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# Vectors and Parallel Data

Vectors (denoted by `Vec`) are used to store discrete PDE solutions, right-hand sides for linear systems, etc. Users can create and manipulate entries in vectors directly with a basic, low-level interface or they can use the PETSc `DM` objects to connect actions on vectors to the type of discretization and grid that they are working with. These higher-level interfaces handle much of the details of the interactions with vectors and hence, are preferred in most situations. This chapter is organized as follows:

- Creating Vectors

  - User managed

  - DMDA - Creating vectors for structured grids

  - DMSTAG - Creating vectors for staggered grids

  - DMPLEX - Creating vectors for unstructured grids

  - DMNETWORK - Creating vectors for networks

- Setting vector values

  - For generic vectors

  - DMDA - Setting vector values

  - DMSTAG - Setting vector values

  - DMPLEX - Setting vector values

  - DMNETWORK - Setting vector values

- Basic Vector Operations

- Local/global vectors and communicating between vectors

- Low-level Vector Communication

  - Local to global mappings

  - Global Vectors with locations for ghost values

- Application Orderings

## Creating Vectors

PETSc provides many ways to create vectors. The most basic, where the user is responsible for managing the parallel distribution of the vector entries, and a variety of higher-level approaches, based on `DM`, for classes of problems such as structured grids, staggered grids, unstructured grids, networks, and particles.

The most basic way to create a vector with a local size of `m` and a global size of `M`, is to use

    VecCreate(MPI_Comm comm,Vec *v);
    VecSetSizes(Vec v, PetscInt m, PetscInt M);
    VecSetFromOptions(Vec v);

which automatically generates the appropriate vector type (sequential or parallel) over all processes in `comm`. The option `-vec_type <type>` can be used in conjunction with `VecSetFromOptions()` to specify the use of a particular type of vector. For example, for NVIDIA GPU CUDA, use `cuda`. The GPU-based vectors allow one to set values on either the CPU or GPU but do their computations on the GPU.

We emphasize that all processes in `comm` must call the vector creation routines since these routines are collective on all processes in the communicator. If you are unfamiliar with MPI communicators, see the discussion in Writing PETSc Programs. In addition, if a sequence of creation routines is used, they must be called in the same order for each process in the communicator.

Instead of, or before calling `VecSetFromOptions()`, one can call

    VecSetType(Vec v,VecType <VECCUDA, VECHIP, VECKOKKOS etc>)

One can create vectors whose entries are stored on GPUs using the convenience routine,

    VecCreateMPICUDA(MPI_Comm comm,PetscInt m,PetscInt M,Vec *x);

There are convenience creation routines for almost all vector types; we recommend using the more verbose form because it allows selecting CPU or GPU simulations at runtime.

For applications running in parallel that involve multi-dimensional structured grids, unstructured grids, networks, etc, it is cumbersome for users to explicitly manage the needed local and global sizes of the vectors. Hence, PETSc provides a powerful abstract object called the `DM` to help manage the vectors and matrices needed for such applications. Parallel vectors can be created easily with

    DMCreateGlobalVector(DM dm,Vec *v)

The `DM` object, see DMDA - Creating vectors for structured grids, DMSTAG - Creating vectors for staggered grids, and DMPlex: Unstructured Grids for more details on `DM` for structured grids, staggered structured grids, and for unstructured grids, manages creating the correctly sized parallel vectors efficiently. One controls the type of vector that `DM` creates by calling

    DMSetVecType(DM dm,VecType vt)

or by calling `DMSetFromOptions(DM dm)` and using the option `-dm_vec_type <standard or cuda or kokkos etc>`

### DMDA - Creating vectors for structured grids

Each `DM` type is suitable for a family of problems. The first of these, `DMDA` are intended for use with logically structured rectangular grids when communication of nonlocal data is needed before certain local computations can occur. `DMDA` is designed only for the case in which data can be thought of as being stored in a standard multidimensional array; thus, `DMDA` are not intended for parallelizing unstructured grid problems, etc.

For example, a typical situation one encounters in solving PDEs in parallel is that, to evaluate a local function, `f(x)`, each process requires its local portion of the vector `x` as well as its ghost points (the bordering portions of the vector that are owned by neighboring processes). Figure Ghost Points for Two Stencil Types on the Seventh Process illustrates the ghost points for the seventh process of a two-dimensional, structured parallel grid. Each box represents a process; the ghost points for the seventh process’s local part of a parallel array are shown in gray.

Fig. 3 Ghost Points for Two Stencil Types on the Seventh Process

The `DMDA` object contains parallel data layout information and communication information and is used to create vectors and matrices with the proper layout.

One creates a `DMDA` two dimensions with the convenience routine

    DMDACreate2d(MPI_Comm comm,DMBoundaryType xperiod,DMBoundaryType yperiod,DMDAStencilType st,PetscInt M, PetscInt N,PetscInt m,PetscInt n,PetscInt dof,PetscInt s,PetscInt *lx,PetscInt *ly,DM *da);

The arguments `M` and `N` indicate the global numbers of grid points in each direction, while `m` and `n` denote the process partition in each direction; `m*n` must equal the number of processes in the MPI communicator, `comm`. Instead of specifying the process layout, one may use `PETSC_DECIDE` for `m` and `n` so that PETSc will select the partition. The type of periodicity of the array is specified by `xperiod` and `yperiod`, which can be `DM_BOUNDARY_NONE` (no periodicity), `DM_BOUNDARY_PERIODIC` (periodic in that direction), `DM_BOUNDARY_TWIST` (periodic in that direction, but identified in reverse order), `DM_BOUNDARY_GHOSTED` , or `DM_BOUNDARY_MIRROR`. The argument `dof` indicates the number of degrees of freedom at each array point, and `s` is the stencil width (i.e., the width of the ghost point region). The optional arrays `lx` and `ly` may contain the number of nodes along the x and y axis for each cell, i.e. the dimension of `lx` is `m` and the dimension of `ly` is `n`; alternately, `NULL` may be passed in.

Two types of `DMDA` communication data structures can be created, as specified by `st`. Star-type stencils that radiate outward only in the coordinate directions are indicated by `DMDA_STENCIL_STAR`, while box-type stencils are specified by `DMDA_STENCIL_BOX`. For example, for the two-dimensional case, `DMDA_STENCIL_STAR` with width 1 corresponds to the standard 5-point stencil, while `DMDA_STENCIL_BOX` with width 1 denotes the standard 9-point stencil. In both instances, the ghost points are identical, the only difference being that with star-type stencils, certain ghost points are ignored, substantially decreasing the number of messages sent. Note that the `DMDA_STENCIL_STAR` stencils can save interprocess communication in two and three dimensions.

These `DMDA` stencils have nothing directly to do with a specific finite difference stencil one might choose to use for discretization; they only ensure that the correct values are in place for the application of a user-defined finite difference stencil (or any other discretization technique).

The commands for creating `DMDA` in one and three dimensions are analogous:

    DMDACreate1d(MPI_Comm comm,DMBoundaryType xperiod,PetscInt M,PetscInt w,PetscInt s,PetscInt *lc,DM *inra);

    DMDACreate3d(MPI_Comm comm,DMBoundaryType xperiod,DMBoundaryType yperiod,DMBoundaryType zperiod, DMDAStencilType stencil_type,PetscInt M,PetscInt N,PetscInt P,PetscInt m,PetscInt n,PetscInt p,PetscInt w,PetscInt s,PetscInt *lx,PetscInt *ly,PetscInt *lz,DM *inra);

The routines to create a `DM` are collective so that all processes in the communicator `comm` must call the same creation routines in the same order.

