[petsc-users] Is there a way to do row/column scaling of jacobian matrix

[Gong Ding]

I known the diagonal scaling. And I will try it tomorrow.
Thank to slepc, I can monitor the eigen values as an approximation of condition number.  
The original problem as condition number about 1e20, which defeat any iterative solver.
I hope I can reduce it as much as possible.

Further more, can I use MC64, which permute and scale a sparse unsymmetric matrix to 
put large entries on the diagonal?

 
>    The literature is unclear to me, but I don't think these scalings are done in this way to improve the conditioning of the matrix. They are done to change the relative importance of different entries in the vector to determine stopping conditions and search directions in Newton's method. For example, if you consider getting the first vector entry in the residual/error small more important than the other entries you would use the scaling vector like [bignumber 1 1 1 1 ....]. In some way the scaling vectors reflect working with a different norm to measure the residual. Since PETSc does not support providing these scaling vectors you can get the same effect if you define your a new function (and hence also new Jacobian) that weights the various entries the way you want based on their importance. In other words newF(x)   = diagonalscaling1* oldF( diagonalscaling2 * y) then if x* is the solution to the new problem, y* = inv(diagonalscaling2*x*) is the solution to the original problem.  In some cases this transformation can correspond to working in "dimensionless coordinates" but all that language is over my head.
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>    If you just want to scale the matrix to have ones on the diagonal before forming the preconditioner (on the theory that it is better to solve problems with a "well-scaled" matrix) you can use the run time options -ksp_diagonal_scale -ksp_diagonal_scale_fix or in the code with KSPSetDiagonalScale() KSPSetDiagonalScaleFix().
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>    Barry
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> On Apr 4, 2011, at 10:52 PM, Barry Smith wrote:
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> >   If you are looking for something like this:
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> > When solving F(x) = 0, I would like to be able to scale both the solution
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> > vector x and the residual function vector F, simply by specifying scaling
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> > vectors, sx and sf, say. (These vectors would be the diagonal entries of
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> > scaling matrices Dx and Df.)
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> > I realize this can be achieved, at least in part, within the user residual
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> > function.
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> > This is what I had been doing, until I looked at Denis and Schnabel (sp?),
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> > Brown and Saad, and the KINSOL user guide. It seems one has to take the
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> > scaling matrices into account when computing various norms, when applying the
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> > preconditioner, and when computing the step size, \sigma. No doubt there
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> > are other things I have missed that also need to be done.
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> > http://www.mcs.anl.gov/petsc/petsc-as/developers/projects.html
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> > we don't have support for this (nor do I understand it). Anyways it has been on the "projects to do list" for a very long time; suspect it would require a good amount of futzing around in the source code to add.
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> >   Barry
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> > On Apr 4, 2011, at 10:16 PM, Gong Ding wrote:
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> >> Hi,
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> >> I'd like to scaling the jacobian matrix as if the condition number can be improved.
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> >> That is scaling J by Dl*J*Dr. The scaling diagonal matrix will be changed in each nonlinear iteration.
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> >> Does SNES already exist some interface to do this?
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