Re: [deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-24 Thread Thomas WICK 

Dear Wolfgang,


Am 21.05.21 um 14:48 schrieb Wolfgang Bangerth:

On 5/21/21 2:23 AM, Sebastian Kinnewig wrote:


The problem you are considering, i.e.
     curl ( curl ( E ) ) - k^2 E = 0
is respectively sometimes called the ill-posed Maxwell's equations or 
the indefinite Maxwell's equations.
When you are considering only the 2D case a direct solver should be 
enough.
But for the 3D case you will need more advanced methods. The most 
common approach to solve these equations is via domain decomposition 
method (ddm). The basic idea of the ddm is to divide the the domain 
into smaller subdomains, where each subdomain becomes small enough so 
it can be handled by a direct solver. The ddm for Maxwell is 
explained in greater detail in this paper:A quasi-optimal domain 
decomposition algorithm for the time-harmonic Maxwell's equations 
. 
But it is not easy to build a Maxwell solver based on ddm in deal.II 
(I know this, since I am currently working on such a solver by myself).

I am sorry, there is no easy way to do this.


This is somewhat outside of my realm of experience, but I'll chime in 
anyway:


Domain decomposition methods conceptually originated from the desire 
to use serial codes that were developed in the 1990s and re-use them 
in a parallel context. The question was how to connect them into a 
parallel code. deal.II does not follow this paradigm: it just 
distributes *all* data structures (mesh, matrices, vectors, etc). As a 
consequence, it is difficult to press deal.II into service for DD 
methods because they philosophically just don't quite fit into the 
deal.II world view.


But, I believe you also don't have to because essentially everything 
that has been developed in the DD world is also available in the 
fully-distributed world deal.II lives in. In the case you describe 
above, the key piece is the direct solver that can be run on each 
subdomain sequentially. But there are good parallel direct solvers 
that one could use -- specifically, I would suggest to explore MUMPS, 
which you can access via its PETSc interfaces. (I think I recall that 
there is also a Trilinos parallel direct solver, but I've forgotten 
its name and we don't have an interface to it.) If you were to 
consider something like step-40, where the domain is split into 
subdomains, the matrix is split into corresponding sub-matrices, then 
a parallel direct solver such as MUMPS can conceptually be thought of 
as inverting the diagonal blocks of the matrix corresponding to each 
subdomain -- i.e., not so different from what a direct solver applied 
to one subdomain in the DD context does. The difference is that in the 
DD context, you then have to think of how to stitch the individual 
subdomains back together (the boundary or "transmission" conditions 
between subdomains) whereas in the fully-distributed, 
one-global-linear-system world this is unnecessary.




Thanks for your email concerning DD and deal.II. In general, we agree 
that DD in the deal.II context
is not really necessary. Just using a parallel sparse direct solver 
(e.g. MUMPS) is not an option because

of memory.

Maxwell is indeed very special and one cannot simply apply well-known
concepts. Of course, Hiptmair proposed in 1998 a multigrid method for 
Maxwell. The key is a special
construction of the smoother (Gauß-Seidel-like / multiplicative Schwarz) 
and a decomposition of the function spaces.


Of course this paper is highly cited and used, but it does not seem that 
idea resolves all difficulties one may have

in Maxwell because of the indefinite structure.

Recently, over the last years, DD seems to be more competitive (e.g., 
Bouajaj et al. JCP 2015) and for exactly this reason, we started

developing a DD solution within deal.II. Our current results are promising.

We need to couple this later with Navier-Stokes and/or solids and for 
this reason,
we decided to go with deal.II, despite that other packages (ngsolve for 
instance) could

have been better for Maxwell alone.

Any further comments are specifically welcome of course.

