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HYPRE Overview

HYPRE Fortran Framework

Introduction​

HYPRE is a flexible and scalable software library for solving large-scale scientific and engineering problems. It provides a wide range of linear solvers, preconditioners, and interfaces to other software packages. HYPRE is written in Fortran and is designed to be efficient on parallel computers.

History​

HYPRE was first developed in the late 1990s by researchers at the Lawrence Livermore National Laboratory (LLNL). It was originally designed to solve linear systems arising from the simulation of physical phenomena, such as fluid flow and structural mechanics. Over the years, HYPRE has been continuously enhanced and expanded to address a broader range of scientific and engineering applications.

Features​

HYPRE offers a rich set of features that make it a powerful tool for solving large-scale linear systems. Some of the notable features include:

  1. Parallel Support: HYPRE is designed to efficiently utilize parallel computing resources, such as distributed memory systems and multi-core processors. It employs various parallel algorithms and data structures to achieve high performance.

  2. Linear Solvers: HYPRE provides a collection of linear solvers, including iterative and direct methods. These solvers can be used to solve both symmetric and nonsymmetric systems of equations. The library supports various preconditioning techniques to improve the convergence of iterative solvers.

  3. Preconditioners: HYPRE offers a wide range of preconditioners, such as Jacobi, Gauss-Seidel, and multigrid methods. These preconditioners can be used to accelerate the convergence of iterative solvers by transforming the original system into a more easily solvable form.

  4. Interface to Other Libraries: HYPRE provides interfaces to other widely used software packages, such as PETSc and Trilinos. These interfaces allow users to seamlessly integrate HYPRE with other tools and take advantage of their capabilities.

  5. Extensibility: HYPRE is designed to be extensible, allowing users to add their own custom algorithms and preconditioners. This flexibility enables researchers and developers to tailor HYPRE to their specific needs and experiment with new solution techniques.

Examples​

Example 1: Solving a Linear System​

program hypre_example
  use HYPRE
  implicit none
  
  integer :: ierr, n, i
  integer, parameter :: size = 100
  real(kind=8), allocatable :: A(:,:), x(:), b(:)
  type(HYPRE_StructMatrix) :: A_hypre
  type(HYPRE_StructVector) :: x_hypre, b_hypre
  type(HYPRE_StructSolver) :: solver
  
  ! Initialize HYPRE
  call HYPRE_Init(ierr)
  
  ! Set the size of the problem
  n = size*size
  
  ! Allocate memory for the matrix and vectors
  allocate(A(n, n), x(n), b(n))
  
  ! Initialize the matrix and vectors
  A = 0.0
  x = 0.0
  b = 1.0
  
  ! Create the HYPRE matrix and vectors
  call HYPRE_StructMatrixCreate(MPI_COMM_WORLD, A_hypre, ierr)
  call HYPRE_StructVectorCreate(MPI_COMM_WORLD, x_hypre, ierr)
  call HYPRE_StructVectorCreate(MPI_COMM_WORLD, b_hypre, ierr)
  
  ! Set the matrix and vectors values
  call HYPRE_StructMatrixSetValues(A_hypre, n, n, [1,1], [n,n], A, ierr)
  call HYPRE_StructVectorSetValues(x_hypre, n, [1,1], [n,n], x, ierr)
  call HYPRE_StructVectorSetValues(b_hypre, n, [1,1], [n,n], b, ierr)
  
  ! Assemble the matrix and vectors
  call HYPRE_StructMatrixAssemble(A_hypre, ierr)
  call HYPRE_StructVectorAssemble(x_hypre, ierr)
  call HYPRE_StructVectorAssemble(b_hypre, ierr)
  
  ! Create the solver
  call HYPRE_StructPCGCreate(MPI_COMM_WORLD, solver, ierr)
  
  ! Set the solver parameters
  call HYPRE_StructPCGSetTol(solver, 1.0E-6, ierr)
  call HYPRE_StructPCGSetMaxIter(solver, 1000, ierr)
  
  ! Set the preconditioner
  call HYPRE_StructPCGSetPrecond(solver, HYPRE_StructPFMGSetup, HYPRE_StructPFMGSolve, ierr)
  
  ! Solve the linear system
  call HYPRE_StructPCGSetup(solver, A_hypre, b_hypre, x_hypre, ierr)
  call HYPRE_StructPCGSolve(solver, A_hypre, b_hypre, x_hypre, ierr)
  
  ! Get the solution vector
  call HYPRE_StructVectorGetValues(x_hypre, n, [1,1], [n,n], x, ierr)
  
  ! Clean up
  call HYPRE_StructMatrixDestroy(A_hypre, ierr)
  call HYPRE_StructVectorDestroy(x_hypre, ierr)
  call HYPRE_StructVectorDestroy(b_hypre, ierr)
  call HYPRE_StructPCGDestroy(solver, ierr)
  
  ! Finalize HYPRE
  call HYPRE_Finalize(ierr)
  
  ! Print the solution
  do i = 1, n
    print *, x(i)
  end do
  
end program hypre_example

In this example, we use HYPRE to solve a linear system of equations using the preconditioned conjugate gradient (PCG) method. We first initialize HYPRE and set the size of the problem. Then, we allocate memory for the matrix and vectors and initialize their values. Next, we create the HYPRE matrix and vectors, set their values, and assemble them. We create the PCG solver, set its parameters, and specify the preconditioner. Finally, we solve the linear system, retrieve the solution vector, and clean up the HYPRE objects. The solution is then printed.

