[petsc-users] [petsc-maint] norm L2 problemQuestion about changing the norm used in nonlinear solvers (L2 Euclidean vs. L2 Lebesgue)

Matthew Knepley knepley at gmail.com
Mon Jul 28 11:55:04 CDT 2025 

On Mon, Jul 28, 2025 at 11:00 AM Ali ALI AHMAD <ali.ali_ahmad at 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!ZSlAlCgNHU3fRLXzgg5EFEe53RPc1qNdYZxn7EdOoqgcUzNXDLCWTu2r4nJaeRsjD5zTKLeJcfm7BpSaFmx-$ 
>    <https://urldefense.us/v3/__https://arxiv.org/abs/1607.04254__;!!G_uCfscf7eWS!ep85s6aXge00Uc1_kixR9yGEU0FBO9W8HTZ6LsVYzy9n9Jel5_r26mPMP138k1Ilhi2CAN6cR4DGe4kKZG997W1E0q3AIw$>.
>    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

>
>    - 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...
>
> 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.

>
>    -
>
>    For the convergence test as well, in the linear system for finding the
>    direction *d* (A · d = b),
>
> You want to use the inner product that generates your weighted L_2 norm?

>
>    -
>
>    and also when we search for a good step using the line search formula:
>    xk+1=xk+step×d.
>
> You want to minimize your weighted L_2 norm. That should work in the same
way.

  Thanks,

     Matt


> I hope this explanation is clear for you.
>
> Best regards,
> Ali ALI AHMAD
>
> ------------------------------
> *De: *"Barry Smith" <bsmith at petsc.dev>
> *À: *"Ali ALI AHMAD" <ali.ali_ahmad at utt.fr>
> *Cc: *"petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <
> petsc-maint at 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)
>
>
>
> On Jun 20, 2025, at 5:10 AM, Ali ALI AHMAD <ali.ali_ahmad at 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.
>
>
>    Ok, could you point to sample code (any language) or written algorithms
> where a different norm is used in the line search?
>
>
> * 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.
>
>
>    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
>
>
> 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" <bsmith at petsc.dev>
> *À: *"Ali ALI AHMAD" <ali.ali_ahmad at utt.fr>
> *Cc: *"petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <
> petsc-maint at 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.
>
>
>
> On Jun 13, 2025, at 4:55 AM, Ali ALI AHMAD <ali.ali_ahmad at 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" <bsmith at petsc.dev>
> *À: *"Ali ALI AHMAD" <ali.ali_ahmad at utt.fr>
> *Cc: *"petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <
> petsc-maint at 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
>
>
> On Jun 12, 2025, at 9:42 AM, Ali ALI AHMAD <ali.ali_ahmad at 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" <ali.ali_ahmad at utt.fr>
> *À: *"Barry Smith" <bsmith at petsc.dev>
> *Cc: *"petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <
> petsc-maint at 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" <bsmith at petsc.dev>
> *À: *"Ali ALI AHMAD" <ali.ali_ahmad at utt.fr>
> *Cc: *"petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <
> petsc-maint at 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
>
>
> On Jun 11, 2025, at 4:45 AM, Ali ALI AHMAD <ali.ali_ahmad at 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
>
>
>
>
>

-- 
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

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