A DM may be created, and its type set with

    DMCreate(comm,&dm);
    DMSetType(dm,"Typename");  // for example, "DMDA"

Then `DMType` specific operations can be performed to provide information from which the specifics of the `DM` will be provided. For example,

    DMSetDimension(dm, 1);
    DMDASetSizes(dm, M, 1, 1));
    DMDASetDof(dm, 1));
    DMSetUp(dm);

We now very briefly introduce a few more `DMType`.

### DMSTAG - Creating vectors for staggered grids

For structured grids with staggered data (living on elements, faces, edges, and/or vertices), the `DMSTAG` object is available. It behaves much like `DMDA`. See DMSTAG: Staggered, Structured Grid for discussion of creating vectors with `DMSTAG`.

### DMPLEX - Creating vectors for unstructured grids

See DMPlex: Unstructured Grids for a discussion of creating vectors with `DMPLEX`.

### DMNETWORK - Creating vectors for networks

See Networks for discussion of creating vectors with `DMNETWORK`.

## Common vector functions and operations

One can examine (print out) a vector with the command

    VecView(Vec x,PetscViewer v);

To print the vector to the screen, one can use the viewer `PETSC_VIEWER_STDOUT_WORLD`, which ensures that parallel vectors are printed correctly to `stdout`. To display the vector in an X-window, one can use the default X-windows viewer `PETSC_VIEWER_DRAW_WORLD`, or one can create a viewer with the routine `PetscViewerDrawOpen()`. A variety of viewers are discussed further in Viewers: Looking at PETSc Objects.

To create a new vector of the same format and parallel layout as an existing vector, use

    VecDuplicate(Vec old,Vec *new);

To create several new vectors of the same format as an existing vector, use

    VecDuplicateVecs(Vec old,PetscInt n,Vec **new);

This routine creates an array of pointers to vectors. The two routines are useful because they allow one to write library code that does not depend on the particular format of the vectors being used. Instead, the subroutines can automatically create work vectors based on the specified existing vector. As discussed in Duplicating Multiple Vectors, the Fortran interface for `VecDuplicateVecs()` differs slightly.

When a vector is no longer needed, it should be destroyed with the command

    VecDestroy(Vec *x);

To destroy an array of vectors, use the command

    VecDestroyVecs(PetscInt n,Vec **vecs);

Note that the Fortran interface for `VecDestroyVecs()` differs slightly, as described in Duplicating Multiple Vectors.

It is also possible to create vectors that use an array the user provides rather than having PETSc internally allocate the array space. Such vectors can be created with the routines such as

    VecCreateSeqWithArray(PETSC_COMM_SELF,PetscInt bs,PetscInt n,PetscScalar *array,Vec *V);

    VecCreateMPIWithArray(MPI_Comm comm,PetscInt bs,PetscInt n,PetscInt N,PetscScalar *array,Vec *V);

    VecCreateMPICUDAWithArray(MPI_Comm comm,PetscInt bs,PetscInt n,PetscInt N,PetscScalar *array,Vec *V);

The `array` pointer should be a GPU memory location for GPU vectors.

Note that here, one must provide the value `n`; it cannot be `PETSC_DECIDE` and the user is responsible for providing enough space in the array; `n*sizeof(PetscScalar)`.

## Assembling (putting values in) vectors

One can assign a single value to all components of a vector with

    VecSet(Vec x,PetscScalar value);

Assigning values to individual vector components is more complicated to make it possible to write efficient parallel code. Assigning a set of components on a CPU is a two-step process: one first calls

    VecSetValues(Vec x,PetscInt n,PetscInt *indices,PetscScalar *values,INSERT_VALUES);

any number of times on any or all of the processes. The argument `n` gives the number of components being set in this insertion. The integer array `indices` contains the global component indices, and `values` is the array of values to be inserted at those global component index locations. Any process can set any vector components; PETSc ensures that they are automatically stored in the correct location. Once all of the values have been inserted with `VecSetValues()`, one must call

    VecAssemblyBegin(Vec x);

followed by

    VecAssemblyEnd(Vec x);

to perform any needed message passing of nonlocal components. In order to allow the overlap of communication and calculation, the user’s code can perform any series of other actions between these two calls while the messages are in transition.

Example usage of `VecSetValues()` may be found in src/vec/vec/tutorials/ex2.c or src/vec/vec/tutorials/ex2f.F90.

Rather than inserting elements in a vector, one may wish to add values. This process is also done with the command

    VecSetValues(Vec x,PetscInt n,PetscInt *indices, PetscScalar *values,ADD_VALUES);

Again, one must call the assembly routines `VecAssemblyBegin()` and `VecAssemblyEnd()` after all of the values have been added. Note that addition and insertion calls to `VecSetValues()` cannot be mixed. Instead, one must add and insert vector elements in phases, with intervening calls to the assembly routines. This phased assembly procedure overcomes the nondeterministic behavior that would occur if two different processes generated values for the same location, with one process adding while the other is inserting its value. (In this case, the addition and insertion actions could be performed in either order, thus resulting in different values at the particular location. Since PETSc does not allow the simultaneous use of `INSERT_VALUES` and `ADD_VALUES` this nondeterministic behavior will not occur in PETSc.)

You can call `VecGetValues()` to pull local values from a vector (but not off-process values).

For vectors obtained with `DMCreateGlobalVector()`, on can use `VecSetValuesLocal()` to set values into a global vector but using the local (ghosted) vector indexing of the vector entries. See also Local to global mappings that allows one to provide arbitrary local-to-global mapping when not working with a `DM`.

It is also possible to interact directly with the arrays that the vector values are stored in. The routine `VecGetArray()` returns a pointer to the elements local to the process:

    VecGetArray(Vec v,PetscScalar **array);

When access to the array is no longer needed, the user should call

    VecRestoreArray(Vec v, PetscScalar **array);

If the values do not need to be modified, the routines

    VecGetArrayRead(Vec v, const PetscScalar **array);
    VecRestoreArrayRead(Vec v, const PetscScalar **array);

should be used instead.

Listing: SNES Tutorial src/snes/tutorials/ex1.c

    PetscErrorCode FormFunction1(SNES snes, Vec x, Vec f, void *ctx)
    {
      const PetscScalar *xx;
      PetscScalar       *ff;

      PetscFunctionBeginUser;
      /*
       Get pointers to vector data.
          - For default PETSc vectors, VecGetArray() returns a pointer to
            the data array.  Otherwise, the routine is implementation dependent.
          - You MUST call VecRestoreArray() when you no longer need access to
            the array.
       */
      PetscCall(VecGetArrayRead(x, &xx));
      PetscCall(VecGetArray(f, &ff));

      /* Compute function */
      ff[0] = xx[0] * xx[0] + xx[0] * xx[1] - 3.0;
      ff[1] = xx[0] * xx[1] + xx[1] * xx[1] - 6.0;

      /* Restore vectors */
      PetscCall(VecRestoreArrayRead(x, &xx));
      PetscCall(VecRestoreArray(f, &ff));
      PetscFunctionReturn(PETSC_SUCCESS);

Minor differences exist in the Fortran interface for `VecGetArray()` and `VecRestoreArray()`, as discussed in Routines that Return Fortran Allocatable Arrays. It is important to note that `VecGetArray()` and `VecRestoreArray()` do not copy the vector elements; they merely give users direct access to the vector elements. Thus, these routines require essentially no time to call and can be used efficiently.

