Structured-Grid System Interface (Struct)

In order to get access to the most efficient and scalable solvers for scalar structured-grid applications, users should use the Struct interface described in this chapter. This interface will also provide access (this is not yet supported) to solvers in hypre that were designed for unstructured-grid applications and sparse linear systems in general. These additional solvers are usually provided via the unstructured-grid interface (FEI) or the linear-algebraic interface (IJ) described in Chapters Finite Element Interface and Linear-Algebraic System Interface (IJ).

Figure Structured Grid Example gives an example of the type of grid currently supported by the Struct interface. The interface uses a finite-difference or finite-volume style, and currently supports only scalar PDEs (i.e., one unknown per gridpoint).

_images/figStructExample1.svg

Structured Grid Example

An example 2D structured grid, distributed accross two processors.

There are four basic steps involved in setting up the linear system to be solved:

  1. set up the grid,

  2. set up the stencil,

  3. set up the matrix,

  4. set up the right-hand-side vector.

To describe each of these steps in more detail, consider solving the 2D Laplacian problem

(1)\[\begin{split}\left \{ \begin{array}{ll} \nabla^2 u = f , & \mbox{in the domain}, \\ u = 0, & \mbox{on the boundary}. \end{array} \right .\end{split}\]

Assume (1) is discretized using standard 5-pt finite-volumes on the uniform grid pictured in Structured Grid Example, and assume that the problem data is distributed across two processes as depicted.

Setting Up the Struct Grid

The grid is described via a global index space, i.e., via integer singles in 1D, tuples in 2D, or triples in 3D (see Figure Boxes in Index Space).

_images/figStructGridBoxes.svg

Boxes in Index Space

A box is a collection of abstract cell-centered indices, described by its minimum and maximum indices. Here, two boxes are illustrated.

The integers may have any value, negative or positive. The global indexes allow hypre to discern how data is related spatially, and how it is distributed across the parallel machine. The basic component of the grid is a box: a collection of abstract cell-centered indices in index space, described by its “lower” and “upper” corner indices. The scalar grid data is always associated with cell centers, unlike the more general SStruct interface which allows data to be associated with box indices in several different ways.

Each process describes that portion of the grid that it “owns”, one box at a time. For example, the global grid in Figure Structured Grid Example can be described in terms of three boxes, two owned by process 0, and one owned by process 1. The following is the code (with visual annotations) for setting up the grid on process 0 (the code for process 1 is similar).

1: figStructGrid1

2: figStructGrid2

3: figStructGrid3

4: figStructGrid4

    HYPRE_StructGrid grid;
    int ndim        = 2;
    int ilower[][2] = {{-3,1}, {0,1}};
    int iupper[][2] = {{-1,2}, {2,4}};

    /* Create the grid object */
1:  HYPRE_StructGridCreate(MPI_COMM_WORLD, ndim, &grid);

    /* Set grid extents for the first box */
2:  HYPRE_StructGridSetExtents(grid, ilower[0], iupper[0]);

    /* Set grid extents for the second box */
3:  HYPRE_StructGridSetExtents(grid, ilower[1], iupper[1]);

    /* Assemble the grid */
4:  HYPRE_StructGridAssemble(grid);

The images along the top illustrate the result of the numbered lines of code. The Create() routine creates an empty 2D grid object that lives on the MPI_COMM_WORLD communicator. The SetExtents() routine adds a new box to the grid. The Assemble() routine is a collective call (i.e., must be called on all processes from a common synchronization point), and finalizes the grid assembly, making the grid “ready to use”.

Setting Up the Struct Stencil

The geometry of the discretization stencil is described by an array of indexes, each representing a relative offset from any given gridpoint on the grid. For example, the geometry of the 5-pt stencil for the example problem being considered can be represented by the list of index offsets shown in Figure Figure 4a.

_images/figStructStenc0.svg

Figure 4a

Representation of the 5-point discretization stencil for the example problem.

_images/figStructStenc7.svg

Figure 4b

Need to combine this with 4a.

Here, the \((0,0)\) entry represents the “center” coefficient, and is the 0th stencil entry. The \((0,-1)\) entry represents the “south” coefficient, and is the 3rd stencil entry. And so on.

On process 0 or 1, the following code (with visual annotations) will set up the stencil in Figure Figure 4a. The stencil must be the same on all processes.

1: figStructStenc1

2: figStructStenc2

3: figStructStenc3

4: figStructStenc4

5: figStructStenc5

6: figStructStenc6

      HYPRE_StructStencil stencil;
      int ndim         = 2;
      int size         = 5;
      int entry;
      int offsets[][2] = {{0,0}, {-1,0}, {1,0}, {0,-1}, {0,1}};

      /* Create the stencil object */
  1:  HYPRE_StructStencilCreate(ndim, size, &stencil);

      /* Set stencil entries */
      for (entry = 0; entry < size; entry++)
      {
2-6:     HYPRE_StructStencilSetElement(stencil, entry, offsets[entry]);
      }

      /* Thats it!  There is no assemble routine */

The Create() routine creates an empty 2D, 5-pt stencil object. The SetElement() routine defines the geometry of the stencil and assigns the stencil numbers for each of the stencil entries. None of the calls are collective calls.

