How to Keep Certain Solution Components Fixed in KINSOL?

[swiesheier]

For finite element simulations, it's common to solve nonlinear systems involving multiple blocks in a staggered fashion.

My question is:
Is it possible to configure KINSOL to update only a subset of the solution vector while keeping the remaining components fixed?

I’m aware of the functionality to impose upper and lower bounds (e.g., non-negativity), but I couldn’t find a way to fix individual components of the solution vector --- something that would be very useful for staggered solvers.

Any suggestions or workarounds would be greatly appreciated!

[drreynolds]

One possibility for this might be to create multiple KINSOL instances, one for each block. These could take the same NVector as input, but would only define nonlinear residuals for their respective block. You could then call each KINSOL instance sequentially, so that the first instance would solve the nonlinear system for its block, and so on. Since each block would define zero-valued residuals for non-owned blocks, each KINSOL solve would "update" non-owned solution components with zeros, thereby leaving them unchanged.

You could then wrap these sequential KINSOL solves in an outer iteration to ensure convergence of the overall nonlinear system.

A fundamental question, however, is how to ensure sufficient sufficient coupling from one block solve to the next, so that the eventual decoupled solution procedure solves the original coupled system of equations. There is a multitude of research on this topic (overlap regions, transmission boundary conditions), and the best approach is likely problem-dependent. If you're unfamiliar, then I recommend the long-running "Domain Decomposition" conference series, e.g. https://www.dd29.polimi.it/

[swiesheier]

Since each block would define zero-valued residuals for non-owned blocks, each KINSOL solve would "update" non-owned solution components with zeros, thereby leaving them unchanged.

I understand the idea of defining zero-valued residuals for non-owned solution components. But to ensure also zero-valued updates for those components, do we not have to modify the Jacobian matrix as well, e.g., by zeroing out the rows/columns (except diagonal) for those components?

[drreynolds]

This depends on how you construct the Jacobian matrix for KINSOL. If you provide your own functions to compute the Jacobian matrices for each KINSOL instance, then you should be able to enforce this structure yourself. I agree that the structure you describe should be sufficient.

What kind of linear solver are you using with KINSOL, and if it is a direct solver, are you providing a Jacobian function or are you asking KINSOL to approximate the Jacobian internally via difference quotients (i.e., you do not provide a function to compute the Jacobian)?

[swiesheier]

I interface with KINSOL through a wrapper from deal.II. The Jacobian matrix is assembled analytically. A good approach, as you suggested, is to allow KINSOL to work on the full solution and residual vector, setting the residuals of the non-owned solution components to zero. For the linear solver with the Jacobian, I then extract the relevant Jacobian block and set the Newton updates for the non-owned blocks to zero.

[drreynolds]

That sounds good to me. If you're good with this, then can we close this issue?