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From: Benjamin Kirk <benjamin.kirk@na...>  20080201 16:42:19

> This could be useful to solve Euler equation with a DG method that > employs a Riemann solver to compute the numerical fluxes > (discontinuous at element facesedges) without dealing with Godunov > fluxes to compute the jacobian... Am I wrong? Absolutely. And a common trick from the finite volume community would be to precondition with a firstorder implicit system operator, while you evaluate the residual to higher order. In DG this equates to piecewise constant, For perfect gas in 3D this would give you 5*n_cells*n_faces_per_cell (assuming all your elements have the same number of faces) storage for the preconditioner instead of (5*n_cells*n_faces_per_cell*n_dofs_per_var) Ben 
From: Lorenzo Botti <bottilorenzo@gm...>  20080201 16:13:41

This could be useful to solve Euler equation with a DG method that employs a Riemann solver to compute the numerical fluxes (discontinuous at element facesedges) without dealing with Godunov fluxes to compute the jacobian... Am I wrong? Lorenzo 2008/2/1, Benjamin Kirk <benjamin.kirk@...>: > > > Let me rewrite the expression you wrote as > > > > J*v = (F(u+epsilon*v)F(u)) / epsilon > > > > Where epsilon is a small perturbation and F(u) is the nonlinear residual > > function and J is the jacobian matrix of the nonlinear system. The above > > formula compute the action of a Jacobian on a given vector, or more > > specifically the Krylov vector if you are using say GMRES or CG to solve > > your system. > > > That's absolutely true. To expound a little more, by definition > > F = F(U) > J = dF/dU > > J*v = [dF/dU]*v (1) > > which is simply the directional derivative of F (in the direction of v. As > specified above, you can approximate this matrixvector product as > > J*v ~ (F(u+epsilon*v)F(u)) / epsilon (2) > > In Krylov subspace methods the operation (1) occurs repeatedly at each > iteration. So, if you are willing to approximate it, you can use (2) > instead. > > There are a couple of reasons why you might want to do this: >  obviously, this finite difference of vectors eliminates the need > to store J, hence "matrix free" >  not to whine, but sometimes computing an accurate J is *hard* and/or > computationally intensive. It can be errorprone. Using (2) you get > > Now of course any hard problem will need preconditioning in the Krylov > solver. You can accomplish this in several ways... One would be to build > an approximate Jacobian and use ILU or the like. Now you have one matrix > instead of two. Also, this could be a much simpler matrix (blockdiagonal > for certain cases), thus limiting the requirements. > > But there are other options too, allowing for a matrix free preconditioner > as well. These include GaussSiedel, Geometric multigrid, another Krylov > technique, etc... > > Ben > > >  > This SF.net email is sponsored by: Microsoft > Defy all challenges. Microsoft(R) Visual Studio 2008. > http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ > _______________________________________________ > Libmeshusers mailing list > Libmeshusers@... > https://lists.sourceforge.net/lists/listinfo/libmeshusers > 
From: Benjamin Kirk <benjamin.kirk@na...>  20080201 13:20:56

> Let me rewrite the expression you wrote as > > J*v = (F(u+epsilon*v)F(u)) / epsilon > > Where epsilon is a small perturbation and F(u) is the nonlinear residual > function and J is the jacobian matrix of the nonlinear system. The above > formula compute the action of a Jacobian on a given vector, or more > specifically the Krylov vector if you are using say GMRES or CG to solve > your system. That's absolutely true. To expound a little more, by definition F = F(U) J = dF/dU J*v = [dF/dU]*v (1) which is simply the directional derivative of F (in the direction of v. As specified above, you can approximate this matrixvector product as J*v ~ (F(u+epsilon*v)F(u)) / epsilon (2) In Krylov subspace methods the operation (1) occurs repeatedly at each iteration. So, if you are willing to approximate it, you can use (2) instead. There are a couple of reasons why you might want to do this:  obviously, this finite difference of vectors eliminates the need to store J, hence "matrix free"  not to whine, but sometimes computing an accurate J is *hard* and/or computationally intensive. It can be errorprone. Using (2) you get Now of course any hard problem will need preconditioning in the Krylov solver. You can accomplish this in several ways... One would be to build an approximate Jacobian and use ILU or the like. Now you have one matrix instead of two. Also, this could be a much simpler matrix (blockdiagonal for certain cases), thus limiting the requirements. But there are other options too, allowing for a matrix free preconditioner as well. These include GaussSiedel, Geometric multigrid, another Krylov technique, etc... Ben 
From: Vijay M <unknownreference@gm...>  20080201 01:19:28