For GPU vectors, one can access either the values on the CPU as described above or one can call, for example,

    VecCUDAGetArray(Vec v, PetscScalar **array);

Listing: SNES Tutorial src/snes/tutorials/ex47cu.cu

        PetscCall(VecCUDAGetArrayRead(xlocal, &xarray));
        PetscCall(VecCUDAGetArrayWrite(f, &farray));
        if (rank) xstartshift = 1;
        else xstartshift = 0;
        if (rank != size - 1) xendshift = 1;
        else xendshift = 0;
        PetscCall(VecGetOwnershipRange(f, &fstart, NULL));
        PetscCall(VecGetLocalSize(x, &lsize));
        // clang-format off
        try {
          thrust::for_each(
            thrust::make_zip_iterator(
              thrust::make_tuple(
                thrust::device_ptr<PetscScalar>(farray),
                thrust::device_ptr<const PetscScalar>(xarray + xstartshift),
                thrust::device_ptr<const PetscScalar>(xarray + xstartshift + 1),
                thrust::device_ptr<const PetscScalar>(xarray + xstartshift - 1),
                thrust::counting_iterator<int>(fstart),
                thrust::constant_iterator<int>(Mx),
                thrust::constant_iterator<PetscScalar>(hx))),
            thrust::make_zip_iterator(
              thrust::make_tuple(
                thrust::device_ptr<PetscScalar>(farray + lsize),
                thrust::device_ptr<const PetscScalar>(xarray + lsize - xendshift),
                thrust::device_ptr<const PetscScalar>(xarray + lsize - xendshift + 1),
                thrust::device_ptr<const PetscScalar>(xarray + lsize - xendshift - 1),
                thrust::counting_iterator<int>(fstart) + lsize,
                thrust::constant_iterator<int>(Mx),
                thrust::constant_iterator<PetscScalar>(hx))),
            ApplyStencil());
        }

or

    VecGetArrayAndMemType(Vec v, PetscScalar **array,PetscMemType *mtype);

which, in the first case, returns a GPU memory address and, in the second case, returns either a CPU or GPU memory address depending on the type of the vector. One can then launch a GPU kernel function that accesses the vector’s memory for usage with GPUs. When computing on GPUs, `VecSetValues()` is not used! One always accesses the vector’s arrays and passes them to the GPU code.

It can also be convenient to treat the vector entries as a Kokkos view. One first creates Kokkos vectors and then calls

    VecGetKokkosView(Vec v, Kokkos::View<const PetscScalar*,MemorySpace> *kv)

to set or access the vector entries.

Of course, to provide the correct values to a vector, one must know what parts of the vector are owned by each MPI process. For parallel vectors, either CPU or GPU-based, it is possible to determine a process’s local range with the routine

    VecGetOwnershipRange(Vec vec,PetscInt *start,PetscInt *end);

The argument `start` indicates the first component owned by the local process, while `end` specifies one more than the last owned by the local process. This command is useful, for instance, in assembling parallel vectors.

If the Vec was obtained from a DM with DMCreateGlobalVector(), then the range values are determined by the specific DM. If the Vec was created directly, the range values are determined by the local size passed to VecSetSizes() or VecCreateMPI(). If PETSC_DECIDE was passed as the local size, then the vector uses default values for the range using PetscSplitOwnership(). For certain DM, such as DMDA, it is better to use DM specific routines, such as DMDAGetGhostCorners(), to determine the local values in the vector.

Very occasionally, all MPI processes need to know all the range values, these can be obtained with

    VecGetOwnershipRanges(Vec vec,PetscInt range[]);

The number of elements stored locally can be accessed with

    VecGetLocalSize(Vec v,PetscInt *size);

The global vector length can be determined by

    VecGetSize(Vec v,PetscInt *size);

### DMDA - Setting vector values

PETSc provides an easy way to set values into the `DMDA` vectors and access them using the natural grid indexing. This is done with the routines

    DMDAVecGetArray(DM da,Vec l,void *array);
    ... use the array indexing it with 1, 2, or 3 dimensions ...
    ... depending on the dimension of the DMDA ...
    DMDAVecRestoreArray(DM da,Vec l,void *array);
    DMDAVecGetArrayRead(DM da,Vec l,void *array);
    ... use the array indexing it with 1, 2, or 3 dimensions ...
    ... depending on the dimension of the DMDA ...
    DMDAVecRestoreArrayRead(DM da,Vec l,void *array);

where `array` is a multidimensional C array with the same dimension as `da`, and

    DMDAVecGetArrayDOF(DM da,Vec l,void *array);
    ... use the array indexing it with 2, 3, or 4 dimensions ...
    ... depending on the dimension of the DMDA ...
    DMDAVecRestoreArrayDOF(DM da,Vec l,void *array);
    DMDAVecGetArrayDOFRead(DM da,Vec l,void *array);
    ... use the array indexing it with 2, 3, or 4 dimensions ...
    ... depending on the dimension of the DMDA ...
    DMDAVecRestoreArrayDOFRead(DM da,Vec l,void *array);

where `array` is a multidimensional C array with one more dimension than `da`. The vector `l` can be either a global vector or a local vector. The `array` is accessed using the usual global indexing on the entire grid, but the user may only refer to this array’s local and ghost entries as all other entries are undefined. For example, for a scalar problem in two dimensions, one could use

    PetscScalar **f,**u;
    ...
    DMDAVecGetArrayRead(DM da,Vec local,&u);
    DMDAVecGetArray(DM da,Vec global,&f);
    ...
      f[i][j] = u[i][j] - ...
    ...
    DMDAVecRestoreArrayRead(DM da,Vec local,&u);
    DMDAVecRestoreArray(DM da,Vec global,&f);

Listing: SNES Tutorial src/snes/tutorials/ex3.c

    PetscErrorCode FormFunction(SNES snes, Vec x, Vec f, void *ctx)
    {
      ApplicationCtx    *user = (ApplicationCtx *)ctx;
      DM                 da   = user->da;
      PetscScalar       *ff, d;
      const PetscScalar *xx, *FF;
      PetscInt           i, M, xs, xm;
      Vec                xlocal;

      PetscFunctionBeginUser;
      PetscCall(DMGetLocalVector(da, &xlocal));
      /*
         Scatter ghost points to local vector, using the 2-step process
            DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
         By placing code between these two statements, computations can
         be done while messages are in transition.
      */
      PetscCall(DMGlobalToLocalBegin(da, x, INSERT_VALUES, xlocal));
      PetscCall(DMGlobalToLocalEnd(da, x, INSERT_VALUES, xlocal));

      /*
         Get pointers to vector data.
           - The vector xlocal includes ghost point; the vectors x and f do
             NOT include ghost points.
           - Using DMDAVecGetArray() allows accessing the values using global ordering
      */
      PetscCall(DMDAVecGetArrayRead(da, xlocal, (void *)&xx));
      PetscCall(DMDAVecGetArray(da, f, &ff));
      PetscCall(DMDAVecGetArrayRead(da, user->F, (void *)&FF));

      /*
         Get local grid boundaries (for 1-dimensional DMDA):
           xs, xm  - starting grid index, width of local grid (no ghost points)
      */
      PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
      PetscCall(DMDAGetInfo(da, NULL, &M, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));

      /*
         Set function values for boundary points; define local interior grid point range:
            xsi - starting interior grid index
            xei - ending interior grid index
      */
      if (xs == 0) { /* left boundary */
        ff[0] = xx[0];
        xs++;
        xm--;
      }
      if (xs + xm == M) { /* right boundary */
        ff[xs + xm - 1] = xx[xs + xm - 1] - 1.0;
        xm--;
      }

      /*
         Compute function over locally owned part of the grid (interior points only)
      */
      d = 1.0 / (user->h * user->h);
      for (i = xs; i < xs + xm; i++) ff[i] = d * (xx[i - 1] - 2.0 * xx[i] + xx[i + 1]) + xx[i] * xx[i] - FF[i];

      /*
         Restore vectors
      */
      PetscCall(DMDAVecRestoreArrayRead(da, xlocal, (void *)&xx));
      PetscCall(DMDAVecRestoreArray(da, f, &ff));
      PetscCall(DMDAVecRestoreArrayRead(da, user->F, (void *)&FF));
      PetscCall(DMRestoreLocalVector(da, &xlocal));
      PetscFunctionReturn(PETSC_SUCCESS);

The recommended approach for multi-component PDEs is to declare a `struct` representing the fields defined at each node of the grid, e.g.

    typedef struct {
      PetscScalar u,v,omega,temperature;
    } Node;

and write the residual evaluation using

    Node **f,**u;
    DMDAVecGetArray(DM da,Vec local,&u);
    DMDAVecGetArray(DM da,Vec global,&f);
     ...
        f[i][j].omega = ...
     ...
    DMDAVecRestoreArray(DM da,Vec local,&u);
    DMDAVecRestoreArray(DM da,Vec global,&f);

The `DMDAVecGetArray` routines are also provided for GPU access with CUDA, HIP, and Kokkos. For example,

    DMDAVecGetKokkosOffsetView(DM da,Vec vec,Kokkos::View<const PetscScalar*XX*,MemorySpace> *ov)

where `*XX*` can contain any number of *. This allows one to write very natural Kokkos multi-dimensional parallel for kernels that act on the local portion of `DMDA` vectors.