Best regards,

Thomas

---
Thomas Wick
Leibniz University Hannover
https://thomaswick.org








Best
Wolfgang



--
---

Prof. Dr. Thomas Wick
Leibniz Universität Hannover (LUH)
Institut für Angewandte Mathematik (IfAM)
Arbeitsgruppe Wissenschaftliches Rechnen (GWR)

Welfengarten 1
30167 Hannover, Germany

Tel.:   +49 511 762 3360
Email:  thomas.w...@ifam.uni-hannover.de
www:https://ifam.uni-hannover.de/wick
www:https://

Re: [deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-21 Thread Wolfgang Bangerth 

On 5/21/21 2:23 AM, Sebastian Kinnewig wrote:


The problem you are considering, i.e.
     curl ( curl ( E ) ) - k^2 E = 0
is respectively sometimes called the ill-posed Maxwell's equations or the 
indefinite Maxwell's equations.

When you are considering only the 2D case a direct solver should be enough.
But for the 3D case you will need more advanced methods. The most common 
approach to solve these equations is via domain decomposition method (ddm). 
The basic idea of the ddm is to divide the the domain into smaller subdomains, 
where each subdomain becomes small enough so it can be handled by a direct 
solver. The ddm for Maxwell is explained in greater detail in this paper: A 
quasi-optimal domain decomposition algorithm for the time-harmonic Maxwell's 
equations 
. 
But it is not easy to build a Maxwell solver based on ddm in deal.II (I know 
this, since I am currently working on such a solver by myself).

I am sorry, there is no easy way to do this.


This is somewhat outside of my realm of experience, but I'll chime in anyway:

Domain decomposition methods conceptually originated from the desire to use 
serial codes that were developed in the 1990s and re-use them in a parallel 
context. The question was how to connect them into a parallel code. deal.II 
does not follow this paradigm: it just distributes *all* data structures 
(mesh, matrices, vectors, etc). As a consequence, it is difficult to press 
deal.II into service for DD methods because they philosophically just don't 
quite fit into the deal.II world view.


But, I believe you also don't have to because essentially everything that has 
been developed in the DD world is also available in the fully-distributed 
world deal.II lives in. In the case you describe above, the key piece is the 
direct solver that can be run on each subdomain sequentially. But there are 
good parallel direct solvers that one could use -- specifically, I would 
suggest to explore MUMPS, which you can access via its PETSc interfaces. (I 
think I recall that there is also a Trilinos parallel direct solver, but I've 
forgotten its name and we don't have an interface to it.) If you were to 
consider something like step-40, where the domain is split into subdomains, 
the matrix is split into corresponding sub-matrices, then a parallel direct 
solver such as MUMPS can conceptually be thought of as inverting the diagonal 
blocks of the matrix corresponding to each subdomain -- i.e., not so different 
from what a direct solver applied to one subdomain in the DD context does. The 
difference is that in the DD context, you then have to think of how to stitch 
the individual subdomains back together (the boundary or "transmission" 
conditions between subdomains) whereas in the fully-distributed, 
one-global-linear-system world this is unnecessary.


Best
 Wolfgang

--

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

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[deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-21 Thread Sebastian Kinnewig 

Hello Abbas,

okay, I see, previously there was just a sign error in your stack overflow 
post, but the sign is very important in this case. So let's sum it up:

The problem you previously stated in your stack overflow post, i.e.
curl ( curl ( E ) ) + k^2 E = 0
is sometimes called the well-posed Maxwell's equations or the definite 
Maxwell's equations. As the name suggests this leads to a positive definite 
system and is therefore easier to solve. 
For example, the definite Maxwell's equations can be solved efficiently via 
an iterative solver (e.g. GMRES) together with the block-preconditioner 
given in the in the "Large scale 3D geomag... 
" paper.
Also AMS is only a solver for the definite Maxwell's equations.