>> Can you write me the exact expression of the problem you're solving? The example I forwarded you earlier by Benjamin Kirk solves the LaplaceYoung problem. I have linked a paper by C.F. Scott et al on "Computation of capillary surfaces for the LaplaceYoung equation". This should help whoever is interested in knowing the exact system that the = code solves for. (http://eprints.maths.ox.ac.uk/304/01/QJMAM0416.pdf as of 01/31/08) Matrix free technique: Yes, Knoll's paper is a fantastic introduction to obtain a good grasp on JFNK technique. I would recommend that to anyone who wants to solve nonlinear problems in a matrix free way. Let me rewrite the expression you wrote as J*v =3D (F(u+epsilon*v)F(u)) / epsilon Where epsilon is a small perturbation and F(u) is the nonlinear residual function and J is the jacobian matrix of the nonlinear system. The above formula compute the action of a Jacobian on a given vector, or more specifically the Krylov vector if you are using say GMRES or CG to solve your system. F(u) is the right hand side because the Newton linearization yields the system=20 J*delu =3D residual =3D F(u) But since the whole point is not to form the matrix J, you could just = find the action of the matrix on a vector to build the Krylov subspace and = solve the system. Hope that helps. Original Message From: libmeshusersbounces@... [mailto:libmeshusersbounces@...] On Behalf Of li pan Sent: Thursday, January 31, 2008 8:05 AM To: Vijay S. Mahadevan Cc: libmeshusers@... Subject: Re: [Libmeshusers] matrix free scheme thanx a lot! Can you write me the exact expression of the problem you're solving? Recently, I read the paper of Knoll about JFNK method (Jacobianfree Newton=96Krylov methods: a survey of approaches and applications). I'm not sure whether I understood well. Please correct me. If r_0 is the first residual, J is the jacobian matrix, In the approximation=20 J*r_0 =3D (F(u+\epsilon r_0)F(u)) / \epsilon F(u) is the residual, right? But in Knoll's paper F(u) seems the right hand side. Or I am wrong. pan  "Vijay S. Mahadevan" <vijay.m@...> wrote: > I've attached the message from Ben Kirk. Also > attached is the example > program that he sent out. See below for his original > mail. >=20 > On a side note, I have written a working code with > Matrixfree technique to > solve a nonlinear, transient diffusionreaction > problem. It still needs few > refinements and I will send it here or maybe upload > it somewhere soon for > all those interested. >=20 > Original Message > From: Benjamin Kirk [mailto:benjamin.kirk@...]=20 > Sent: Wednesday, January 23, 2008 9:40 AM > To: Vijay M; libmeshdevel@... > Subject: Re: [Libmeshusers] Support for Matrixfree > algorithms >=20 > Here is a really, really raw example, the comments > are not clear right now, > but I wanted to keep you informed. This requires > the latest svn branch to > work. >=20 > Unpack it in the ./examples directory and run make. >=20 > Run it as >=20 > $ ./ex19dbg snes_view r 4 > for a successive approximation which will converge > linearly, and >=20 > $ ./ex19dbg snes_view r 4 snes_mf_operator > for a matrixfree approach in the iterative solver > which will converge > quadratically. >=20 > > Anyway, I do have a question regarding using PETSc > object with LibMesh. I > > have been trying to use Petsc objects Mats, Vecs > and SNES solver with > > Libmesh but the one thing I cant seem figure out > is how to set the > > nonlinear_solver public attribute of say a > NonlinearImplicitSystem object > to > > a PETSc SNES object which I have created and > initialized separately. > Since, > > the SNES object used in the wrapper > PetscNonlinearSolver is private, I > don=B9t > > understand how this can be done. > >=20 > > Have I missed something and taken a completely > wrong path on this ? I > would > > very much appreciate any comments that you can > provide to help me out > here. >=20 > The user interface is totally up for discussion > since I am the only one who > as exercised it to date. (I am sure Roy will have > some comments!) It seems > to me the right approach will be to add a method > which gives the user access > to the SNES object? From there the KSP, Mat, Vec, > PC, etc... can be > accessed. This would be similar to the approach > used in the > PetscLinearSolver. >=20 > Ben >=20 >=20 > Original Message > From: li pan [mailto:li76pan@...]=20 > Sent: Wednesday, January 30, 2008 3:25 AM > To: libmeshusers@... > Cc: vijay.m@... > Subject: matrix free scheme >=20 > Dear all, > I remember that there was a discussion about matrix > free scheme with libmesh before X'mas. I'd like to > ask > if somebody has got a example code for this. >=20 > thanx >=20 > pan >=20 >=20 > =20 > _________________________________________________________________________= ___ > ________ > Never miss a thing. Make Yahoo your home page.=20 > http://www.yahoo.com/r/hs >=20 > No virus found in this incoming message. > Checked by AVG Free Edition.=20 > Version: 7.5.516 / Virus Database: 269.19.16/1251  > Release Date: 1/30/2008 > 9:29 AM > =20 >=20 > No virus found in this outgoing message. > Checked by AVG Free Edition.=20 > Version: 7.5.516 / Virus Database: 269.19.16/1251  > Release Date: 1/30/2008 > 9:29 AM > =20 > =20 >=20 =20 _________________________________________________________________________= ___ ________ Be a better friend, newshound, and=20 knowitall with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=3DAhu06i62sR8HDtDypao8Wcj9tAcJ=20 = This SF.net email is sponsored by: Microsoft Defy all challenges. Microsoft(R) Visual Studio 2008. http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ _______________________________________________ Libmeshusers mailing list Libmeshusers@... https://lists.sourceforge.net/lists/listinfo/libmeshusers No virus found in this incoming message. Checked by AVG Free Edition.=20 Version: 7.5.516 / Virus Database: 269.19.17/1253  Release Date: = 1/31/2008 9:09 AM =20 No virus found in this outgoing message. Checked by AVG Free Edition.=20 Version: 7.5.516 / Virus Database: 269.19.17/1253  Release Date: = 1/31/2008 9:09 AM =20 