Listing: SNES Tutorial src/snes/tutorials/ex3k.kokkos.cxx

    PetscErrorCode KokkosFunction(SNES snes, Vec x, Vec r, void *ctx)
    {
      ApplicationCtx                  *user = (ApplicationCtx *)ctx;
      DM                               da   = user->da;
      PetscScalar                      d;
      PetscInt                         M;
      Vec                              xl;
      PetscScalarKokkosOffsetView      R;
      ConstPetscScalarKokkosOffsetView X, F;

      PetscFunctionBeginUser;
      PetscCall(DMGetLocalVector(da, &xl));
      PetscCall(DMGlobalToLocal(da, x, INSERT_VALUES, xl));
      d = 1.0 / (user->h * user->h);
      PetscCall(DMDAGetInfo(da, NULL, &M, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL));
      PetscCall(DMDAVecGetKokkosOffsetView(da, xl, &X));      /* read only */
      PetscCall(DMDAVecGetKokkosOffsetViewWrite(da, r, &R));  /* write only */
      PetscCall(DMDAVecGetKokkosOffsetView(da, user->F, &F)); /* read only */
      Kokkos::parallel_for(
        Kokkos::RangePolicy<>(R.begin(0), R.end(0)), KOKKOS_LAMBDA(int i) {
          if (i == 0) R(0) = X(0);                                                 /* left boundary */
          else if (i == M - 1) R(i) = X(i) - 1.0;                                  /* right boundary */
          else R(i) = d * (X(i - 1) - 2.0 * X(i) + X(i + 1)) + X(i) * X(i) - F(i); /* interior */
        });
      PetscCall(DMDAVecRestoreKokkosOffsetView(da, xl, &X));
      PetscCall(DMDAVecRestoreKokkosOffsetViewWrite(da, r, &R));
      PetscCall(DMDAVecRestoreKokkosOffsetView(da, user->F, &F));
      PetscCall(DMRestoreLocalVector(da, &xl));
      PetscFunctionReturn(PETSC_SUCCESS);

The global indices of the lower left corner of the local portion of vectors obtained from `DMDA` as well as the local array size can be obtained with the commands

    DMDAGetCorners(DM da,PetscInt *x,PetscInt *y,PetscInt *z,PetscInt *m,PetscInt *n,PetscInt *p);
    DMDAGetGhostCorners(DM da,PetscInt *x,PetscInt *y,PetscInt *z,PetscInt *m,PetscInt *n,PetscInt *p);

These values can then be used as loop bounds for local function evaluations as demonstrated in the function examples above.

The first version excludes ghost points, while the second includes them. The routine `DMDAGetGhostCorners()` deals with the fact that subarrays along boundaries of the problem domain have ghost points only on their interior edges, but not on their boundary edges.

When either type of stencil is used, `DMDA_STENCIL_STAR` or `DMDA_STENCIL_BOX`, the local vectors (with the ghost points) represent rectangular arrays, including the extra corner elements in the `DMDA_STENCIL_STAR` case. This configuration provides simple access to the elements by employing two- (or three–) dimensional indexing. The only difference between the two cases is that when `DMDA_STENCIL_STAR` is used, the extra corner components are not scattered between the processes and thus contain undefined values that should not be used.

### DMSTAG - Setting vector values

For structured grids with staggered data (living on elements, faces, edges, and/or vertices), the `DMStag` object is available. It behaves like `DMDA`; see the `DMSTAG` manual page for more information.

Listing: SNES Tutorial src/dm/impls/stag/tutorials/ex6.c

    static PetscErrorCode UpdateVelocity_2d(const Ctx *ctx, Vec velocity, Vec stress, Vec buoyancy)
    {
      Vec                  velocity_local, stress_local, buoyancy_local;
      PetscInt             ex, ey, startx, starty, nx, ny;
      PetscInt             slot_coord_next, slot_coord_element, slot_coord_prev;
      PetscInt             slot_vx_left, slot_vy_down, slot_buoyancy_down, slot_buoyancy_left;
      PetscInt             slot_txx, slot_tyy, slot_txy_downleft, slot_txy_downright, slot_txy_upleft;
      const PetscScalar  **arr_coord_x, **arr_coord_y;
      const PetscScalar ***arr_stress, ***arr_buoyancy;
      PetscScalar       ***arr_velocity;

      PetscFunctionBeginUser;
      /* Prepare direct access to buoyancy data */
      PetscCall(DMStagGetLocationSlot(ctx->dm_buoyancy, DMSTAG_LEFT, 0, &slot_buoyancy_left));
      PetscCall(DMStagGetLocationSlot(ctx->dm_buoyancy, DMSTAG_DOWN, 0, &slot_buoyancy_down));
      PetscCall(DMGetLocalVector(ctx->dm_buoyancy, &buoyancy_local));
      PetscCall(DMGlobalToLocal(ctx->dm_buoyancy, buoyancy, INSERT_VALUES, buoyancy_local));
      PetscCall(DMStagVecGetArrayRead(ctx->dm_buoyancy, buoyancy_local, (void *)&arr_buoyancy));

      /* Prepare read-only access to stress data */
      PetscCall(DMStagGetLocationSlot(ctx->dm_stress, DMSTAG_ELEMENT, 0, &slot_txx));
      PetscCall(DMStagGetLocationSlot(ctx->dm_stress, DMSTAG_ELEMENT, 1, &slot_tyy));
      PetscCall(DMStagGetLocationSlot(ctx->dm_stress, DMSTAG_UP_LEFT, 0, &slot_txy_upleft));
      PetscCall(DMStagGetLocationSlot(ctx->dm_stress, DMSTAG_DOWN_LEFT, 0, &slot_txy_downleft));
      PetscCall(DMStagGetLocationSlot(ctx->dm_stress, DMSTAG_DOWN_RIGHT, 0, &slot_txy_downright));
      PetscCall(DMGetLocalVector(ctx->dm_stress, &stress_local));
      PetscCall(DMGlobalToLocal(ctx->dm_stress, stress, INSERT_VALUES, stress_local));
      PetscCall(DMStagVecGetArrayRead(ctx->dm_stress, stress_local, (void *)&arr_stress));

      /* Prepare read-write access to velocity data */
      PetscCall(DMStagGetLocationSlot(ctx->dm_velocity, DMSTAG_LEFT, 0, &slot_vx_left));
      PetscCall(DMStagGetLocationSlot(ctx->dm_velocity, DMSTAG_DOWN, 0, &slot_vy_down));
      PetscCall(DMGetLocalVector(ctx->dm_velocity, &velocity_local));
      PetscCall(DMGlobalToLocal(ctx->dm_velocity, velocity, INSERT_VALUES, velocity_local));
      PetscCall(DMStagVecGetArray(ctx->dm_velocity, velocity_local, &arr_velocity));

