgs::solve)
• optional: restore the original order of your system (see mcgs::revertReorder)
• deallocate partitions (see mcgs::destroyPartition)

C++ Example Snippet

#include "mcgs/mcgs.hpp"

...

// Any CSR matrix will do but for the sake of familiarity, let's assume you're using Eigen.
Eigen::SparseMatrix<double,Eigen::RowMajor> A;  // <== left hand side matrix
Eigen::Matrix<double,Eigen::Dynamic,1> b;       // <== right hand side vector

// Construct an adaptor for your matrix.
mcgs::CSRAdaptor</*index type=*/ int, /*value type=*/double> adaptor;
adaptor.rowCount        = A.rows();
adaptor.columnCount     = A.cols();
adaptor.entryCount      = A.nonZeros();
adaptor.pRowExtents     = A.outerIndexPtr();
adaptor.pColumnIndices  = A.innerIndexPtr();
adaptor.pEntries        = A.innerNonZeroPtr();

// Color the rows of your matrix.
std::vector<unsigned> colors(adaptor.rowCount);
mcgs::color(colors.data(),
            adaptor,
            mcgs::ColorSettings {});

// Construct a partition for your matrix with respect to the coloring,
// and reorder the system accordingly. Note that this mutates your original matrix!
auto pPartition = mcgs::makePartition(colors.data(), adaptor.rowCount);
auto pReorderedPartition = mcgs::reorder(A.rows(), A.cols(), A.nonZeros(),
                                         A.outerIndexPtr(), A.innerIndexPtr(), A.innerNonZeroPtr(),
                                         b.data());

// Do 10 Gauss-Seidel iterations.
std::vector<double> x(adaptor.columnCount);
mcgs::SolveSettings</*index type=*/int, /*value type=*/double> settings;
settings.maxIterations = 10;
settings.parallelization = mcgs::Parallelization::RowWise; // <== default parallelization strategy, check out the other ones as well.
mcgs::solve(x.data(), adaptor, b.data(), pReorderedPartition, settings);

// Optional: if you need to recover your original system,
//           you need to undo the reordering.
// See mcgs::revertReorder

// Cleanup
mcgs::destroyPartition(pPartition);
mcgs::destroyPartition(pReorderedPartition);

Requirements

• C++ compiler with full C++17 support (GCC, Clang and MSVC are tested)
• CMake version 3.15 or later
• [optional] OpenMP 2.0 or later for shared memory parallelization
• Linux, MacOS or Windows

Installation

MCGS is written in C++ and uses CMake as its build system. Building produces a single shared library and matching header.

Build and Install

cmake <path-to-mcgs-source>                       \
      -B<path-to-build-dir>                       \
      -DCMAKE_INSTALL_PREFIX=<your-install-dir>   \
      -DCMAKE_BUILD_TYPE=Release                  \
      -DMCGS_SHARED_MEMORY_PARALLELISM=OpenMP

cmake <path-to-build-dir> --build --target install

Build options

• CMake options for MCGS_SHARED_MEMORY_PARALLELISM
  • None: no parallelization
  • OpenMP: use OpenMP for shared memory parallelization

Include MCGS in a CMake project

find_package(MCGS REQUIRED)
target_link_libraries(<your_target> PRIVATE mcgs)