From m_wens01 at wwu.de  Wed Mar 20 12:15:30 2019
From: m_wens01 at wwu.de (Michael Wenske)
Date: Wed, 20 Mar 2019 12:15:30 +0100
Subject: [dune-pdelab] Global Integral condition (Lagrange Multiplier) for
 diffusive problem
Message-ID: <e5a4e66b-404d-f64a-176f-5a5798d73c55@wwu.de>

Hello experts,

I want to impose an integral condition upon a diffusive problem in
dune-PDElab.
As an example, assume a box of unity with a circle in the middle
(\Omega_{int}) which is
constrained to u=1. Outside of the circle (\Omega / \Omega_{int}) the
profile is ruled by:

-\Delta u = f(u)  in \Omega / \Omega_{int}, f(u)=u(1-u)

I do not impose any constraints at the outside of the box. Now, without
any further constraints
the only solution to the problem is a profile of u=1 everywhere, due to
the logistic growth
in the reaction term, the flattening by the diffusive operator and the
constraint in the middle.

I want to impose a further constraint to gain a non-trivial solution to
the profile.

\int_{\Omega} u dx = V_{total}

which basically expresses that I am only interested in a solution which
has a certain total
volume in the domain. Please consider the image for visualization.

Now the elegant way (correct me if I'm wrong) to do that would be via a
Lagrange multiplier:

r(u,v) = a(u,v) +l(u,v) + \lambda* (Volume - V_{tot})

Here, r(u,v) is the residuum to be minimized by my newton solver, a(u,v)
is the bilinear form and l(u,v) the
discretisation of the reaction term (as usual).

How would I go about implementing such a constraint in a clean way? The
penalty terms in DG- methods only need local information, so they can be
easily added to the
methods in the local operator. The methods assembling the different
parts of the residuum in the Localoperator only have the local dof's as
arguments but what I need is not a constraint on any single dof, but a
global one
as stated above.

How would I add such a constraint within dune-PDELab? I would also be
thankful if anyone has a hint to
relevant literature for -integral- constraints.

Thanks in advance,

Michael






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From m_wens01 at wwu.de  Wed Mar 20 13:36:59 2019
From: m_wens01 at wwu.de (Michael Wenske)
Date: Wed, 20 Mar 2019 13:36:59 +0100
Subject: [dune-pdelab] Fwd: Global Integral condition (Lagrange Multiplier)
 for diffusive problem
In-Reply-To: <758823e8-5402-6135-b436-1b26b189d2ac@mi.fu-berlin.de>
References: <758823e8-5402-6135-b436-1b26b189d2ac@mi.fu-berlin.de>
Message-ID: <015986a9-e859-49d2-3786-c9e3963220e5@wwu.de>

Thank you Carsten for answering!

(I take the liberty to forward your reply to the mailing list.)

You are correct, it is a different problem which I solve.

My take on it is, that penalty terms and lagrange multipliers are quite
similar.
In the one case you have a functional f(x) and a condition g(x) = c
which you combine
to the new Lagrange functional to be minimized: L(x,lambda) = f(x) -
lambda*(g(x)-c).
Penalty terms enforcing boundary conditions work in much the same way it
seems.

Anyhow, how would I approach implementing global volume constraints like
this in pdelab?

thanks,

Michael


-------- Forwarded Message --------
Subject: 	Re: [dune-pdelab] Global Integral condition (Lagrange
Multiplier) for diffusive problem
Date: 	Wed, 20 Mar 2019 12:43:39 +0100
From: 	Carsten Gräser <graeser at mi.fu-berlin.de>
To: 	Michael Wenske <m_wens01 at wwu.de>



Hi Michael,
I'm not an expert on PDElab, but I can give some comments
on the outlined approach:

Am 20.03.19 um 12:15 schrieb Michael Wenske:
[...]
> I do not impose any constraints at the outside of the box. Now, without
> any further constraints
> the only solution to the problem is a profile of u=1 everywhere, due to
> the logistic growth
> in the reaction term, the flattening by the diffusive operator and the
> constraint in the middle.
>
> I want to impose a further constraint to gain a non-trivial solution to
> the profile.
This cannot work out. If it's correct that there's only one
solution without constraints, then you cannot generate others
by imposing further constraints. You may, however, get other
solutions for an associated minimization problem with constraints,
but this will solve a different PDE.

[...]
> Now the elegant way (correct me if I'm wrong) to do that would be via a
> Lagrange multiplier:
>
> r(u,v) = a(u,v) +l(u,v) + \lambda* (Volume - V_{tot})
While I'm not sure what exactly you want to solve, this
does not look like a dual approach with lambda being
a Lagrange multiplier but more like a penalty approach
with penalty factor lambda.