      /* Prepare read-only access to coordinate data */
      PetscCall(DMStagGetProductCoordinateLocationSlot(ctx->dm_velocity, DMSTAG_LEFT, &slot_coord_prev));
      PetscCall(DMStagGetProductCoordinateLocationSlot(ctx->dm_velocity, DMSTAG_RIGHT, &slot_coord_next));
      PetscCall(DMStagGetProductCoordinateLocationSlot(ctx->dm_velocity, DMSTAG_ELEMENT, &slot_coord_element));
      PetscCall(DMStagGetProductCoordinateArrays(ctx->dm_velocity, (void *)&arr_coord_x, (void *)&arr_coord_y, NULL));

      /* Iterate over interior of the domain, updating the velocities */
      PetscCall(DMStagGetCorners(ctx->dm_velocity, &startx, &starty, NULL, &nx, &ny, NULL, NULL, NULL, NULL));
      for (ey = starty; ey < starty + ny; ++ey) {
        for (ex = startx; ex < startx + nx; ++ex) {
          /* Update y-velocity */
          if (ey > 0) {
            const PetscScalar dx = arr_coord_x[ex][slot_coord_next] - arr_coord_x[ex][slot_coord_prev];
            const PetscScalar dy = arr_coord_y[ey][slot_coord_element] - arr_coord_y[ey - 1][slot_coord_element];
            const PetscScalar B  = arr_buoyancy[ey][ex][slot_buoyancy_down];

            arr_velocity[ey][ex][slot_vy_down] += B * ctx->dt * ((arr_stress[ey][ex][slot_txy_downright] - arr_stress[ey][ex][slot_txy_downleft]) / dx + (arr_stress[ey][ex][slot_tyy] - arr_stress[ey - 1][ex][slot_tyy]) / dy);
          }

          /* Update x-velocity */

### DMPLEX - Setting vector values

See DMPlex: Unstructured Grids for a discussion on setting vector values with `DMPLEX`.

### DMNETWORK - Setting vector values

See Networks for a discussion on setting vector values with `DMNETWORK`.

## Basic Vector Operations

Table 2 PETSc Vector Operations
Function Name | Operation
--- | ---
`VecAXPY(Vec y,PetscScalar a,Vec x);` | \(y = y + a*x\)
`VecAYPX(Vec y,PetscScalar a,Vec x);` | \(y = x + a*y\)
`VecWAXPY(Vec w,PetscScalar a,Vec x,Vec y);` | \(w = a*x + y\)
`VecAXPBY(Vec y,PetscScalar a,PetscScalar b,Vec x);` | \(y = a*x + b*y\)
`VecAXPBYPCZ(Vec z,PetscScalar a,PetscScalar b, PetscScalar c,Vec x,Vec y);` | \(z = a*x + b*y + c*z\)
`VecScale(Vec x, PetscScalar a);` | \(x = a*x\)
`VecDot(Vec x, Vec y, PetscScalar *r);` | \(r = \bar{x}^T*y\)
`VecTDot(Vec x, Vec y, PetscScalar *r);` | \(r = x'*y\)
`VecNorm(Vec x, NormType type, PetscReal *r);` | \(r = ||x||_{type}\)
`VecSum(Vec x, PetscScalar *r);` | \(r = \sum x_{i}\)
`VecCopy(Vec x, Vec y);` | \(y = x\)
`VecSwap(Vec x, Vec y);` | \(y = x\) while \(x = y\)
`VecPointwiseMult(Vec w,Vec x,Vec y);` | \(w_{i} = x_{i}*y_{i}\)
`VecPointwiseDivide(Vec w,Vec x,Vec y);` | \(w_{i} = x_{i}/y_{i}\)
`VecMDot(Vec x,PetscInt n,Vec y[],PetscScalar *r);` | \(r[i] = \bar{x}^T*y[i]\)
`VecMTDot(Vec x,PetscInt n,Vec y[],PetscScalar *r);` | \(r[i] = x^T*y[i]\)
`VecMAXPY(Vec y,PetscInt n, PetscScalar *a, Vec x[]);` | \(y = y + \sum_i a_{i}*x[i]\)
`VecMax(Vec x, PetscInt *idx, PetscReal *r);` | \(r = \max x_{i}\)
`VecMin(Vec x, PetscInt *idx, PetscReal *r);` | \(r = \min x_{i}\)
`VecAbs(Vec x);` | \(x_i = |x_{i}|\)
`VecReciprocal(Vec x);` | \(x_i = 1/x_{i}\)
`VecShift(Vec x,PetscScalar s);` | \(x_i = s + x_{i}\)
`VecSet(Vec x,PetscScalar alpha);` | \(x_i = \alpha\)

As the table lists, we have chosen certain basic vector operations to support within the PETSc vector library. These operations were selected because they often arise in application codes. The `NormType` argument to `VecNorm()` is one of `NORM_1`, `NORM_2`, or `NORM_INFINITY`. The 1-norm is \(\sum_i |x_{i}|\), the 2-norm is \(( \sum_{i} x_{i}^{2})^{1/2}\) and the infinity norm is \(\max_{i} |x_{i}|\).

In addition to `VecDot()` and `VecMDot()` and `VecNorm()`, PETSc provides split phase versions of this functionality that allow several independent inner products and/or norms to share the same communication (thus improving parallel efficiency). For example, one may have code such as

    VecDot(Vec x,Vec y,PetscScalar *dot);
    VecMDot(Vec x,PetscInt nv, Vec y[],PetscScalar *dot);
    VecNorm(Vec x,NormType NORM_2,PetscReal *norm2);
    VecNorm(Vec x,NormType NORM_1,PetscReal *norm1);

This code works fine, but it performs four separate parallel communication operations. Instead, one can write

    VecDotBegin(Vec x,Vec y,PetscScalar *dot);
    VecMDotBegin(Vec x, PetscInt nv,Vec y[],PetscScalar *dot);
    VecNormBegin(Vec x,NormType NORM_2,PetscReal *norm2);
    VecNormBegin(Vec x,NormType NORM_1,PetscReal *norm1);
    VecDotEnd(Vec x,Vec y,PetscScalar *dot);
    VecMDotEnd(Vec x, PetscInt nv,Vec y[],PetscScalar *dot);
    VecNormEnd(Vec x,NormType NORM_2,PetscReal *norm2);
    VecNormEnd(Vec x,NormType NORM_1,PetscReal *norm1);

With this code, the communication is delayed until the first call to `VecxxxEnd()` at which a single MPI reduction is used to communicate all the values. It is required that the calls to the `VecxxxEnd()` are performed in the same order as the calls to the `VecxxxBegin()`; however, if you mistakenly make the calls in the wrong order, PETSc will generate an error informing you of this. There are additional routines `VecTDotBegin()` and `VecTDotEnd()`, `VecMTDotBegin()`, `VecMTDotEnd()`.

For GPU vectors (like CUDA), the numerical computations will, by default, run on the GPU. Any scalar output, like the result of a `VecDot()` are placed in CPU memory.

## Local/global vectors and communicating between vectors

Many PDE problems require ghost (or halo) values in each MPI process or even more general parallel communication of vector values. These values are needed to perform function evaluation on that MPI process. The exact structure of the ghost values needed depends on the type of grid being used. `DM` provides a uniform API for communicating the needed values. We introduce the concept in detail for `DMDA`.

Each `DM` object defines the layout of two vectors: a distributed global vector and a local vector that includes room for the appropriate ghost points. The `DM` object provides information about the size and layout of these vectors. The user can create vector objects that use the `DM` layout information with the routines

    DMCreateGlobalVector(DM da,Vec *g);
    DMCreateLocalVector(DM da,Vec *l);

These vectors will generally serve as the building blocks for local and global PDE solutions, etc. If additional vectors with such layout information are needed in a code, they can be obtained by duplicating `l` or `g` via `VecDuplicate()` or `VecDuplicateVecs()`.

