Scaling And Reordering of Jacobian Free GMRES

[metu_aee]

Hello guys,

I have to code Jacobian Free version of GMRES with scaling and reordering algorithms separately. But I have serious problems about the convergence of inner gmres iterations and I have doubts on my formulation about jacobian-vector product for scaled equations since its bookkeeping gets harder.
Please notice me if the following is wrong; (For simplicity let's forget reordering now)
For row(S1) and column(S2) scalings of right preconditioned jacobian-vector product;

(1) (S1*A*S2) * (S1*M*S2)^-1 * (S1*v) = S1 * [F(Q+eps*(S1*M*S2)^-1 * (S1*v)) - F(Q)]/eps

where eps is perturbation epsilon, v is search direction, F(Q) is the function, M is right preconditioner matrix and A is jacobian matrix. Notation is similar to Yousef Saad's notation. I'm pretty sure about my formulation and application on code is right. But I'm not sure if I have to scale my residual vector. If answer is yes, how? Can someone share good references,

Thank you all.

 

[jvicens]

Hello,

Thank you for sharing your question and the details of your work so far. From what I understand, you are trying to implement a Jacobian-Free version of the GMRES algorithm with scaling and reordering techniques. However, you are facing convergence issues with the inner GMRES iterations and are unsure about the formulation of the Jacobian-vector product for scaled equations.

Firstly, I would suggest checking your implementation of the scaling and reordering algorithms. It is possible that there may be some errors in the code that are affecting the convergence of the inner GMRES iterations. You could also try implementing simpler versions of the algorithm to test for convergence and then gradually adding in the scaling and reordering techniques.

Regarding your formulation, it is always good to double-check and validate your equations with established references. I would recommend looking into papers or books by Yousef Saad, as you have mentioned in your post. You could also refer to other sources such as the SIAM Journal on Scientific Computing or the Journal of Computational Physics for relevant articles and references.

In terms of scaling the residual vector, it would depend on the specific problem you are working on. Generally, it is not necessary to scale the residual vector. However, if you are working with a particularly ill-conditioned problem, scaling the residual vector could help improve the convergence of the algorithm. Again, I would suggest consulting relevant references for more information on this.

I hope this helps and wish you all the best with your work. Don't hesitate to reach out if you have any further questions.