Re: [firedrake] Returning eigenvectors of a solution to a PDE at each node

20 Mar 2020

      Hi Andrew, Koki,

To solve local eigenproblems I currently call Eigen within par_loops. After specifying a mesh, a tensor field M (which you seek to compute the eigendecompositon of) and your PETSC_ARCH, the following should work. Extension to 3d just means changing the kernel_str appropriately.

Cheers,
Joe

PETSC_ARCH = ...
mesh = ...
P1_ten = TensorFunctionSpace(mesh, "CG", 1)
M = interpolate(..., P1_ten)  # Tensor field to compute eigendecomposition of
evec = Function(P1_ten)  # Tensor field to hold eigenvectors
eval = Function(P1_ten)  # Tensor field to hold eigenvalues

kernel_str = """
#include <Eigen/Dense>

using namespace Eigen;

void get_eigendecomposition(double EVecs_[4], double EVals_[2], const double * M_) {
  Map<Matrix<double, 2, 2, RowMajor> > EVecs((double *)EVecs_);
  Map<Vector2d> EVals((double *)EVals_);
  Map<Matrix<double, 2, 2, RowMajor> > M((double *)M_);
  SelfAdjointEigenSolver<Matrix<double, 2, 2, RowMajor>> eigensolver(M);
  Matrix<double, 2, 2, RowMajor> Q = eigensolver.eigenvectors();
  Vector2d D = eigensolver.eigenvalues();
  EVecs = Q;
  EVals = D;
}
"""
kernel = op2.Kernel(kernel_str, 'get_eigendecomposition', cpp=True, include_dirs= ["%s/include/eigen3" % PETSC_ARCH])
op2.par_loop(kernel, P1_ten.node_set, evec.dat(op2.RW), eval.dat(op2.RW), M.dat(op2.READ))

________________________________
From: firedrake-bounces@imperial.ac.uk <firedrake-bounces@imperial.ac.uk> on behalf of Sagiyama, Koki <k.sagiyama@imperial.ac.uk>
Sent: 20 March 2020 10:52
To: Andrew Hicks <ahick17@lsu.edu>; firedrake <firedrake@imperial.ac.uk>
Subject: Re: [firedrake] Returning eigenvectors of a solution to a PDE at each node

Dear Andrew,

I'm afraid that I don't know the most efficient way to do this sort of many small eigensolves, but you could get your solution in `np.ndarray` using `q_soln.dat.data` (`q_soln.dat` returns `PyOP2.Dat` object: https://op2.github.io/PyOP2/user.html#pyop2.Dat).
You could also get the mesh data calling `mesh.coordinates.dat.data` .

For instance:
...
...
...
from firedrake import *
mesh = UnitSquareMesh(2,2)
V = VectorFunctionSpace(mesh, "CG", 1)
f = Function(V)
f.dat.data
array([[0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.],
           [0., 0.]])
mesh.coordinates.dat.data
array([[0. , 0. ],
          [0. , 0.5],
          [0.5, 0. ],
          [0.5, 0.5],
          [1. , 0. ],
          [0. , 1. ],
          [1. , 0.5],
          [0.5, 1. ],
          [1. , 1. ]])
I think you can use these data in your context.

Taking a glance at the documentation, `numpy.linalg.eig` seems to take multiple matrices, so I think you can solve the node-wise eigen problems all at once.

Thank you,
Koki

________________________________
From: firedrake-bounces@imperial.ac.uk <firedrake-bounces@imperial.ac.uk> on behalf of Andrew Hicks <ahick17@lsu.edu>
Sent: Friday, March 20, 2020 4:23 AM
To: firedrake <firedrake@imperial.ac.uk>
Subject: [firedrake] Returning eigenvectors of a solution to a PDE at each node

Hello,

I am currently using firedrake to solve a PDE. The solution to this PDE is vector-valued, and I have no problem finding the solution. However, I’d like to go a step further. I have named the solution “q_soln” and this is a 2-dimesional vector. I have used this vector to construct a 2x2 matrix at the node [0.5,0.5]. I then extracted an eigenvector from this matrix, and then I printed it:

import numpy as np

from numpy import linalg as la

E1 = (1 / np.sqrt(2)) * np.array([[1,0],[0,-1]])

E2 = (1 / np.sqrt(2)) * np.array([[0,1],[1,0]])

# as an example, we evaluate the solution at the point [0.5,0.5], and pick the first eigenvector

q_soln_atpoint = q_soln.at([0.5,0.5])

Q_soln = q_soln_atpoint[0] * E1 + q_soln_atpoint[1] * E2

evals, evecs = la.eig(Q_soln) # evec is a 2x2 array with the columns being the e-vectors

q_soln_evec = np.zeros((1,2))

q_soln_evec += evecs[0,:]

print(q_soln_evec)

My question is this: how can I get firedrake to do this automatically for every node in my mesh, and then store it in as a Function “q_soln_evec”? The only reason I used numpy to get this one point is because I’m unfamiliar with firedrake. I bet firedrake has some built-in way of doing what I need here.

Thanks,

Andrew Hicks

Re: [firedrake] Returning eigenvectors of a solution to a PDE at each node

Wallwork, Joe