We emphasize that a `DM` provides the information needed to communicate the ghost value information between processes. In most cases, several different vectors can share the same communication information (or, in other words, can share a given `DM`). The design of the `DM` object makes this easy, as each `DM` operation may operate on vectors of the appropriate size, as obtained via `DMCreateLocalVector()` and `DMCreateGlobalVector()` or as produced by `VecDuplicate()`.

At certain stages of many applications, there is a need to work on a local portion of the vector that includes the ghost points. This may be done by scattering a global vector into its local parts by using the two-stage commands

    DMGlobalToLocalBegin(DM da,Vec g,InsertMode iora,Vec l);
    DMGlobalToLocalEnd(DM da,Vec g,InsertMode iora,Vec l);

which allows the overlap of communication and computation. Since the global and local vectors, given by `g` and `l`, respectively, must be compatible with the `DM`, `da`, they should be generated by `DMCreateGlobalVector()` and `DMCreateLocalVector()` (or be duplicates of such a vector obtained via `VecDuplicate()`). The `InsertMode` can be `ADD_VALUES` or `INSERT_VALUES` among other possible values.

One can scatter the local vectors into the distributed global vector with the command

    DMLocalToGlobal(DM da,Vec l,InsertMode mode,Vec g);

or the commands

    DMLocalToGlobalBegin(DM da,Vec l,InsertMode mode,Vec g);
    /* (Computation to overlap with communication) */
    DMLocalToGlobalEnd(DM da,Vec l,InsertMode mode,Vec g);

In general this is used with an `InsertMode` of `ADD_VALUES`, because if one wishes to insert values into the global vector, they should access the global vector directly and put in the values.

A third type of `DM` scatter is from a local vector (including ghost points that contain irrelevant values) to a local vector with correct ghost point values. This scatter may be done with the commands

    DMLocalToLocalBegin(DM da,Vec l1,InsertMode iora,Vec l2);
    DMLocalToLocalEnd(DM da,Vec l1,InsertMode iora,Vec l2);

Since both local vectors, `l1` and `l2`, must be compatible with `da`, they should be generated by `DMCreateLocalVector()` (or be duplicates of such vectors obtained via `VecDuplicate()`). The `InsertMode` can be either `ADD_VALUES` or `INSERT_VALUES`.

In most applications, the local ghosted vectors are only needed temporarily during user “function evaluations”. PETSc provides an easy, light-weight (requiring essentially no CPU time) way to temporarily obtain these work vectors and return them when no longer needed. This is done with the routines

    DMGetLocalVector(DM da,Vec *l);
    ... use the local vector l ...
    DMRestoreLocalVector(DM da,Vec *l);

## Low-level Vector Communication

Most users of PETSc who can utilize a `DM` will not need to utilize the lower-level routines discussed in the rest of this section and should skip ahead to Matrices.

To facilitate creating general vector scatters and gathers used, for example, in updating ghost points for problems for which no `DM` currently exists PETSc employs the concept of an index set, via the `IS` class. An index set, a generalization of a set of integer indices, is used to define scatters, gathers, and similar operations on vectors and matrices. Much of the underlying code that implements `DMGlobalToLocal` communication is built on the infrastructure discussed below.

The following command creates an index set based on a list of integers:

    ISCreateGeneral(MPI_Comm comm,PetscInt n,PetscInt *indices,PetscCopyMode mode, IS *is);

When `mode` is `PETSC_COPY_VALUES`, this routine copies the `n` indices passed to it by the integer array `indices`. Thus, the user should be sure to free the integer array `indices` when it is no longer needed, perhaps directly after the call to `ISCreateGeneral()`. The communicator, `comm`, should include all processes using the `IS`.

Another standard index set is defined by a starting point (`first`) and a stride (`step`), and can be created with the command

    ISCreateStride(MPI_Comm comm,PetscInt n,PetscInt first,PetscInt step,IS *is);

The meaning of `n`, `first`, and `step` correspond to the MATLAB notation `first:step:first+n*step`.

Index sets can be destroyed with the command

    ISDestroy(IS &is);

On rare occasions, the user may need to access information directly from an index set. Several commands assist in this process:

    ISGetSize(IS is,PetscInt *size);
    ISStrideGetInfo(IS is,PetscInt *first,PetscInt *stride);
    ISGetIndices(IS is,PetscInt **indices);

The function `ISGetIndices()` returns a pointer to a list of the indices in the index set. For certain index sets, this may be a temporary array of indices created specifically for the call. Thus, once the user finishes using the array of indices, the routine

    ISRestoreIndices(IS is, PetscInt **indices);

should be called to ensure that the system can free the space it may have used to generate the list of indices.

A blocked version of index sets can be created with the command

    ISCreateBlock(MPI_Comm comm,PetscInt bs,PetscInt n,PetscInt *indices,PetscCopyMode mode, IS *is);

This version is used for defining operations in which each element of the index set refers to a block of `bs` vector entries. Related routines analogous to those described above exist as well, including `ISBlockGetIndices()`, `ISBlockGetSize()`, `ISBlockGetLocalSize()`, `ISGetBlockSize()`.

Most PETSc applications use a particular `DM` object to manage the communication details needed for their grids. In some rare cases, however, codes need to directly set up their required communication patterns. This is done using PETSc’s `VecScatter` and `PetscSF` (for more general data than vectors). One can select any subset of the components of a vector to insert or add to any subset of the components of another vector. We refer to these operations as generalized scatters, though they are a combination of scatters and gathers.

To copy selected components from one vector to another, one uses the following set of commands:

    VecScatterCreate(Vec x,IS ix,Vec y,IS iy,VecScatter *ctx);
    VecScatterBegin(VecScatter ctx,Vec x,Vec y,INSERT_VALUES,SCATTER_FORWARD);
    VecScatterEnd(VecScatter ctx,Vec x,Vec y,INSERT_VALUES,SCATTER_FORWARD);
    VecScatterDestroy(VecScatter *ctx);

Here `ix` denotes the index set of the first vector, while `iy` indicates the index set of the destination vector. The vectors can be parallel or sequential. The only requirements are that the number of entries in the index set of the first vector, `ix`, equals the number in the destination index set, `iy`, and that the vectors be long enough to contain all the indices referred to in the index sets. If both `x` and `y` are parallel, their communicator must have the same set of processes, but their process order can differ. The argument `INSERT_VALUES` specifies that the vector elements will be inserted into the specified locations of the destination vector, overwriting any existing values. To add the components, rather than insert them, the user should select the option `ADD_VALUES` instead of `INSERT_VALUES`. One can also use `MAX_VALUES` or `MIN_VALUES` to replace the destination with the maximal or minimal of its current value and the scattered values.

To perform a conventional gather operation, the user makes the destination index set, `iy`, be a stride index set with a stride of one. Similarly, a conventional scatter can be done with an initial (sending) index set consisting of a stride. The scatter routines are collective operations (i.e. all processes that own a parallel vector must call the scatter routines). When scattering from a parallel vector to sequential vectors, each process has its own sequential vector that receives values from locations as indicated in its own index set. Similarly, in scattering from sequential vectors to a parallel vector, each process has its own sequential vector that contributes to the parallel vector.

Caution: When `INSERT_VALUES` is used, if two different processes contribute different values to the same component in a parallel vector, either value may be inserted. When `ADD_VALUES` is used, the correct sum is added to the correct location.

In some cases, one may wish to “undo” a scatter, that is, perform the scatter backward, switching the roles of the sender and receiver. This is done by using

    VecScatterBegin(VecScatter ctx,Vec y,Vec x,INSERT_VALUES,SCATTER_REVERSE);
    VecScatterEnd(VecScatter ctx,Vec y,Vec x,INSERT_VALUES,SCATTER_REVERSE);

Note that the roles of the first two arguments to these routines must be swapped whenever the `SCATTER_REVERSE` option is used.