Example 2: Using a Preconditioner​

program hypre_example
  use HYPRE
  implicit none
  
  integer :: ierr, n, i
  integer, parameter :: size = 100
  real(kind=8), allocatable :: A(:,:), x(:), b(:)
  type(HYPRE_StructMatrix) :: A_hypre
  type(HYPRE_StructVector) :: x_hypre, b_hypre
  type(HYPRE_StructSolver) :: solver
  type(HYPRE_StructSolver) :: preconditioner
  
  ! Initialize HYPRE
  call HYPRE_Init(ierr)
  
  ! Set the size of the problem
  n = size*size
  
  ! Allocate memory for the matrix and vectors
  allocate(A(n, n), x(n), b(n))
  
  ! Initialize the matrix and vectors
  A = 0.0
  x = 0.0
  b = 1.0
  
  ! Create the HYPRE matrix and vectors
  call HYPRE_StructMatrixCreate(MPI_COMM_WORLD, A_hypre, ierr)
  call HYPRE_StructVectorCreate(MPI_COMM_WORLD, x_hypre, ierr)
  call HYPRE_StructVectorCreate(MPI_COMM_WORLD, b_hypre, ierr)
  
  ! Set the matrix and vectors values
  call HYPRE_StructMatrixSetValues(A_hypre, n, n, [1,1], [n,n], A, ierr)
  call HYPRE_StructVectorSetValues(x_hypre, n, [1,1], [n,n], x, ierr)
  call HYPRE_StructVectorSetValues(b_hypre, n, [1,1], [n,n], b, ierr)
  
  ! Assemble the matrix and vectors
  call HYPRE_StructMatrixAssemble(A_hypre, ierr)
  call HYPRE_StructVectorAssemble(x_hypre, ierr)
  call HYPRE_StructVectorAssemble(b_hypre, ierr)
  
  ! Create the solver
  call HYPRE_StructPCGCreate(MPI_COMM_WORLD, solver, ierr)
  
  ! Set the solver parameters
  call HYPRE_StructPCGSetTol(solver, 1.0E-6, ierr)
  call HYPRE_StructPCGSetMaxIter(solver, 1000, ierr)
  
  ! Create the preconditioner
  call HYPRE_StructPFMGCreate(MPI_COMM_WORLD, preconditioner, ierr)
  
  ! Set the preconditioner parameters
  call HYPRE_StructPFMGSetTol(preconditioner, 0.01, ierr)
  call HYPRE_StructPFMGSetMaxIter(preconditioner, 100, ierr)
  
  ! Set the preconditioner options
  call HYPRE_StructPFMGSetRelaxType(preconditioner, 1, ierr)
  call HYPRE_StructPFMGSetNumPreRelax(preconditioner, 1, ierr)
  call HYPRE_StructPFMGSetNumPostRelax(preconditioner, 1, ierr)
  
  ! Set the preconditioner
  call HYPRE_StructPCGSetPrecond(solver, HYPRE_StructPFMGSetup, HYPRE_StructPFMGSolve, ierr)
  
  ! Solve the linear system
  call HYPRE_StructPCGSetup(solver, A_hypre, b_hypre, x_hypre, ierr)
  call HYPRE_StructPCGSolve(solver, A_hypre, b_hypre, x_hypre, ierr)
  
  ! Get the solution vector
  call HYPRE_StructVectorGetValues(x_hypre, n, [1,1], [n,n], x, ierr)
  
  ! Clean up
  call HYPRE_StructMatrixDestroy(A_hypre, ierr)
  call HYPRE_StructVectorDestroy(x_hypre, ierr)
  call HYPRE_StructVectorDestroy(b_hypre, ierr)
  call HYPRE_StructPCGDestroy(solver, ierr)
  call HYPRE_StructPFMGDestroy(preconditioner, ierr)
  
  ! Finalize HYPRE
  call HYPRE_Finalize(ierr)
  
  ! Print the solution
  do i = 1, n
    print *, x(i)
  end do
  
end program hypre_example

In this example, we use HYPRE to solve a linear system of equations using the PCG method with a preconditioner. The overall structure of the code is similar to Example 1, but we add the creation and setup of a preconditioner before solving the linear system. The preconditioner used here is the geometric multigrid method (HYPRE_StructPFMG). The parameters of the preconditioner, such as tolerance and relaxation type, are set before solving the system.

Conclusion​

HYPRE is a powerful Fortran framework for solving large-scale linear systems. It offers a wide range of linear solvers, preconditioners, and interfaces to other software packages. Its parallel support and extensibility make it a valuable tool for researchers and developers working on scientific and engineering problems. For more information and detailed documentation, you can visit the official HYPRE website.