Regards,
Carsten

>
> Here, r(u,v) is the residuum to be minimized by my newton solver, a(u,v)
> is the bilinear form and l(u,v) the
> discretisation of the reaction term (as usual).
>
> How would I go about implementing such a constraint in a clean way? The
> penalty terms in DG- methods only need local information, so they can be
> easily added to the
> methods in the local operator. The methods assembling the different
> parts of the residuum in the Localoperator only have the local dof's as
> arguments but what I need is not a constraint on any single dof, but a
> global one
> as stated above.
>
> How would I add such a constraint within dune-PDELab? I would also be
> thankful if anyone has a hint to
> relevant literature for -integral- constraints.
>
> Thanks in advance,
>
> Michael


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From graeser at mi.fu-berlin.de  Wed Mar 20 15:36:56 2019
From: graeser at mi.fu-berlin.de (=?UTF-8?Q?Carsten_Gr=c3=a4ser?=)
Date: Wed, 20 Mar 2019 15:36:56 +0100
Subject: [dune-pdelab] Fwd: Global Integral condition (Lagrange
 Multiplier) for diffusive problem
In-Reply-To: <015986a9-e859-49d2-3786-c9e3963220e5@wwu.de>
References: <758823e8-5402-6135-b436-1b26b189d2ac@mi.fu-berlin.de>
 <015986a9-e859-49d2-3786-c9e3963220e5@wwu.de>
Message-ID: <c18e3dde-d364-8a81-b245-160862638517@mi.fu-berlin.de>

Am 20.03.19 um 13:36 schrieb Michael Wenske:
> Thank you Carsten for answering!
> 
> (I take the liberty to forward your reply to the mailing list.)
> 
> You are correct, it is a different problem which I solve.
> 
> My take on it is, that penalty terms and lagrange multipliers are quite
> similar.
Unfortunately they are not:

> In the one case you have a functional f(x) and a condition g(x) = c
> which you combine
> to the new Lagrange functional to be minimized: L(x,lambda) = f(x) -
> lambda*(g(x)-c).
Yes, that's a classical formulation with a Lagrange multiplier.
Here you would look for saddle point of L which is a (x,lambda)
pair. In the penalty formulation you would look for a minimizer x of

  J(x) = f(x) + lambda*|g(x)-c|^2

for fixed scalar lambda. This essentially corresponds to your nonlinear
residual r(.,.).

> Penalty terms enforcing boundary conditions work in much the same way it
> seems.
> 
> Anyhow, how would I approach implementing global volume constraints like
> this in pdelab?
I think it would be helpful to know which of those formulations you aim
at, since they are _very_ different.

Best,
Carsten

> 
> thanks,
> 
> Michael
> 
> 
> -------- Forwarded Message --------
> Subject: 	Re: [dune-pdelab] Global Integral condition (Lagrange
> Multiplier) for diffusive problem
> Date: 	Wed, 20 Mar 2019 12:43:39 +0100
> From: 	Carsten Gräser <graeser at mi.fu-berlin.de>
> To: 	Michael Wenske <m_wens01 at wwu.de>
> 
> 
> 
> Hi Michael,
> I'm not an expert on PDElab, but I can give some comments
> on the outlined approach:
> 
> Am 20.03.19 um 12:15 schrieb Michael Wenske:
> [...]
>> I do not impose any constraints at the outside of the box. Now, without
>> any further constraints
>> the only solution to the problem is a profile of u=1 everywhere, due to
>> the logistic growth
>> in the reaction term, the flattening by the diffusive operator and the
>> constraint in the middle.
>>
>> I want to impose a further constraint to gain a non-trivial solution to
>> the profile.
> This cannot work out. If it's correct that there's only one
> solution without constraints, then you cannot generate others
> by imposing further constraints. You may, however, get other
> solutions for an associated minimization problem with constraints,
> but this will solve a different PDE.
> 
> [...]
>> Now the elegant way (correct me if I'm wrong) to do that would be via a
>> Lagrange multiplier:
>>
>> r(u,v) = a(u,v) +l(u,v) + \lambda* (Volume - V_{tot})
> While I'm not sure what exactly you want to solve, this
> does not look like a dual approach with lambda being
> a Lagrange multiplier but more like a penalty approach
> with penalty factor lambda.
> 
> Regards,
> Carsten
> 
>>
>> Here, r(u,v) is the residuum to be minimized by my newton solver, a(u,v)
>> is the bilinear form and l(u,v) the
>> discretisation of the reaction term (as usual).
>>
>> How would I go about implementing such a constraint in a clean way? The
>> penalty terms in DG- methods only need local information, so they can be
>> easily added to the
>> methods in the local operator. The methods assembling the different
>> parts of the residuum in the Localoperator only have the local dof's as
>> arguments but what I need is not a constraint on any single dof, but a
>> global one
>> as stated above.
>>
>> How would I add such a constraint within dune-PDELab? I would also be
>> thankful if anyone has a hint to
>> relevant literature for -integral- constraints.
>>
>> Thanks in advance,
>>
>> Michael
> 
> 
> 
> 
> _______________________________________________
> dune-pdelab mailing list
> dune-pdelab at lists.dune-project.org
> https://lists.dune-project.org/mailman/listinfo/dune-pdelab
> 


-- 
Prof. Dr. Carsten Gräser
Freie Universität Berlin
Institut für Mathematik
Arnimallee 6
14195 Berlin, Germany
phone: +49 30 838 72637
fax  : +49 30 838 472637
email: graeser at mi.fu-berlin.de
URL  : http://page.mi.fu-berlin.de/graeser

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