* As I mentioned earlier, the weighted norm I am referring to is:
||f||^2_2 = \int_{\Omega} |f(x)|^2 \, dx \approx \sum_{K \in \Omega} \sum_q
|f(G_q)|^2 \, area(K)
where G_q is the centroid of triangle K , computed using a first-order Gaussian
quadrature rule.
Yes, I am using the inexact Newton method, where the method stops when
||F(x_k) + J_{x_k} \, d_k|| \leq \nu_k \, ||F(x_k)||.
Here, I would also like to change the norm to see whether this modification
affects the quality of the descent direction. I know that these norms are
theoretically equivalent, but I would like to test them to observe any
difference between the two. My initial expectation is that the method may
converge faster when using the L^2 norm.
Best regards,
Ali ALI AHMAD
De: "Matthew Knepley" <knep...@gmail.com>
À: "Ali ALI AHMAD" <ali.ali_ah...@utt.fr>
Cc: "Barry Smith" <bsm...@petsc.dev>, "petsc-users" <petsc-users@mcs.anl.gov>,
"petsc-maint" <petsc-ma...@mcs.anl.gov>
Envoyé: Lundi 28 Juillet 2025 18:55:04
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
On Mon, Jul 28, 2025 at 11:00 AM Ali ALI AHMAD < [ mailto:ali.ali_ah...@utt.fr
| ali.ali_ah...@utt.fr ] > wrote:
I’m sorry for getting back to you so late. Thank you for your patience and
understanding.
*
For example, when using L2 algorithms where a different norm is applied in the
line search, see Line_search_L2.png as an example from this reference: [
https://urldefense.us/v3/__https://arxiv.org/abs/1607.04254__;!!G_uCfscf7eWS!ep85s6aXge00Uc1_kixR9yGEU0FBO9W8HTZ6LsVYzy9n9Jel5_r26mPMP138k1Ilhi2CAN6cR4DGe4kKZG997W1E0q3AIw$
|
https://urldefense.us/v3/__https://arxiv.org/abs/1607.04254__;!!G_uCfscf7eWS!fiC1n5vGsbqOyw1uxsDvcoyysDPSuMVX-X60t2a22q6RxvgToFh_JTCqhACBSTB33XlfjUocJoQh741gmrYyKTsWaDMi_Q$
] . Here, we can change the norm from the L2 Euclidean to the L2 Lebesgue
norm.
What does "L2 Lebesgue" mean here? I know what I mean by L_2. It is
||f||^2_2 = \int_Omega |f(x)|^2 dx \approx \sum_q |f(x_q)|^2 w_q
Oh, from below it seems you want a weight function wt(x) in the norm. So we
would have
\sum_q |f(x_q)|^2 wt(x_q) w_q
BQ_BEGIN
* For GMRES, we need to replace NORM_2 (L2 Euclidean) in your code with
weighted_NormL2 (L2 Lebesgue) everywhere, including all the details such as in
the Arnoldi algorithm...
BQ_END
I do not quite understand here, because GMRES does not use the L:_2 norm. It
uses the l_2 norm, which is
||v||^2_2 = \sum |v_i|^2
The things in the vector are coefficients of basis functions. I guess, if you
had an interpolatory element, you could interpret this as quadrature rule,
meaning you would have
\sum wt(x_i) |v_i|^2
where x_i were the coordinates of the dual basis evaluation functionals.
BQ_BEGIN
*
For the convergence test as well, in the linear system for finding the
direction d (A · d = b),
BQ_END
You want to use the inner product that generates your weighted L_2 norm?
BQ_BEGIN
*
and also when we search for a good step using the line search formula:
xk+1=xk+step×d.
BQ_END
You want to minimize your weighted L_2 norm. That should work in the same way.
Thanks,
Matt
BQ_BEGIN
I hope this explanation is clear for you.
Best regards,
Ali ALI AHMAD
De: "Barry Smith" < [ mailto:bsm...@petsc.dev | bsm...@petsc.dev ] >
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ah...@utt.fr | ali.ali_ah...@utt.fr ] >
Cc: "petsc-users" < [ mailto:petsc-users@mcs.anl.gov | petsc-users@mcs.anl.gov
] >, "petsc-maint" < [ mailto:petsc-ma...@mcs.anl.gov | petsc-ma...@mcs.anl.gov
] >
Envoyé: Vendredi 20 Juin 2025 15:50:17
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
BQ_BEGIN
On Jun 20, 2025, at 5:10 AM, Ali ALI AHMAD < [ mailto:ali.ali_ah...@utt.fr |
ali.ali_ah...@utt.fr ] > wrote:
* Yes, I am indeed using an inexact Newton method in my code. The descent
direction is computed by solving a linear system involving the Jacobian, so the
update follows the classical formula "J(un)^{-1}d(un)=-F(un)" I'm also trying
to use a line search strategy based on a weighted L2 norm (in the Lebesgue
sense), which a priori should lead to better accuracy and faster convergence in
anisotropic settings.
