How to use custom preconditioner with Eigen

问题

I am trying to use a custom preconditioner for an iterative solver (CG for instance) with Eigen. Specifically, I have to solve a similar problem multiple times: the matrix changes slightly but stays close to a mean matrix. I would like to compute a Cholesky decomposition of my mean matrix and then use this as a preconditioner.

What I had in mind is something like:

ConjugateGradient< SparseMatrix<double>, Lower, CholmodSupernodalLLT<SparseMatrix<double>> > solver(meanMatrix);
solver.preconditioner().compute(meanMatrix);
// Loop on n similar matrices
for(int i = 0; i < n; i++){
  // create matrix: it is similar (in structure and in values) to meanMatrix
  SparseMatrix<double> matrix = ...;
  // create right-hand-side
  VectorXd rhs = ...;
  // update matrix reference for solver
  solver.compute(matrix);
  // solve using the preconditioned CG
  solver.solve(rhs);
}

The problem is that calling solver.compute(matrix) actually causes ConjugateGradient (in fact IterativeSolverBase) to call compute on its preconditioner (see l. 111 of IterativeSolverBase.h, Eigen 3.2.9):

m_preconditioner.compute(*mp_matrix);

In other words, the preconditioner based on the mean matrix is replaced by the Cholesky decomposition of the new matrix, and thus the CG solve converges in 1 iteration. On the contrary, I would like to keep the same preconditioner (the Cholesky decomposition of the mean matrix, computed once and for all before the loop), and solve for the different matrices using the preconditioned CG.

Is there an easy way to achieve what I am trying to do?

Many thanks in advance for your help! I hope this makes sense. If not, please do not hesitate to ask me to clarify.

回答1:

One way would be to write a small wrapper around CholmodSupernodalLLT exhibiting the required preconditioner API and whose compute() method would be a no-op. This can be done in 10-15 lines of code taking inspiration from the IdentityPreconditioner. The only required changes is to store a CholmodSupernodalLLT object (or reference) and implement solve to return m_llt.solve(b);.

来源：https://stackoverflow.com/questions/39980074/how-to-use-custom-preconditioner-with-eigen

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c++

eigen

eigen3