The problem you are considering, i.e.
curl ( curl ( E ) ) - k^2 E = 0
is respectively sometimes called the ill-posed Maxwell's equations or the 
indefinite Maxwell's equations.
When you are considering only the 2D case a direct solver should be enough. 
But for the 3D case you will need more advanced methods. The most common 
approach to solve these equations is via domain decomposition method (ddm). 
The basic idea of the ddm is to divide the the domain into smaller 
subdomains, where each subdomain becomes small enough so it can be handled 
by a direct solver. The ddm for Maxwell is explained in greater detail in 
this paper: A quasi-optimal domain decomposition algorithm for the 
time-harmonic Maxwell's equations 
. But 
it is not easy to build a Maxwell solver based on ddm in deal.II (I know 
this, since I am currently working on such a solver by myself). 
I am sorry, there is no easy way to do this.

Best,
Sebastian
Abbas schrieb am Donnerstag, 20. Mai 2021 um 23:33:22 UTC+2:

> Thank you Sebestian, 
>
> In case someone comes across this thread in the future 
> It seems that the problem that is being solved in the "Large scale 3D 
> geomag... " paper is 
> different than mine. 
> The paper's: curl x curl x E + sigma E = J   , sigma > 0 
> Mine is: curl x curl x E  - k^2 E  = 0 
> and because of that it looks Hypre's AMS 
>  solver can only 
> work with problems as given in the paper
> Am I correct here? 
>
> best,
> Abbas 
>
> On Wednesday, May 19, 2021 at 12:29:13 PM UTC-4 sebastian...@gmail.com 
> wrote:
>
>>
>> Hi Abbas,
>>
>> as you already noticed, standard preconditioners do not yield very good 
>> convergence for Maxwell's equations, for more details see Why it is 
>> Difficult to Solve Helmholtz Problems with Classical Iterative Methods 
>> . Since 
>> the scalar Helmholtz problem is closely related to the Maxwell's equations, 
>> the same arguments from the Helmholtz problem also hold true for the 
>> Maxwell's equations. So not even your plan B would work (see Geometric 
>> Multigrid Methods for Maxwell’s Equations 
>> , Chapter 6.2).
>>
>> Therefore you need to apply specially suited preconditioners for the 
>> Maxwell's equations. For the problem you are considering, I would recommend 
>> the block-preconditioner, which is presented in this paper: Large-scale 
>> 3D geoelectromagnetic modeling using parallel adaptive high-order finite 
>> element method . As 
>> a starting point for your implementation of that block-preconditioner you 
>> can use step-22 
>> . 
>>
>> An other approach to solve the Maxwell's equations is via domain 
>> decomposition method (for example see: A quasi-optimal domain 
>> decomposition algorithm for the time-harmonic Maxwell's equations 
>> ).
>>
>> I hope this helps you, best regards
>> Sebastian
>>   
>> Abbas schrieb am Mittwoch, 19. Mai 2021 um 15:45:35 UTC+2:
>>
>>> This isn't a dealii problem and I am sorry if this isn't the right place 
>>> to post to, but if there is someone who could give me some hints on how I 
>>> would approach this then it would be you guys. 
>>>
>>> I have been trying to solve the time harmonic maxwell equation 
>>> discritized with the use of Nedelec elements. Details are in This stack 
>>> overflow post 
>>> 
>>> . 
>>>
>>> I have an issue with the linear system, since using any black box 
>>> Trilinos preconditioner yields no convergence or worse convergence than a 
>>> simple Jaccobi or an identity preconditioner and even in that case the 
>>> restarted GMRES solver needs more than 400 
>>> vectors to converge and it gets worse with a refined mesh. My system

[deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-21 Thread Alexander 
Hi Abbas
AMS requires the matrix to be (semi-)positive definite. The problem you 
stated is not going to satisfy this if I understand it right.