BQ_END
Ok, could you point to sample code (any language) or written algorithms where a
different norm is used in the line search?
BQ_BEGIN
* During the subsequent iterations, I apply the Eisenstat–Walker method to
adapt the tolerance, which should also involve modifying the norm used in the
algorithm.
* The current implementation still uses the standard Euclidean L2 norm in
PETSc's linear solver and in GMRES. I believe this should ideally be replaced
by a weighted L2 norm consistent with the discretization. However, I haven't
yet succeeded in modifying the norm used internally by the linear solver in
PETSc, so, I'm not yet sure how much impact this change would have on the
overall convergence, but I suspect it could improve robustness, especially for
highly anisotropic problems. I would greatly appreciate any guidance on how to
implement this properly in PETSc.
BQ_END
Norms are used in multiple ways in GMRES.
1) defining convergence
2) as part of preconditioning
Again can you point to sample code (any language) or written algorithms that
describe exactly what you would like to accomplish.
Barry
BQ_BEGIN
Do not hesitate to contact me again if anything remains unclear or if you need
further information.
Best regards,
Ali ALI AHMAD
De: "Barry Smith" < [ mailto:bsm...@petsc.dev | bsm...@petsc.dev ] >
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ah...@utt.fr | ali.ali_ah...@utt.fr ] >
Cc: "petsc-users" < [ mailto:petsc-users@mcs.anl.gov | petsc-users@mcs.anl.gov
] >, "petsc-maint" < [ mailto:petsc-ma...@mcs.anl.gov | petsc-ma...@mcs.anl.gov
] >
Envoyé: Samedi 14 Juin 2025 01:06:52
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
I appreciate the clarification. I would call 3) preconditioning.
To increase my understanding, you are already using Newton's method? That is,
you compute the Jacobian of the function and use - J^{-1}(u^n) F(u^n) as your
update direction?
When you switch the inner product (or precondition) how will the search
direction be different?
Thanks
Barry
The case you need support for is becoming important to PETSc so we need to
understand it well and support it well which is why I am asking these (perhaps
to you) trivial questions.
BQ_BEGIN
On Jun 13, 2025, at 4:55 AM, Ali ALI AHMAD < [ mailto:ali.ali_ah...@utt.fr |
ali.ali_ah...@utt.fr ] > wrote:
Thank you for your message.
To answer your question: I would like to use the L2 norm in the sense of
Lebesgue for all three purposes , especially the third one .
1- For displaying residuals during the nonlinear iterations, I would like to
observe the convergence behavior using a norm that better reflects the physical
properties of the problem.
2- For convergence testing , I would like the stopping criterion to be based on
a weighted L2 norm that accounts for the geometry of the mesh (since I am
working with unstructured, anisotropic triangular meshes).
3 - Most importantly , I would like to modify the inner product used in the
algorithm so that it aligns with the weighted L2 norm (since I am working with
unstructured, anisotropic triangular meshes).
Best regards,
Ali ALI AHMAD
De: "Barry Smith" < [ mailto:bsm...@petsc.dev | bsm...@petsc.dev ] >
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ah...@utt.fr | ali.ali_ah...@utt.fr ] >
Cc: "petsc-users" < [ mailto:petsc-users@mcs.anl.gov | petsc-users@mcs.anl.gov
] >, "petsc-maint" < [ mailto:petsc-ma...@mcs.anl.gov | petsc-ma...@mcs.anl.gov
] >
Envoyé: Vendredi 13 Juin 2025 03:14:06
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
You haven't answered my question. Where (conceptually) and for what purpose do
you want to use the L2 norm.
1) displaying norms to observe the convergence behavior
2) in the convergence testing to determine when to stop
3) changing the "inner product" in the algorithm which amounts to
preconditioning.
Barry
BQ_BEGIN
On Jun 12, 2025, at 9:42 AM, Ali ALI AHMAD < [ mailto:ali.ali_ah...@utt.fr |
ali.ali_ah...@utt.fr ] > wrote:
Thank you for your answer.
I am currently working with the nonlinear solvers newtonls (with bt , l2 ,
etc.) and newtontr (using newton , cauchy , and dogleg strategies) combined
with the linear solver gmres and the ILU preconditioner, since my Jacobian
matrix is nonsymmetric.
I also use the Eisenstat-Walker method for newtonls , as my initial guess is
often very far from the exact solution.
What I would like to do now is to replace the standard Euclidean L2 norm with
the L2 norm in the Lebesgue sense in the above numerical algorithm , because my
problem is defined on an unstructured, anisotropic triangular mesh where a
weighted norm would be more physically appropriate.
