Solver capabilities for Indefinite linear system

[Tom Mathew]

Dear all,

I would like your opinion and advice on solving indefinite problems with dealii. Does dealii has any solver classes (iterative) to deal with these problems ? Does the adaptive multigrid techniques be a good strategy; as presently I use Two level Optimized Schwarz methods, which seems to be working fine for my indefinite Maxwell's equations. Have anyone solved any indefinite problems iteratively with dealii before ? This is a problem where even the state of the art for Maxwell's: Hypre's Auxillary Maxwell Space (AMS) solver also seems obsolete on clusters. And, to mention, my requirement is to solve physical problems on a very large scale or 1000s of cluster MPI processes. 

Also, I'm hoping to try implement the aforesaid ORAS schemes with dealii data structures. But this needs some time as I'm a Mechanical Engineer, and also the mathematics of these methods involving restriction and partition of unity from geometric partitioning or domain decomposition seems quite different from the MPI parallelizing strategy (which I guess is algebraic) of dealii. Please correct me if have some information stated wrongly. Some previous work where an iterative solver (similar to the one I have stated) was cooked with the solver classes of dealii also would be a perfect starting point.

see the method here: https://arxiv.org/abs/1705.08138

Hoping to receive some feedback,

Thankfully,

Tom Mathew

[Wolfgang Bangerth]

Tom,

> I would like your opinion and advice on solving indefinite problems with
> dealii. Does dealii has any solver classes (iterative) to deal with these
> problems ?

Yes, plenty in fact. There are BiCGStab, GMRES, Minres, to list just a couple.

> Does the adaptive multigrid techniques be a good strategy; as
> presently I use Two level Optimized Schwarz methods, which seems to be working
> fine for my indefinite Maxwell's equations. Have anyone solved any indefinite
> problems iteratively with dealii before ?

Yes, many of the tutorial programs solve indefinite problems. Take a look, for
example, at the advection problems, or the saddle point problems, or the
coupled multiphysics problems, all of which have indefinite matrices.

> Also, I'm hoping to try implement the aforesaid ORAS schemes with dealii data
> structures. But this needs some time as I'm a Mechanical Engineer, and also
> the mathematics of these methods involving restriction and partition of unity
> from geometric partitioning or domain decomposition seems quite different from
> the MPI parallelizing strategy (which I guess is algebraic) of dealii. Please
> correct me if have some information stated wrongly. Some previous work where
> an iterative solver (similar to the one I have stated) was cooked with the
> solver classes of dealii also would be a perfect starting point.
> see the method here: https://arxiv.org/abs/1705.08138

> <https://nam01.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fabs%2F1705.08138&data=02%7C01%7CWolfgang.Bangerth%40colostate.edu%7C42883fea63d14f1fba0708d7731b3cee%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C637104433135935173&sdata=nhQdMUArCdSVEYLP%2FEZfusmJB8KZsSTovnjc8fWKSus%3D&reserved=0>

We've generally tried to always think of our linear systems as *global*, where
each processor only happens to store a certain number of rows. This has proven
to be an easier perspective to work with than the domain decomposition approach.

In your context, I would try to think about how you would solve the problem on
a single processor first. (In the context of the DD approach, looking at how
the system is solved for each subdomain may give a hint.) One can then try and
generalize this approach to a parallel setting.

Best
W.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/