# Interpolation¶

Firedrake offers highly flexible capabilities for interpolating expressions
(functions of space) into finite element `Function`

s.
Interpolation is often used to set up initial conditions and/or boundary
conditions. Mathematically, if \(e(x)\) is a function of space and
\(V\) is a finite element functionspace then
\(\operatorname{interpolate}(e, V)\) is the `Function`

\(v_i \phi_i\in V\) such that:

where \(\bar{\phi}^*_i\) is the \(i\)-th dual basis function to \(V\) suitably extended such that its domain encompasses \(e\).

Note

The extension of dual basis functions to \(e\) usually follows from the definition of the dual basis. For example, point evaluation and integral nodes can naturally be extended to any expression which is evaluatable at the relevant points, or integrable over that domain.

Firedrake will not impose any constraints on the expression to be interpolated beyond that its value shape matches that of the space into which it is interpolated. If the user interpolates an expression for which the nodes are not well defined (for example point evaluation at a discontinuity), the result is implementation-dependent.

## The interpolate operator¶

Note

The semantics for interpolation in Firedrake are in the course of changing. The documentation provided here is for the new behaviour, in which the interpolate operator is symbolic. In order to access the behaviour documented here (which is recommended), users need to use the following import line:

```
from firedrake.__future__ import interpolate
```

The basic syntax for interpolation is:

```
# create a UFL expression for the interpolation operation.
f_i = interpolate(expression, V)
# numerically evaluate the interpolation to create a new Function
f = assemble(f_i)
```

It is also possible to interpolate an expression directly into an existing
`Function`

:

```
f = Function(V)
f.interpolate(expression)
```

This is a numerical operation, equivalent to:

```
f = Function(V)
f.assign(assemble(interpolate(expression, V)))
```

The source expression can be any UFL expression with the correct shape. UFL produces clear error messages in case of syntax or type errors, yet UFL expressions have good run-time performance, since they are translated to C interpolation kernels using TSFC technology. Moreover, UFL offers a rich language for describing expressions, including:

The coordinates: in physical space as

`SpatialCoordinate`

, and in reference space as`ufl.geometry.CellCoordinate`

.Firedrake

`Function`

s, derivatives of`Function`

s, and`Constant`

s.Literal numbers, basic arithmetic operations, and also mathematical functions such as

`sin`

,`cos`

,`sqrt`

,`abs`

, etc.Conditional expressions using UFL

`conditional`

.Compound expressions involving any of the above.

Here is an example demonstrating some of these features:

```
# g is a vector-valued Function, e.g. on an H(div) function space
f = assemble(interpolate(sqrt(3.2 * div(g)), V))
```

This also works as expected when interpolating into a a space defined on the facets of the mesh:

```
trace = FunctionSpace(mesh, "HDiv Trace", 0)
f = assemble(interpolate(expression, trace))
```

Note

Interpolation is supported into most, but not all, of the elements that Firedrake provides. In particular it is not currently possible to interpolate into spaces defined by higher-continuity elements such as Argyris and Hermite.

## Interpolation across meshes¶

The interpolation API supports interpolation between meshes where the target function space has finite elements (as given in the list of supported elements)

**Lagrange/CG**(also known a Continuous Galerkin or P elements),**Q**(i.e. Lagrange/CG on lines, quadrilaterals and hexahedra),**Discontinuous Lagrange/DG**(also known as Discontinuous Galerkin or DP elements) and**DQ**(i.e. Discontinuous Lagrange/DG on lines, quadrilaterals and hexahedra).

Vector, tensor and mixed function spaces can also be interpolated into from other meshes as long as they are constructed from these spaces.

Note

The list of supported elements above is only for *target* function spaces.
Function spaces on the *source* mesh can be built from most of the supported
elements.

There are few constraints on the meshes involved: the target mesh can have a different cell shape, topological dimension, or resolution to the source mesh. There are many use cases for this: For example, two solutions to the same problem calculated on meshes with different resolutions or cell shapes can be interpolated onto one another, or onto a third, finer mesh, and be directly compared.

### Interpolating onto sub-domain meshes¶

The target mesh for a cross-mesh interpolation need not cover the full domain
of the source mesh. Volume, surface and line integrals can therefore be
calculated by interpolating onto the mesh or
immersed manifold which defines the volume,
surface or line of interest in the domain. The integral itself is calculated
by calling `assemble()`

on an approriate form over the target mesh
function space:

```
# Start with a simple field exactly represented in the function space over
# the unit square domain.
m = UnitSquareMesh(2, 2)
V = FunctionSpace(m, "CG", 2)
x, y = SpatialCoordinate(m)
f = Function(V).interpolate(x * y)
# We create a 1D mesh immersed 2D from (0, 0) to (1, 1) which we call "line".
# Note that it only has 1 cell
cells = np.asarray([[0, 1]])
vertex_coords = np.asarray([[0.0, 0.0], [1.0, 1.0]])
plex = mesh.plex_from_cell_list(1, cells, vertex_coords, comm=m.comm)
line = mesh.Mesh(plex, dim=2)
```

```
# We want to calculate the line integral of f along it. To do this we
# create a function space on the line mesh...
V_line = FunctionSpace(line, "CG", 2)
# ... and interpolate our function f onto it.
f_line = assemble(interpolate(f, V_line))
# The integral of f along the line is then a simple form expression which
# we assemble:
assemble(f_line * dx) # this outputs sqrt(2) / 3
```

For more on forms, see this section of the manual.