Once a `VecScatter` object has been created, it may be used with any vectors that have the same parallel data layout. That is, one can call `VecScatterBegin()` and `VecScatterEnd()` with different vectors than used in the call to `VecScatterCreate()` as long as they have the same parallel layout (the number of elements on each process are the same). Usually, these “different” vectors would have been obtained via calls to `VecDuplicate()` from the original vectors used in the call to `VecScatterCreate()`.

`VecGetValues()` can only access local values from the vector. To get off-process values, the user should create a new vector where the components will be stored and then perform the appropriate vector scatter. For example, if one desires to obtain the values of the 100th and 200th entries of a parallel vector, `p`, one could use a code such as that below. In this example, the values of the 100th and 200th components are placed in the array values. In this example, each process now has the 100th and 200th component, but obviously, each process could gather any elements it needed, or none by creating an index set with no entries.

    Vec         p, x;         /* initial vector, destination vector */
    VecScatter  scatter;      /* scatter context */
    IS          from, to;     /* index sets that define the scatter */
    PetscScalar *values;
    PetscInt    idx_from[] = {100,200}, idx_to[] = {0,1};

    VecCreateSeq(PETSC_COMM_SELF,2,&x);
    ISCreateGeneral(PETSC_COMM_SELF,2,idx_from,PETSC_COPY_VALUES,&from);
    ISCreateGeneral(PETSC_COMM_SELF,2,idx_to,PETSC_COPY_VALUES,&to);
    VecScatterCreate(p,from,x,to,&scatter);
    VecScatterBegin(scatter,p,x,INSERT_VALUES,SCATTER_FORWARD);
    VecScatterEnd(scatter,p,x,INSERT_VALUES,SCATTER_FORWARD);
    VecGetArray(x,&values);
    ISDestroy(&from);
    ISDestroy(&to);
    VecScatterDestroy(&scatter);

The scatter comprises two stages to allow for the overlap of communication and computation. The introduction of the `VecScatter` context allows the communication patterns for the scatter to be computed once and reused repeatedly. Generally, even setting up the communication for a scatter requires communication; hence, it is best to reuse such information when possible.

Scatters provide a very general method for managing the communication of required ghost values for unstructured grid computations. One scatters the global vector into a local “ghosted” work vector, performs the computation on the local work vectors, and then scatters back into the global solution vector. In the simplest case, this may be written as

    VecScatterBegin(VecScatter scatter,Vec globalin,Vec localin,InsertMode INSERT_VALUES, ScatterMode SCATTER_FORWARD);
    VecScatterEnd(VecScatter scatter,Vec globalin,Vec localin,InsertMode INSERT_VALUES,ScatterMode SCATTER_FORWARD);
    /* For example, do local calculations from localin to localout */
    ...
    VecScatterBegin(VecScatter scatter,Vec localout,Vec globalout,InsertMode ADD_VALUES,ScatterMode SCATTER_REVERSE);
    VecScatterEnd(VecScatter scatter,Vec localout,Vec globalout,InsertMode ADD_VALUES,ScatterMode SCATTER_REVERSE);

In this case, the scatter is used in a way similar to the usage of `DMGlobalToLocal()` and `DMLocalToGlobal()` discussed above.

### Local to global mappings

When working with a global representation of a vector (usually on a vector obtained with `DMCreateGlobalVector()`) and a local representation of the same vector that includes ghost points required for local computation (obtained with `DMCreateLocalVector()`). PETSc provides routines to help map indices from a local numbering scheme to the PETSc global numbering scheme, recall their use above for the routine `VecSetValuesLocal()` introduced above. This is done via the following routines

    ISLocalToGlobalMappingCreate(MPI_Comm comm,PetscInt bs,PetscInt N,PetscInt* globalnum,PetscCopyMode mode,ISLocalToGlobalMapping* ctx);
    ISLocalToGlobalMappingApply(ISLocalToGlobalMapping ctx,PetscInt n,PetscInt *in,PetscInt *out);
    ISLocalToGlobalMappingApplyIS(ISLocalToGlobalMapping ctx,IS isin,IS* isout);
    ISLocalToGlobalMappingDestroy(ISLocalToGlobalMapping *ctx);

Here `N` denotes the number of local indices, `globalnum` contains the global number of each local number, and `ISLocalToGlobalMapping` is the resulting PETSc object that contains the information needed to apply the mapping with either `ISLocalToGlobalMappingApply()` or `ISLocalToGlobalMappingApplyIS()`.

Note that the `ISLocalToGlobalMapping` routines serve a different purpose than the `AO` routines. In the former case, they provide a mapping from a local numbering scheme (including ghost points) to a global numbering scheme, while in the latter, they provide a mapping between two global numbering schemes. Many applications may use both `AO` and `ISLocalToGlobalMapping` routines. The `AO` routines are first used to map from an application global ordering (that has no relationship to parallel processing, etc.) to the PETSc ordering scheme (where each process has a contiguous set of indices in the numbering). Then, to perform function or Jacobian evaluations locally on each process, one works with a local numbering scheme that includes ghost points. The mapping from this local numbering scheme back to the global PETSc numbering can be handled with the `ISLocalToGlobalMapping` routines.

If one is given a list of block indices in a global numbering, the routine

    ISGlobalToLocalMappingApplyBlock(ISLocalToGlobalMapping ctx,ISGlobalToLocalMappingMode type,PetscInt nin,PetscInt idxin[],PetscInt *nout,PetscInt idxout[]);

will provide a new list of indices in the local numbering. Again, negative values in `idxin` are left unmapped. But in addition, if `type` is set to `IS_GTOLM_MASK` , then `nout` is set to `nin` and all global values in `idxin` that are not represented in the local to global mapping are replaced by -1. When `type` is set to `IS_GTOLM_DROP`, the values in `idxin` that are not represented locally in the mapping are not included in `idxout`, so that potentially `nout` is smaller than `nin`. One must pass in an array long enough to hold all the indices. One can call `ISGlobalToLocalMappingApplyBlock()` with `idxout` equal to `NULL` to determine the required length (returned in `nout`) and then allocate the required space and call `ISGlobalToLocalMappingApplyBlock()` a second time to set the values.

Often it is convenient to set elements into a vector using the local node numbering rather than the global node numbering (e.g., each process may maintain its own sublist of vertices and elements and number them locally). To set values into a vector with the local numbering, one must first call

    VecSetLocalToGlobalMapping(Vec v,ISLocalToGlobalMapping ctx);

and then call

    VecSetValuesLocal(Vec x,PetscInt n,const PetscInt indices[],const PetscScalar values[],INSERT_VALUES);

Now the `indices` use the local numbering rather than the global, meaning the entries lie in \([0,n)\) where \(n\) is the local size of the vector. Global vectors obtained from a `DM` already have the global to local mapping provided by the `DM`.

One can use global indices with `MatSetValues()` or `MatSetValuesStencil()` to assemble global stiffness matrices. Alternately, the global node number of each local node, including the ghost nodes, can be obtained by calling

    DMGetLocalToGlobalMapping(DM da,ISLocalToGlobalMapping *map);

followed by

    VecSetLocalToGlobalMapping(Vec v,ISLocalToGlobalMapping map);
    MatSetLocalToGlobalMapping(Mat A,ISLocalToGlobalMapping rmapping,ISLocalToGlobalMapping cmapping);

Now, entries may be added to the vector and matrix using the local numbering and `VecSetValuesLocal()` and `MatSetValuesLocal()`.

The example SNES Tutorial ex5 illustrates the use of a `DMDA` in the solution of a nonlinear problem. The analogous Fortran program is SNES Tutorial ex5f90; see SNES: Nonlinear Solvers for a discussion of the nonlinear solvers.

### Global Vectors with locations for ghost values

There are two minor drawbacks to the basic approach described above for unstructured grids:

- the extra memory requirement for the local work vector, `localin`, which duplicates the local values in the memory in `globalin`, and

- the extra time required to copy the local values from `localin` to `globalin`.

