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

Wed Aug 13 14:45:33 CDT 2025

* 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" <knepley at gmail.com> 
À: "Ali ALI AHMAD" <ali.ali_ahmad at utt.fr> 
Cc: "Barry Smith" <bsmith at petsc.dev>, "petsc-users" <petsc-users at mcs.anl.gov>, "petsc-maint" <petsc-maint at 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_ahmad at utt.fr | 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!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 

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    * 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... 

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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. 

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    * 

For the convergence test as well, in the linear system for finding the direction d (A · d = b), 

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You want to use the inner product that generates your weighted L_2 norm? 

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    * 

and also when we search for a good step using the line search formula: xk+1=xk+step×d. 

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You want to minimize your weighted L_2 norm. That should work in the same way. 

Thanks, 

Matt 

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I hope this explanation is clear for you. 

Best regards, 
Ali ALI AHMAD 

De: "Barry Smith" < [ mailto:bsmith at petsc.dev | bsmith at petsc.dev ] > 
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ahmad at utt.fr | ali.ali_ahmad at utt.fr ] > 
Cc: "petsc-users" < [ mailto:petsc-users at mcs.anl.gov | petsc-users at mcs.anl.gov ] >, "petsc-maint" < [ mailto:petsc-maint at mcs.anl.gov | 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) 

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On Jun 20, 2025, at 5:10 AM, Ali ALI AHMAD < [ mailto:ali.ali_ahmad at utt.fr | 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. 

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Ok, could you point to sample code (any language) or written algorithms where a different norm is used in the line search? 

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* 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. 

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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 

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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:bsmith at petsc.dev | bsmith at petsc.dev ] > 
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ahmad at utt.fr | ali.ali_ahmad at utt.fr ] > 
Cc: "petsc-users" < [ mailto:petsc-users at mcs.anl.gov | petsc-users at mcs.anl.gov ] >, "petsc-maint" < [ mailto:petsc-maint at mcs.anl.gov | 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. 

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On Jun 13, 2025, at 4:55 AM, Ali ALI AHMAD < [ mailto:ali.ali_ahmad at utt.fr | 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" < [ mailto:bsmith at petsc.dev | bsmith at petsc.dev ] > 
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ahmad at utt.fr | ali.ali_ahmad at utt.fr ] > 
Cc: "petsc-users" < [ mailto:petsc-users at mcs.anl.gov | petsc-users at mcs.anl.gov ] >, "petsc-maint" < [ mailto:petsc-maint at mcs.anl.gov | 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 

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On Jun 12, 2025, at 9:42 AM, Ali ALI AHMAD < [ mailto:ali.ali_ahmad at utt.fr | 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" < [ mailto:ali.ali_ahmad at utt.fr | ali.ali_ahmad at utt.fr ] > 
À: "Barry Smith" < [ mailto:bsmith at petsc.dev | bsmith at petsc.dev ] > 
Cc: "petsc-users" < [ mailto:petsc-users at mcs.anl.gov | petsc-users at mcs.anl.gov ] >, "petsc-maint" < [ mailto:petsc-maint at mcs.anl.gov | 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" < [ mailto:bsmith at petsc.dev | bsmith at petsc.dev ] > 
À: "Ali ALI AHMAD" < [ mailto:ali.ali_ahmad at utt.fr | ali.ali_ahmad at utt.fr ] > 
Cc: "petsc-users" < [ mailto:petsc-users at mcs.anl.gov | petsc-users at mcs.anl.gov ] >, "petsc-maint" < [ mailto:petsc-maint at mcs.anl.gov | 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 

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On Jun 11, 2025, at 4:45 AM, Ali ALI AHMAD < [ mailto:ali.ali_ahmad at utt.fr | 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 

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-- 
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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