Building the matrix associated with pointwise evaluation of basis functions

[douglas-brinkerhoff]

I am in need of the sparse matrix associated with the evaluation of a firedrake function at a collection of arbitrary coordinates. In principle, I can build the matrix that I'm looking for column-wise with the following:

import firedrake as df
import numpy as np

mesh = df.UnitSquareMesh(25,25)

V_cg = df.FunctionSpace(mesh,"CG",1)
u = df.Function(V_cg)

interpolation_points = np.random.rand(100,2)

basis_values = []

for i in range(V_cg.dim()):
    u.dat.data[:] = 0.0
    u.dat.data[i] = 1.0
    basis_values.append(u.at(interpolation_points))

interpolation_operator = np.column_stack(basis_values)  ## This is the matrix I'm after
g = df.interpolate(df.cos(df.SpatialCoordinate(mesh)[0]*2*np.pi),V_cg)
g_at_points = interpolation_operator @ g.dat.data[:]

Obviously, this is horribly inefficient, and doesn't respect the sparsity of the problem. I also recognize that I can get the product of this matrix with a vector via the .at() or VertexOnlyMesh functionality - however I need the matrix itself (and its transpose) in order to solve a regularized least-squares problem for the function coefficients given some pointwise observations.

Is there a clever way to achieve this?

Thanks very much in advance to everyone for their insight.

[dham]

Hi Doug,

There is no way to do this at present. Leo Collins is currently working on this but he's just spinning up as a PhD student, so I don't expect immediate results.

Could you possibly use a matrix-free solver to solve the problem you are trying to solve?

[dham]

On MTW, I think you can evaluate that at a point, can't you? What we can't do is interpolate into that space. That's due to inadequacies in our finite element representation that @indiamai is working on, but it will be a while.

[douglas-brinkerhoff]

Thank you - your sketch for "transpose interpolation" is extremely helpful.

Regarding MTW, it looks like interpolation from the MTW space to a VOM isn't implemented. For example:

import firedrake as fd
import numpy as np

mesh = fd.UnitSquareMesh(5,5)

V_mtw = fd.FunctionSpace(mesh,"MTW",3)
u = fd.Function(V_mtw)

interpolation_points = np.random.rand(3,2)
vom = fd.VertexOnlyMesh(mesh,interpolation_points)
V_vom = fd.VectorFunctionSpace(vom,"DG",0)

fd.interpolate(u,V_vom)

throws the error
NotImplementedError: TODO: not yet thought about it
and similarly for evaluating a MTW function with .at().

However, I just did some experimentation and it looks like interpolation from the MTW space to - say - a vector valued CG space does work.

[douglas-brinkerhoff]

Hi All, I just wanted to bump this regarding point evaluation of the MTW element. Doesn't seem to be implemented currently, but perhaps one of you clever folks know of a workaround. Thanks!

[pbrubeck]

Your request has been heard! Here are the PRs firedrakeproject/fiat#144 and #4213. It might take a few days before they make it into the master branch, but you can test the branches.

[douglas-brinkerhoff]

I just tested the development branches that you linked, and they work perfectly for my use case. Thank you very much for working on this - it's tremendously helpful!