An alternative approach is to allocate global vectors with space preallocated for the ghost values.

    VecCreateGhost(MPI_Comm comm,PetscInt n,PetscInt N,PetscInt nghost,PetscInt *ghosts,Vec *vv)

or

    VecCreateGhostWithArray(MPI_Comm comm,PetscInt n,PetscInt N,PetscInt nghost,PetscInt *ghosts,PetscScalar *array,Vec *vv)

Here `n` is the number of local vector entries, `N` is the number of global entries (or `NULL`), and `nghost` is the number of ghost entries. The array `ghosts` is of size `nghost` and contains the global vector location for each local ghost location. Using `VecDuplicate()` or `VecDuplicateVecs()` on a ghosted vector will generate additional ghosted vectors.

In many ways, a ghosted vector behaves like any other MPI vector created by `VecCreateMPI()`. The difference is that the ghosted vector has an additional “local” representation that allows one to access the ghost locations. This is done through the call to

    VecGhostGetLocalForm(Vec g,Vec *l);

The vector `l` is a sequential representation of the parallel vector `g` that shares the same array space (and hence numerical values); but allows one to access the “ghost” values past “the end of the” array. Note that one accesses the entries in `l` using the local numbering of elements and ghosts, while they are accessed in `g` using the global numbering.

A common usage of a ghosted vector is given by

    VecGhostUpdateBegin(Vec globalin,InsertMode INSERT_VALUES, ScatterMode SCATTER_FORWARD);
    VecGhostUpdateEnd(Vec globalin,InsertMode INSERT_VALUES, ScatterMode SCATTER_FORWARD);
    VecGhostGetLocalForm(Vec globalin,Vec *localin);
    VecGhostGetLocalForm(Vec globalout,Vec *localout);
    ...  Do local calculations from localin to localout ...
    VecGhostRestoreLocalForm(Vec globalin,Vec *localin);
    VecGhostRestoreLocalForm(Vec globalout,Vec *localout);
    VecGhostUpdateBegin(Vec globalout,InsertMode ADD_VALUES, ScatterMode SCATTER_REVERSE);
    VecGhostUpdateEnd(Vec globalout,InsertMode ADD_VALUES, ScatterMode SCATTER_REVERSE);

The routines `VecGhostUpdateBegin()` and `VecGhostUpdateEnd()` are equivalent to the routines `VecScatterBegin()` and `VecScatterEnd()` above, except that since they are scattering into the ghost locations, they do not need to copy the local vector values, which are already in place. In addition, the user does not have to allocate the local work vector since the ghosted vector already has allocated slots to contain the ghost values.

The input arguments `INSERT_VALUES` and `SCATTER_FORWARD` cause the ghost values to be correctly updated from the appropriate process. The arguments `ADD_VALUES` and `SCATTER_REVERSE` update the “local” portions of the vector from all the other processes’ ghost values. This would be appropriate, for example, when performing a finite element assembly of a load vector. One can also use `MAX_VALUES` or `MIN_VALUES` with `SCATTER_REVERSE`.

`DMPLEX` does not yet support ghosted vectors sharing memory with the global representation. This is a work in progress; if you are interested in this feature, please contact the PETSc community members.

Partitioning discusses the important topic of partitioning an unstructured grid.

## Application Orderings

When writing parallel PDE codes, there is extra complexity caused by having multiple ways of indexing (numbering) and ordering objects such as vertices and degrees of freedom. For example, a grid generator or partitioner may renumber the nodes, requiring adjustment of the other data structures that refer to these objects; see Figure Natural Ordering and PETSc Ordering for a 2D Distributed Array (Four Processes). PETSc provides various tools to help manage the mapping amongst the various numbering systems. The most basic is the `AO` (application ordering), which enables mapping between different global (cross-process) numbering schemes.

In many applications, it is desirable to work with one or more “orderings” (or numberings) of degrees of freedom, cells, nodes, etc. Doing so in a parallel environment is complicated by the fact that each process cannot keep complete lists of the mappings between different orderings. In addition, the orderings used in the PETSc linear algebra routines (often contiguous ranges) may not correspond to the “natural” orderings for the application.

PETSc provides certain utility routines that allow one to deal cleanly and efficiently with the various orderings. To define a new application ordering (called an `AO` in PETSc), one can call the routine

    AOCreateBasic(MPI_Comm comm,PetscInt n,const PetscInt apordering[],const PetscInt petscordering[],AO *ao);

The arrays `apordering` and `petscordering`, respectively, contain a list of integers in the application ordering and their corresponding mapped values in the PETSc ordering. Each process can provide whatever subset of the ordering it chooses, but multiple processes should never contribute duplicate values. The argument `n` indicates the number of local contributed values.

For example, consider a vector of length 5, where node 0 in the application ordering corresponds to node 3 in the PETSc ordering. In addition, nodes 1, 2, 3, and 4 of the application ordering correspond, respectively, to nodes 2, 1, 4, and 0 of the PETSc ordering. We can write this correspondence as

\[\{ 0, 1, 2, 3, 4 \} \to \{ 3, 2, 1, 4, 0 \}. \]

The user can create the PETSc `AO` mappings in several ways. For example, if using two processes, one could call

    AOCreateBasic(PETSC_COMM_WORLD,2,{0,3},{3,4},&ao);

on the first process and

    AOCreateBasic(PETSC_COMM_WORLD,3,{1,2,4},{2,1,0},&ao);

on the other process.

Once the application ordering has been created, it can be used with either of the commands

    AOPetscToApplication(AO ao,PetscInt n,PetscInt *indices);
    AOApplicationToPetsc(AO ao,PetscInt n,PetscInt *indices);

Upon input, the `n`-dimensional array `indices` specifies the indices to be mapped, while upon output, `indices` contains the mapped values. Since we, in general, employ a parallel database for the `AO` mappings, it is crucial that all processes that called `AOCreateBasic()` also call these routines; these routines cannot be called by just a subset of processes in the MPI communicator that was used in the call to `AOCreateBasic()`.

An alternative routine to create the application ordering, `AO`, is

    AOCreateBasicIS(IS apordering,IS petscordering,AO *ao);

where index sets are used instead of integer arrays.

The mapping routines

    AOPetscToApplicationIS(AO ao,IS indices);
    AOApplicationToPetscIS(AO ao,IS indices);

will map index sets (`IS` objects) between orderings. Both the `AOXxxToYyy()` and `AOXxxToYyyIS()` routines can be used regardless of whether the `AO` was created with a `AOCreateBasic()` or `AOCreateBasicIS()`.

The `AO` context should be destroyed with `AODestroy(AO *ao)` and viewed with `AOView(AO ao,PetscViewer viewer)`.

Although we refer to the two orderings as “PETSc” and “application” orderings, the user is free to use them both for application orderings and to maintain relationships among a variety of orderings by employing several `AO` contexts.

The `AOxxToxx()` routines allow negative entries in the input integer array. These entries are not mapped; they remain unchanged. This functionality enables, for example, mapping neighbor lists that use negative numbers to indicate nonexistent neighbors due to boundary conditions, etc.

Since the global ordering that PETSc uses to manage its parallel vectors (and matrices) does not usually correspond to the “natural” ordering of a two- or three-dimensional array, the `DMDA` structure provides an application ordering `AO` (see Application Orderings) that maps between the natural ordering on a rectangular grid and the ordering PETSc uses to parallelize. This ordering context can be obtained with the command

    DMDAGetAO(DM da,AO *ao);

In Figure Natural Ordering and PETSc Ordering for a 2D Distributed Array (Four Processes), we indicate the orderings for a two-dimensional `DMDA`, divided among four processes.

Fig. 4 Natural Ordering and PETSc Ordering for a 2D Distributed Array (Four Processes)