Setting Up the Struct Matrix

The matrix is set up in terms of the grid and stencil objects described in Sections Setting Up the Struct Grid and Setting Up the Struct Stencil. The coefficients associated with each stencil entry will typically vary from gridpoint to gridpoint, but in the example problem being considered, they are as follows over the entire grid (except at boundaries; see below):

(2)\[\begin{split}\left [ \begin{array}{ccc} & -1 & \\ -1 & 4 & -1 \\ & -1 & \end{array} \right ] .\end{split}\]

On process 0, the following code sets up matrix values associated with the center (entry 0) and south (entry 3) stencil entries as given by (2) and Figure Figure 4a (boundaries are ignored here temporarily).

HYPRE_StructMatrix  A;
double              values[36];
int                 stencil_indices[2] = {0,3};
int                 i;

HYPRE_StructMatrixCreate(MPI_COMM_WORLD, grid, stencil, &A);
HYPRE_StructMatrixInitialize(A);

for (i = 0; i < 36; i += 2)
{
   values[i]   =  4.0;
   values[i+1] = -1.0;
}

HYPRE_StructMatrixSetBoxValues(A, ilower[0], iupper[0], 2,
                               stencil_indices, values);
HYPRE_StructMatrixSetBoxValues(A, ilower[1], iupper[1], 2,
                               stencil_indices, values);

/* set boundary conditions */
...

HYPRE_StructMatrixAssemble(A);

The Create() routine creates an empty matrix object. The Initialize() routine indicates that the matrix coefficients (or values) are ready to be set. This routine may or may not involve the allocation of memory for the coefficient data, depending on the implementation. The optional Set routines mentioned later in this chapter and in the Reference Manual, should be called before this step. The SetBoxValues() routine sets the matrix coefficients for some set of stencil entries over the gridpoints in some box. Note that the box need not correspond to any of the boxes used to create the grid, but values should be set for all gridpoints that this process “owns”. The Assemble() routine is a collective call, and finalizes the matrix assembly, making the matrix “ready to use”.

Matrix coefficients that reach outside of the boundary should be set to zero. For efficiency reasons, hypre does not do this automatically. The most natural time to insure this is when the boundary conditions are being set, and this is most naturally done after the coefficients on the grid’s interior have been set. For example, during the implementation of the Dirichlet boundary condition on the lower boundary of the grid in Figure Structured Grid Example, the south coefficient must be set to zero. To do this on process 0, the following code could be used:

int  ilower[2] = {-3, 1};
int  iupper[2] = { 2, 1};

/* create matrix and set interior coefficients */
...

/* implement boundary conditions */
...

for (i = 0; i < 12; i++)
{
   values[i] =  0.0;
}

i = 3;
HYPRE_StructMatrixSetBoxValues(A, ilower, iupper, 1, &i, values);

/* complete implementation of boundary conditions */
...

Setting Up the Struct Right-Hand-Side Vector

The right-hand-side vector is set up similarly to the matrix set up described in Section Setting Up the Struct Matrix above. The main difference is that there is no stencil (note that a stencil currently does appear in the interface, but this will eventually be removed).

On process 0, the following code sets up the right-hand-side vector values.

HYPRE_StructVector  b;
double              values[18];
int                 i;

HYPRE_StructVectorCreate(MPI_COMM_WORLD, grid, &b);
HYPRE_StructVectorInitialize(b);

for (i = 0; i < 18; i++)
{
   values[i]   =  0.0;
}

HYPRE_StructVectorSetBoxValues(b, ilower[0], iupper[0], values);
HYPRE_StructVectorSetBoxValues(b, ilower[1], iupper[1], values);

HYPRE_StructVectorAssemble(b);

The Create() routine creates an empty vector object. The Initialize() routine indicates that the vector coefficients (or values) are ready to be set. This routine follows the same rules as its corresponding Matrix routine. The SetBoxValues() routine sets the vector coefficients over the gridpoints in some box, and again, follows the same rules as its corresponding Matrix routine. The Assemble() routine is a collective call, and finalizes the vector assembly, making the vector “ready to use”.

Symmetric Matrices

Some solvers and matrix storage schemes provide capabilities for significantly reducing memory usage when the coefficient matrix is symmetric. In this situation, each off-diagonal coefficient appears twice in the matrix, but only one copy needs to be stored. The Struct interface provides support for matrix and solver implementations that use symmetric storage via the SetSymmetric() routine.

To describe this in more detail, consider again the 5-pt finite-volume discretization of (1) on the grid pictured in Figure Structured Grid Example. Because the discretization is symmetric, only half of the off-diagonal coefficients need to be stored. To turn symmetric storage on, the following line of code needs to be inserted somewhere between the Create() and Initialize() calls.

HYPRE_StructMatrixSetSymmetric(A, 1);

The coefficients for the entire stencil can be passed in as before. Note that symmetric storage may or may not actually be used, depending on the underlying storage scheme. Currently in hypre, the Struct interface always uses symmetric storage.

To most efficiently utilize the Struct interface for symmetric matrices, notice that only half of the off-diagonal coefficients need to be set. To do this for the example being considered, we simply need to redefine the 5-pt stencil of Section Setting Up the Struct Stencil to an “appropriate” 3-pt stencil, then set matrix coefficients (as in Section Setting Up the Struct Matrix) for these three stencil elements only. For example, we could use the following stencil

(3)\[\begin{split}\left [ \begin{array}{ccc} ~~~~~~ & ( 0, 1) & \\ ~~~~~~ & ( 0, 0) & ( 1, 0) \\ ~~~~~~ & & \end{array} \right ] .\end{split}\]

This 3-pt stencil provides enough information to recover the full 5-pt stencil geometry and associated matrix coefficients.