%==========================================================================
\chapter{Structured-Grid System Interface (Struct)}
\label{Structured-Grid System Interface}

In order to get access to the most efficient and scalable solvers for
scalar structured-grid applications, users should use the
\code{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 (\code{FEI}) or
the linear-algebraic interface (\code{IJ}) described in Chapters
\ref{Finite Element Interface} and \ref{Linear-Algebraic System Interface}.

Figure \ref{fig-fv-grid} gives an example of the type of grid
currently supported by the \code{Struct} interface.  The interface
uses a finite-difference or finite-volume style, and currently
supports only scalar PDEs (i.e., one unknown per gridpoint).
\begin{figure}[t]
\centering
\includegraphics[width=4in]{fv_grid.eps}
\caption{%
An example 2D structured grid, distributed accross two processors.}
\label{fig-fv-grid}
\end{figure}
There are four basic steps involved in setting up the linear system
to be solved:
\begin{enumerate}
\item set up the grid,
\item set up the stencil,
\item set up the matrix,
\item set up the right-hand-side vector.
\end{enumerate}
To describe each of these steps in more detail, consider solving the
2D Laplacian problem
\begin{equation}\label{eqn-laplacian}
\left \{
\begin{array}{ll}
\nabla^2 u = f , & \mbox{in the domain}, \\
u = 0,           & \mbox{on the boundary}.
\end{array}
\right .
\end{equation}
Assume (\ref{eqn-laplacian}) is discretized using standard 5-pt
finite-volumes on the uniform grid pictured in \ref{fig-fv-grid}, and
assume that the problem data is distributed across two processes as
depicted.

%==========================================================================

\section{Setting Up the Struct Grid}
\label{Setting Up the Struct Grid}

The grid is described via a global {\em index space}, i.e., via
integer tuples (triples in 3D).  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.  Each process describes that portion of the grid
that it ``owns'', one {\em box} at a time.  For example, in the
figure, the global grid can be described in terms of three boxes, two
owned by process 0, and one owned by process 1.  A box is described in
terms of a lower and upper index.

On process 0, the following code will set up the grid shown in the
figure (the code for process 1 is similar).
\begin{display}
\begin{verbatim}

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

HYPRE_StructGridCreate(MPI_COMM_WORLD, 2, &grid);

HYPRE_StructGridSetExtents(grid, ilower[0], iupper[0]);
HYPRE_StructGridSetExtents(grid, ilower[1], iupper[1]);

HYPRE_StructGridAssemble(grid);

\end{verbatim}
\end{display}
The \code{Create()} routine creates an empty 2D grid object that lives
on the \code{MPI_COMM_WORLD} communicator.  The \code{SetExtents()}
routine adds a new box to the grid.  The \code{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''.

%==========================================================================

\section{Setting Up the Struct Stencil}
\label{Setting Up the Struct Stencil}

The geometry of the discretization stencil is described by an array of
integer tuples in 2D (triples in 3D), each representing a relative
offset (in index space) from some gridpoint on the grid.  For example,
the geometry of the 5-pt stencil for the example problem being
considered can be represented in the following way:
\begin{equation}\label{eqn-stencil-description}
\left [
\begin{array}{ccc}
        & ( 0, 1) &         \\
(-1, 0) & ( 0, 0) & ( 1, 0) \\
        & ( 0,-1) &        
\end{array}
\right ]
\equiv
\left [
\begin{array}{ccc}
    & S_4 &     \\
S_1 & S_0 & S_2 \\
    & S_3 &    
\end{array}
\right ] .
\end{equation}
In (\ref{eqn-stencil-description}), the $(0,0)$ entry represents the
``center'' coefficient, and is the 0th entry in the array ($S_0$).
The $(0,-1)$ entry represents the ``south'' coefficient, and is the
3rd entry in the array ($S_3$).  And so on.

On process 0 or 1, the following code will set up the stencil in
(\ref{eqn-stencil-description}).  The stencil must be the same on all
processes.
\begin{display}
\begin{verbatim}

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

HYPRE_StructStencilCreate(2, 5, &stencil);

for (s = 0; s < 5; s++)
{
   HYPRE_StructStencilSetElement(stencil, s, offsets[s]);
}

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

%==========================================================================

\section{Setting Up the Struct Matrix}
\label{Setting Up the Struct Matrix}

The matrix is set up in terms of the grid and stencil objects
described in Sections
\ref{Setting Up the Struct Grid} and \ref{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):
\begin{equation}\label{eqn-stencil-laplacian}
\left [
\begin{array}{ccc}
    & -1 &    \\
 -1 &  4 & -1 \\
    & -1 &    
\end{array}
\right ] .
\end{equation}

On process 0, the following code will set up matrix values associated
with the center ($S_0$) and south ($S_3$) stencil entries in
(\ref{eqn-stencil-description}) / (\ref{eqn-stencil-laplacian})
(boundaries are ignored here temporarily).
\begin{display}
\begin{verbatim}

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);

\end{verbatim}
\end{display}
The \code{Create()} routine creates an empty matrix object.  The
\code{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 \code{Set} routines mentioned later in
this chapter and in the Reference Manual, should be called before this
step.  The \code{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 \code{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-fv-grid}, the
``south'' coefficient must be set to zero.  To do this on process 0,
the following code could be used:
\begin{display}
\begin{verbatim}

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 */
...

\end{verbatim}
\end{display}

%==========================================================================

\section{Setting Up the Struct Right-Hand-Side Vector}
\label{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{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 will set up the right-hand-side
vector values.
\begin{display}
\begin{verbatim}

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);

\end{verbatim}
\end{display}

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

%==========================================================================

\section{Symmetric Matrices}
\label{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
\code{Struct} interface provides support for matrix and solver
implementations that use symmetric storage via the
\code{SetSymmetric()} routine.

To describe this in more detail, consider again the 5-pt finite-volume
discretization of (\ref{eqn-laplacian}) on the grid pictured in Figure
\ref{fig-fv-grid}.  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 \code{Create()} and \code{Initialize()} calls.
\begin{display}
\begin{verbatim}

HYPRE_StructMatrixSetSymmetric(A, 1);

\end{verbatim}
\end{display}
Note that symmetric storage may or may not actually be used, depending
on the underlying storage scheme.  Currently in \hypre{}, symmetric
storage is always used when indicated.

To most efficiently utilize the \code{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{Setting Up the Struct Stencil} to an ``appropriate'' 3-pt stencil, then
set matrix coefficients (as in Section \ref{Setting Up the Struct Matrix})
for these three stencil elements {\em only}.  For example, we could
use the following stencil
\begin{equation}\label{eqn-symmetric-stencil}
\left [
\begin{array}{ccc}
 & ( 0, 1) &         \\
 & ( 0, 0) & ( 1, 0) \\
 &         &        
\end{array}
\right ]
\equiv
\left [
\begin{array}{ccc}
 & S_2 &     \\
 & S_0 & S_1 \\
 &     &    
\end{array}
\right ] .
\end{equation}
This 3-pt stencil provides enough information to recover the full 5-pt
stencil geometry and associated matrix coefficients.

%==========================================================================