Alexander

On Thursday, May 20, 2021 at 11:33:22 PM UTC+2 Abbas wrote:

> Thank you Sebestian, 
>
> In case someone comes across this thread in the future 
> It seems that the problem that is being solved in the "Large scale 3D 
> geomag... " paper is 
> different than mine. 
> The paper's: curl x curl x E + sigma E = J   , sigma > 0 
> Mine is: curl x curl x E  - k^2 E  = 0 
> and because of that it looks Hypre's AMS 
>  solver can only 
> work with problems as given in the paper
> Am I correct here? 
>
> best,
> Abbas 
>
> On Wednesday, May 19, 2021 at 12:29:13 PM UTC-4 sebastian...@gmail.com 
> wrote:
>
>>
>> Hi Abbas,
>>
>> as you already noticed, standard preconditioners do not yield very good 
>> convergence for Maxwell's equations, for more details see Why it is 
>> Difficult to Solve Helmholtz Problems with Classical Iterative Methods 
>> . Since 
>> the scalar Helmholtz problem is closely related to the Maxwell's equations, 
>> the same arguments from the Helmholtz problem also hold true for the 
>> Maxwell's equations. So not even your plan B would work (see Geometric 
>> Multigrid Methods for Maxwell’s Equations 
>> , Chapter 6.2).
>>
>> Therefore you need to apply specially suited preconditioners for the 
>> Maxwell's equations. For the problem you are considering, I would recommend 
>> the block-preconditioner, which is presented in this paper: Large-scale 
>> 3D geoelectromagnetic modeling using parallel adaptive high-order finite 
>> element method . As 
>> a starting point for your implementation of that block-preconditioner you 
>> can use step-22 
>> . 
>>
>> An other approach to solve the Maxwell's equations is via domain 
>> decomposition method (for example see: A quasi-optimal domain 
>> decomposition algorithm for the time-harmonic Maxwell's equations 
>> ).
>>
>> I hope this helps you, best regards
>> Sebastian
>>   
>> Abbas schrieb am Mittwoch, 19. Mai 2021 um 15:45:35 UTC+2:
>>
>>> This isn't a dealii problem and I am sorry if this isn't the right place 
>>> to post to, but if there is someone who could give me some hints on how I 
>>> would approach this then it would be you guys. 
>>>
>>> I have been trying to solve the time harmonic maxwell equation 
>>> discritized with the use of Nedelec elements. Details are in This stack 
>>> overflow post 
>>> 
>>> . 
>>>
>>> I have an issue with the linear system, since using any black box 
>>> Trilinos preconditioner yields no convergence or worse convergence than a 
>>> simple Jaccobi or an identity preconditioner and even in that case the 
>>> restarted GMRES solver needs more than 400 
>>> vectors to converge and it gets worse with a refined mesh. My system is 
>>> broken into 2x2 block matrices much like step-29  by the way. 
>>>
>>> Creating a multigrid solver like step-16 might solve this issue but I 
>>> would rather use that as plan B. 
>>>
>>> Any hint is appreciated.  
>>>
>>> Thanx :3 
>>> Abbas 
>>>
>>>

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[deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-20 Thread Abbas 
Thank you Sebestian, 

In case someone comes across this thread in the future 
It seems that the problem that is being solved in the "Large scale 3D 
geomag... " paper is 
different than mine. 
The paper's: curl x curl x E + sigma E = J   , sigma > 0 
Mine is: curl x curl x E  - k^2 E  = 0 
and because of that it looks Hypre's AMS 
 solver can only 
work with problems as given in the paper
Am I correct here? 

best,
Abbas 

On Wednesday, May 19, 2021 at 12:29:13 PM UTC-4 sebastian...@gmail.com 
wrote:

