Methods for calculating the symmetric part of a matrix

Question

I am using a multigrid preconditioned GMRES method for a nonsymmetric matrix. The matrix is the discretisation of the derivative of a nonlinear operator. Since multigrid is not the best for nonsymmetric problems I am preconditioning the 'symmetric part' of the matrix, where I am taking the symmetric part of matrix $A$ to be $(A + A^{\text{T}})/2$, where $A^{\text{T}}$ is the transpose of $A$.

My problem is that the 'symmetric part' of the matrix that I get is not positive definite. I was wondering if there is a way to get a symmetric positive definite part of a matrix which is just some manipulation of the original matrix, i.e.

$\tilde{A} = F(A)$

for $F$ some function of $A$

EDIT: From the comments and answers below I realise that I have not included a large part of the important information required for my question, so here is some supplementary information. I am using a standard geometric multigrid implementation with a pointwise Jacobi / Gauss-Seidel smoother. I am aware that using a smoothing operator that is tailored to the specific operator that I am dealing with will give better results, but I am assessing the effectiveness of the standard linear geometric multigrid implementation for some problems I am interested in. As such I ideally want to be applying the multigrid iteration to a symmetric positive definite matrix. I have information about the linear operator that I can use to get a matrix to precondition which is symmetric positive definite and 'close to' the matrix I am interested in inverting, which works quite well. What I am particularly interested in, though, is knowing if there is some 'black box' function which can give a symmetric positive definite matrix for the preconditioning that will be 'good' for a given matrix, without having knowledge of where the given matrix came from

Answer 1

You seem to be under the misconception that multigrid only works for symmetric and positive definite matrices, and that you can apply a multigrid preconditioner if only you can replace the matrix by some SPD matrix. But that's not true. Multigrid works if the error components of your linear system live on all scales and if the matrix allows to construct a smoother that smoothes the error components on the finest levels. But this is not a result of matrix properties such as symmetricity or positive definiteness. Rather, it follows from the underlying differential operator.

Furthermore, if I understand you right, you want to apply the multigrid preconditioner to a matrix $f(A)$ and use this to precondition the solution of a linear system involving the matrix $A$. This will in general only bring any reduction in iteration counts if $f(A)$ and $A$ are in fact closely related. Just because you can find a function $f$ that maps any matrix $A$ to a symmetric and positive matrix $f(A)$ and for which you have a multigrid implementation does not mean that this is then automatically a good preconditioner for $A$.