共轭梯度中使用的Scipy线性代数LinearOperator函数

Question

I am preconditioning a matrix using spilu, however, to pass this preconditioner into cg (the built in conjugate gradient method) it is necessary to use the LinearOperator function, can someone explain to me the parameter matvec, and why I need to use it. Below is my current code

Ainv=scla.spilu(A,drop_tol= 1e-7)
Ainv=scla.LinearOperator(Ainv.shape,matvec=Ainv)
scla.cg(A,b,maxiter=maxIterations, M = Ainv)

However this doesnt work and I am given the error TypeError: 'SuperLU' object is not callable. I have played around and tried

Ainv=scla.LinearOperator(Ainv.shape,matvec=Ainv.solve)

instead. This seems to work but I want to know why matvec needs Ainv.solve rather than just Ainv, and is it the right thing to feed LinearOperator?

Thanks for your time

Answer 1

Without having much experience with this part of scipy, some comments:

• According to the docs you don't have to use LinearOperator , but you might do
  • M : {sparse matrix, dense matrix, LinearOperator} , so you can use explicit matrices too!
  • The idea/advantage of the LinearOperator :
    • Many iterative methods (eg cg, gmres) do not need to know the individual entries of a matrix to solve a linear system A*x=b. Such solvers only require the computation of matrix vector products Many iterative methods (eg cg, gmres) do not need to know the individual entries of a matrix to solve a linear system A*x=b. Such solvers only require the computation of matrix vector products docs
      • Depending on the task, sometimes even matrix-free approaches are available which can be much more efficient
• The working approach you presented is indeed the correct one ( some other source doing it similarily , and some course-materials doing it like that )
  • The idea of not using the inverse matrix, but using solve() here is not to form the inverse explicitly (which might be very costly)
    • A similar idea is very common in BFGS-based optimization algorithms although wiki might not give much insight here
      • scipy has an extra LinearOperator for this not forming the inverse explicitly! (although i think it's only used for statistics / completing/finishing some optimization; but i successfully build some LBFGS-based optimizers with this one)
    • Source @ scicomp.stackexchange discussing this without touching scipy
    • And because of that i would assume spilu is completely going for this too (returning an object with a solve-method)