>
> Hi Abbas,
>
> as you already noticed, standard preconditioners do not yield very good 
> convergence for Maxwell's equations, for more details see Why it is 
> Difficult to Solve Helmholtz Problems with Classical Iterative Methods 
> . Since 
> the scalar Helmholtz problem is closely related to the Maxwell's equations, 
> the same arguments from the Helmholtz problem also hold true for the 
> Maxwell's equations. So not even your plan B would work (see Geometric 
> Multigrid Methods for Maxwell’s Equations 
> , Chapter 6.2).
>
> Therefore you need to apply specially suited preconditioners for the 
> Maxwell's equations. For the problem you are considering, I would recommend 
> the block-preconditioner, which is presented in this paper: Large-scale 
> 3D geoelectromagnetic modeling using parallel adaptive high-order finite 
> element method . As a 
> starting point for your implementation of that block-preconditioner you can 
> use step-22 
> . 
>
> An other approach to solve the Maxwell's equations is via domain 
> decomposition method (for example see: A quasi-optimal domain 
> decomposition algorithm for the time-harmonic Maxwell's equations 
> ).
>
> I hope this helps you, best regards
> Sebastian
>   
> Abbas schrieb am Mittwoch, 19. Mai 2021 um 15:45:35 UTC+2:
>
>> This isn't a dealii problem and I am sorry if this isn't the right place 
>> to post to, but if there is someone who could give me some hints on how I 
>> would approach this then it would be you guys. 
>>
>> I have been trying to solve the time harmonic maxwell equation 
>> discritized with the use of Nedelec elements. Details are in This stack 
>> overflow post 
>> 
>> . 
>>
>> I have an issue with the linear system, since using any black box 
>> Trilinos preconditioner yields no convergence or worse convergence than a 
>> simple Jaccobi or an identity preconditioner and even in that case the 
>> restarted GMRES solver needs more than 400 
>> vectors to converge and it gets worse with a refined mesh. My system is 
>> broken into 2x2 block matrices much like step-29  by the way. 
>>
>> Creating a multigrid solver like step-16 might solve this issue but I 
>> would rather use that as plan B. 
>>
>> Any hint is appreciated.  
>>
>> Thanx :3 
>> Abbas 
>>
>>

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[deal.II] Re: Preconditioner for the time harmonic Maxwell equation

2021-05-19 Thread Sebastian Kinnewig 

Hi Abbas,

as you already noticed, standard preconditioners do not yield very good 
convergence for Maxwell's equations, for more details see Why it is 
Difficult to Solve Helmholtz Problems with Classical Iterative Methods 
. Since the 
scalar Helmholtz problem is closely related to the Maxwell's equations, the 
same arguments from the Helmholtz problem also hold true for the Maxwell's 
equations. So not even your plan B would work (see Geometric Multigrid 
Methods for Maxwell’s Equations 
, Chapter 6.2).

Therefore you need to apply specially suited preconditioners for the 
Maxwell's equations. For the problem you are considering, I would recommend 
the block-preconditioner, which is presented in this paper: Large-scale 3D 
geoelectromagnetic modeling using parallel adaptive high-order finite 
element method . As a 
starting point for your implementation of that block-preconditioner you can 
use step-22 . 

An other approach to solve the Maxwell's equations is via domain 
decomposition method (for example see: A quasi-optimal domain decomposition 
algorithm for the time-harmonic Maxwell's equations 
).

I hope this helps you, best regards
Sebastian
  
Abbas schrieb am Mittwoch, 19. Mai 2021 um 15:45:35 UTC+2:

> This isn't a dealii problem and I am sorry if this isn't the right place 
> to post to, but if there is someone who could give me some hints on how I 
> would approach this then it would be you guys. 
>
> I have been trying to solve the time harmonic maxwell equation discritized 
> with the use of Nedelec elements. Details are in This stack overflow post 
> 
> . 
>
> I have an issue with the linear system, since using any black box Trilinos 
> preconditioner yields no convergence or worse convergence than a simple 
> Jaccobi or an identity preconditioner and even in that case the restarted 
> GMRES solver needs more than 400 
> vectors to converge and it gets worse with a refined mesh. My system is 
> broken into 2x2 block matrices much like step-29  by the way. 
>
> Creating a multigrid solver like step-16 might solve this issue but I 
> would rather use that as plan B. 
>
> Any hint is appreciated.  
>
> Thanx :3 
> Abbas 
>
>

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