### Interpolating onto other meshes¶

Note

Interpolation *from* high-order meshes is
currently not supported.

If the target mesh extends outside the source mesh domain, then cross-mesh
interpolation will raise a `DofNotDefinedError`

.

```
# These meshes only share some of their domain
src_mesh = UnitSquareMesh(2, 2)
dest_mesh = UnitSquareMesh(3, 3, quadrilateral=True)
dest_mesh.coordinates.dat.data_wo[:] *= 2
# We consider a simple function on our source mesh...
x_src, y_src = SpatialCoordinate(src_mesh)
V_src = FunctionSpace(src_mesh, "CG", 2)
f_src = Function(V_src).interpolate(x_src**2 + y_src**2)
# ... and want to interpolate into a function space on our target mesh ...
V_dest = FunctionSpace(dest_mesh, "Q", 2)
```

```
# ... but get a DofNotDefinedError if we try
f_dest = assemble(interpolate(f_src, V_dest)) # raises DofNotDefinedError
```

This can be overriden with the optional `allow_missing_dofs`

keyword
argument:

```
# Setting the allow_missing_dofs keyword allows the interpolation to proceed.
f_dest = assemble(interpolate(f_src, V_dest, allow_missing_dofs=True))
```

```
f_dest = Function(V_dest).interpolate(f_src, allow_missing_dofs=True)
```

In this case, the missing degrees of freedom (DoFs, the global basis function coefficients which could not be set) are, by default, set to zero:

```
f_dest.at(1.5, 1.5) # returns 0.0
```

If we specify an output `Function`

then the missing DoFs are
unmodified.

We can optionally specify a value to use for our missing DoFs. Here
we set them to be `nan`

(‘not a number’) for easy identification:

```
f_dest = assemble(interpolate(
f_src, V_dest, allow_missing_dofs=True, default_missing_val=np.nan
))
f_dest.at(1.5, 1.5) # returns np.nan
```

If we specify an output `Function`

, this overwrites the missing
DoFs.

When using `Interpolator`

s, the `allow_missing_dofs`

keyword
argument is set at construction:

```
interpolator = Interpolator(f_src, V_dest, allow_missing_dofs=True)
```

The `default_missing_val`

keyword argument is then set whenever we call
`interpolate()`

:

```
f_dest = assemble(interpolator.interpolate(default_missing_val=np.nan))
```

If we supply an output `Function`

and don’t set
`default_missing_val`

then any missing DoFs are left as they were prior to
interpolation:

```
x_dest, y_dest = SpatialCoordinate(dest_mesh)
f_dest = Function(V_dest).interpolate(x_dest + y_dest)
f_dest.at(0.5, 0.5) # returns x_dest + y_dest = 1.0
```

```
assemble(interpolator.interpolate(), tensor=f_dest)
f_dest.at(0.5, 0.5) # now returns x_src^2 + y_src^2 = 0.5
```

```
# Similarly, using the interpolate method on a function will not overwrite
# the pre-existing values if default_missing_val is not set
f_dest.interpolate(f_src, allow_missing_dofs=True)
```

## Interpolation from external data¶

Unfortunately, UFL interpolation is not applicable if some of the
source data is not yet available as a Firedrake `Function`

or UFL expression. Here we describe a recipe for moving external to
Firedrake fields.

Let us assume that there is some function `mydata(X)`

which takes as
input an \(n \times d\) array, where \(n\) is the number of
points at which the data values are needed, and \(d\) is the
geometric dimension of the mesh. `mydata(X)`

shall return a
\(n\) long vector of the scalar values evaluated at the points
provided. (Assuming that the target `FunctionSpace`

is
scalar valued, although this recipe can be extended to vector or
tensor valued fields.) Presumably `mydata`

works by interpolating
the external data source, but the precise details are not relevant
now. In this case, interpolation into a target function space `V`

proceeds as follows:

```
# First, grab the mesh.
m = V.mesh()
# Now make the VectorFunctionSpace corresponding to V.
W = VectorFunctionSpace(m, V.ufl_element())
# Next, interpolate the coordinates onto the nodes of W.
X = assemble(interpolate(m.coordinates, W))
# Make an output function.
f = Function(V)
# Use the external data function to interpolate the values of f.
f.dat.data[:] = mydata(X.dat.data_ro)
```

This will also work in parallel, as the interpolation will occur on
each process, and Firedrake will take care of the halo updates before
the next operation using `f`

.

For interaction with external point data, see the corresponding manual section.

## Generating Functions with randomised values¶

The `randomfunctiongen`

module wraps the external numpy package numpy.random,
which gives Firedrake users an easy access to many stochastically sound random number generators,
including `PCG64`

, `Philox`

, and `SFC64`

, which are parallel-safe.
All distribution methods defined in numpy.random,
are made available, and one can pass a `FunctionSpace`

to most of these methods
to generate a randomised `Function`

.

```
mesh = UnitSquareMesh(2,2)
V = FunctionSpace(mesh, "CG", 1)
# PCG64 random number generator
pcg = PCG64(seed=123456789)
rg = RandomGenerator(pcg)
# beta distribution
f_beta = rg.beta(V, 1.0, 2.0)
print(f_beta.dat.data)
# produces:
# [0.56462514 0.11585311 0.01247943 0.398984 0.19097059 0.5446709 0.1078666 0.2178807 0.64848515]
```