.. Copyright (c) 1998 Lawrence Livermore National Security, LLC and other
   HYPRE Project Developers. See the top-level COPYRIGHT file for details.

   SPDX-License-Identifier: (Apache-2.0 OR MIT)


.. _ch-Struct:

******************************************************************************
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 :ref:`ch-FEI` and
:ref:`ch-IJ`.

Figure :ref:`fig-struct-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).

.. _fig-struct-example:

.. figure:: figStructExample1.*
   :align: center

   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:

#. set up the grid,
#. set up the stencil,
#. set up the matrix,
#. set up the right-hand-side vector.

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

.. math::
   :label: eqn-laplacian

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

Assume :eq:`eqn-laplacian` is discretized using standard 5-pt finite-volumes on
the uniform grid pictured in :ref:`fig-struct-example`, and assume that the
problem data is distributed across two processes as depicted.


.. _sec-Struct-Grid:

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 :ref:`fig-struct-boxes`).

.. _fig-struct-boxes:

.. figure:: figStructGridBoxes.*
   :align: center

   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 :ref:`fig-struct-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).

.. |figStructGrid1| image:: figStructGrid1.*
   :width: 100%
.. |figStructGrid2| image:: figStructGrid2.*
   :width: 100%
.. |figStructGrid3| image:: figStructGrid3.*
   :width: 100%
.. |figStructGrid4| image:: figStructGrid4.*
   :width: 100%

+---------------------+---------------------+---------------------+---------------------+
| 1: |figStructGrid1| | 2: |figStructGrid2| | 3: |figStructGrid3| | 4: |figStructGrid4| |
+---------------------+---------------------+---------------------+---------------------+

.. code-block:: c
   
       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".


.. _sec-Struct-Stencil:

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
:ref:`fig-struct-stencil-a`.

.. _fig-struct-stencil-a:

.. figure:: figStructStenc0.*
   :align: center

   Figure 4a

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

.. figure:: figStructStenc7.*
   :align: center

   Figure 4b

   Need to combine this with 4a.

Here, the :math:`(0,0)` entry represents the "center" coefficient, and is the
0th stencil entry.  The :math:`(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 :ref:`fig-struct-stencil-a`.  The stencil must be the same on
all processes.

.. |figStructStenc1| image:: figStructStenc1.*
   :width: 100%
.. |figStructStenc2| image:: figStructStenc2.*
   :width: 100%
.. |figStructStenc3| image:: figStructStenc3.*
   :width: 100%
.. |figStructStenc4| image:: figStructStenc4.*
   :width: 100%
.. |figStructStenc5| image:: figStructStenc5.*
   :width: 100%
.. |figStructStenc6| image:: figStructStenc6.*
   :width: 100%

+----------------------+----------------------+----------------------+
| 1: |figStructStenc1| | 2: |figStructStenc2| | 3: |figStructStenc3| |
+----------------------+----------------------+----------------------+ 
| 4: |figStructStenc4| | 5: |figStructStenc5| | 6: |figStructStenc6| |
+----------------------+----------------------+----------------------+

.. code-block:: c
   
         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.


.. _sec-Struct-Matrix:

Setting Up the Struct Matrix
==============================================================================

The matrix is set up in terms of the grid and stencil objects described in
Sections :ref:`sec-Struct-Grid` and :ref:`sec-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):

.. math::
   :label: eqn-stencil-laplacian

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

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
:eq:`eqn-stencil-laplacian` and Figure :ref:`fig-struct-stencil-a` (boundaries
are ignored here temporarily).

.. _fig-struct-matrix:

.. code-block:: c
   
   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 :ref:`fig-struct-example`, the south
coefficient must be set to zero.  To do this on process 0, the following code
could be used:

.. _fig-struct-matrix-boundary:

.. code-block:: c
   
   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 */
   ...


.. _sec-Struct-RHS:

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 :ref:`sec-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.

.. _fig-struct-rhs:

.. code-block:: c

   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".


.. _sec-Symmetric-Matrices:

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 :eq:`eqn-laplacian` on the grid pictured in Figure
:ref:`fig-struct-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.

.. code-block:: c
   
   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 :ref:`sec-Struct-Stencil` to an "appropriate" 3-pt stencil,
then set matrix coefficients (as in Section :ref:`sec-Struct-Matrix`) for these
three stencil elements *only*.  For example, we could use the following stencil

.. math::
   :label: eqn-symmetric-stencil

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

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