On Mon, Jul 28, 2025 at 11:00 AM Ali ALI AHMAD <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!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" <bsm...@petsc.dev> > *À: *"Ali ALI AHMAD" <ali.ali_ah...@utt.fr> > *Cc: *"petsc-users" <petsc-users@mcs.anl.gov>, "petsc-maint" < > 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) > > > > On Jun 20, 2025, at 5:10 AM, Ali ALI AHMAD <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. > > > 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" <bsm...@petsc.dev> > *À: *"Ali ALI AHMAD" <ali.ali_ah...@utt.fr> > *Cc: *"petsc-users" <petsc-users@mcs.anl.gov>, "petsc-maint" < > 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. > > > > On Jun 13, 2025, at 4:55 AM, Ali ALI AHMAD <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" <bsm...@petsc.dev> > *À: *"Ali ALI AHMAD" <ali.ali_ah...@utt.fr> > *Cc: *"petsc-users" <petsc-users@mcs.anl.gov>, "petsc-maint" < > 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 > > > On Jun 12, 2025, at 9:42 AM, Ali ALI AHMAD <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" <ali.ali_ah...@utt.fr> > *À: *"Barry Smith" <bsm...@petsc.dev> > *Cc: *"petsc-users" <petsc-users@mcs.anl.gov>, "petsc-maint" < > 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" <bsm...@petsc.dev> > *À: *"Ali ALI AHMAD" <ali.ali_ah...@utt.fr> > *Cc: *"petsc-users" <petsc-users@mcs.anl.gov>, "petsc-maint" < > 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 > > > On Jun 11, 2025, at 4:45 AM, Ali ALI AHMAD <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 > > > > > -- 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/__https://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!ZSlAlCgNHU3fRLXzgg5EFEe53RPc1qNdYZxn7EdOoqgcUzNXDLCWTu2r4nJaeRsjD5zTKLeJcfm7BnlDbXYV$ <https://urldefense.us/v3/__http://www.cse.buffalo.edu/*knepley/__;fg!!G_uCfscf7eWS!ZSlAlCgNHU3fRLXzgg5EFEe53RPc1qNdYZxn7EdOoqgcUzNXDLCWTu2r4nJaeRsjD5zTKLeJcfm7BjfatsBE$ >