Would you be able to advise me on how to implement this change properly?
I would deeply appreciate any guidance or suggestions you could provide.
Thank you in advance for your help.
Best regards,
Ali ALI AHMAD
De: "Ali ALI AHMAD" < [ mailto:ali.ali_ah...@utt.fr | ali.ali_ah...@utt.fr ] >
À: "Barry Smith" < [ mailto:bsm...@petsc.dev | bsm...@petsc.dev ] >
Cc: "petsc-users" < [ mailto:petsc-users@mcs.anl.gov | petsc-users@mcs.anl.gov
] >, "petsc-maint" < [ mailto:petsc-ma...@mcs.anl.gov | petsc-ma...@mcs.anl.gov
] >
Envoyé: Jeudi 12 Juin 2025 15:28:02
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
Thank you for your answer.
I am currently working with the nonlinear solvers newtonls (with bt , l2 ,
etc.) and newtontr (using newton , cauchy , and dogleg strategies) combined
with the linear solver gmres and the ILU preconditioner, since my Jacobian
matrix is nonsymmetric.
I also use the Eisenstat-Walker method for newtonls , as my initial guess is
often very far from the exact solution.
What I would like to do now is to replace the standard Euclidean L2 norm with
the L2 norm in the Lebesgue sense , because my problem is defined on an
unstructured, anisotropic triangular mesh where a weighted norm would be more
physically appropriate.
Would you be able to advise me on how to implement this change properly?
I would deeply appreciate any guidance or suggestions you could provide.
Thank you in advance for your help.
Best regards,
Ali ALI AHMAD
De: "Barry Smith" < [ mailto:bsm...@petsc.dev | bsm...@petsc.dev ] >
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ah...@utt.fr | ali.ali_ah...@utt.fr ] >
Cc: "petsc-users" < [ mailto:petsc-users@mcs.anl.gov | petsc-users@mcs.anl.gov
] >, "petsc-maint" < [ mailto:petsc-ma...@mcs.anl.gov | petsc-ma...@mcs.anl.gov
] >
Envoyé: Jeudi 12 Juin 2025 14:57:40
Objet: Re: [petsc-maint] norm L2 problemQuestion about changing the norm used
in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)
Do you wish to use a different norm
1) ONLY for displaying (printing out) the residual norms to track progress
2) in the convergence testing
3) to change the numerical algorithm (for example using the L2 inner product
instead of the usual linear algebra R^N l2 inner product).
For 1) use SNESMonitorSet() and in your monitor function use SNESGetSolution()
to grab the solution and then VecGetArray(). Now you can compute any weighted
norm you want on the solution.
For 2) similar but you need to use SNESSetConvergenceTest
For 3) yes, but you need to ask us specifically.
Barry
BQ_BEGIN
On Jun 11, 2025, at 4:45 AM, Ali ALI AHMAD < [ mailto:ali.ali_ah...@utt.fr |
ali.ali_ah...@utt.fr ] > wrote:
Dear PETSc team,
I hope this message finds you well.
I am currently using PETSc in a C++, where I rely on the nonlinear solvers
`SNES` with either `newtonls` or `newtontr` methods. I would like to ask if it
is possible to change the default norm used (typically the L2 Euclidean norm)
to a custom norm, specifically the L2 norm in the sense of Lebesgue (e.g.,
involving cell-wise weighted integrals over the domain).
My main goal is to define a custom residual norm that better reflects the
physical quantities of interest in my simulation.
Would this be feasible within the PETSc framework? If so, could you point me to
the recommended approach (e.g., redefining the norm manually, using specific
PETSc hooks or options)?
Thank you very much in advance for your help and for the great work on PETSc!
Best regards,
Ali ALI AHMAD
PhD Student
University of Technology of Troyes - UTT - France
GAMMA3 Project - Office H008 - Phone No: +33 7 67 44 68 18
12 rue Marie Curie - CS 42060 10004 TROYES Cedex
BQ_END
BQ_END
BQ_END
BQ_END
BQ_END
--
What most experimenters take for granted before they begin their experiments is
infinitely more interesting than any results to which their experiments lead.
-- Norbert Wiener
[
https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!fiC1n5vGsbqOyw1uxsDvcoyysDPSuMVX-X60t2a22q6RxvgToFh_JTCqhACBSTB33XlfjUocJoQh741gmrYyKTuWi3OAYA$
|
https://urldefense.us/v3/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!fiC1n5vGsbqOyw1uxsDvcoyysDPSuMVX-X60t2a22q6RxvgToFh_JTCqhACBSTB33XlfjUocJoQh741gmrYyKTvgry72Ng$
]
