[petsc-users] SLEPc with offline preconditioner
Jose E. Roman
jroman at dsic.upv.es
Sat May 26 13:30:41 CDT 2012
You are putting options randomly, but always missing a relevant one.
First, have a look at the source code to see what options are set there, then change the required options and use -eps_view to see if they have been changed the way you want.
Jose
$ ./ex11 -n 30 -eps_nev 5 -eps_target 0 -st_type sinvert -st_ksp_type bcgs -st_pc_type bjacobi -st_ksp_rtol 1e-10
Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
Number of iterations of the method: 2
Solution method: krylovschur
Number of requested eigenvalues: 5
Stopping condition: tol=1e-08, maxit=100
Number of converged approximate eigenpairs: 5
k ||Ax-kx||/||kx||
----------------- ------------------
0.010956 9.93776e-09
0.010956 1.00716e-08
0.021912 6.37777e-11
0.043705 9.35519e-10
0.043705 3.18199e-07
El 26/05/2012, a las 15:41, Ye Jianbo Bob escribió:
> Some additional test with different option
> ------------------------------------------------------------------
> ./ex11 -n 30 -eps_nev 5 -eps_target 0 -st_type sinvert -st_pc_type bjacobi
> Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
>
> Number of iterations of the method: 5
> Solution method: krylovschur
>
> Number of requested eigenvalues: 5
> Stopping condition: tol=1e-08, maxit=100
> Number of converged approximate eigenpairs: 5
>
> k ||Ax-kx||/||kx||
> ----------------- ------------------
> 0.457508 0.886784
> 0.482412 0.930914
> 0.492672 0.900046
> 0.513074 0.880722
> 0.526509 0.898048
> ------------------------------------------------------------------
> bobye@:tutorials$ ./ex11 -n 30 -eps_nev 5 -st_type sinvert -st_shift
> 1e-9 -st_pc_type ilu
> Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
>
> Number of iterations of the method: 5
> Solution method: krylovschur
>
> Number of requested eigenvalues: 5
> Stopping condition: tol=1e-08, maxit=100
> Number of converged approximate eigenpairs: 5
>
> k ||Ax-kx||/||kx||
> ----------------- ------------------
> 0.457508 0.886784
> 0.482412 0.930914
> 0.492672 0.900046
> 0.513074 0.880722
> 0.526509 0.898048
>
>
> On Sat, May 26, 2012 at 9:37 PM, Ye Jianbo Bob <yelpoo at gmail.com> wrote:
>> Hi Jose, I found shift-and-invert does not work with an iterative pc
>> solver. Say the examples given in source code
>>
>> /home/bobye/pub/slepc/slepc-3.2-p3/src/eps/examples/tutorials/ex11.c
>> ------------------------------------------------------------------
>> if I use the default setting, in fact I am not sure which pc it is
>> used, even which is coefficient matrix. But it works well. However, in
>> my cases, I am interested in how to reuse manipulation of A for
>> different eigenvalues problem.
>> bobye@:tutorials$ ./ex11 -n 30 -eps_nev 5
>>
>> Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
>>
>> Number of iterations of the method: 20
>> Solution method: krylovschur
>>
>> Number of requested eigenvalues: 5
>> Stopping condition: tol=1e-08, maxit=100
>> Number of converged approximate eigenpairs: 7
>>
>> k ||Ax-kx||/||kx||
>> ----------------- ------------------
>> 0.010956 1.60437e-09
>> 0.021912 1.97096e-09
>> 0.043705 5.3322e-10
>> 0.054661 4.01148e-09
>> 0.087410 8.7469e-10
>> 0.097887 2.0693e-09
>> 0.108843 3.46988e-09
>> ------------------------------------------------------------------
>> if I use direct pc
>> bobye@:tutorials$ ./ex11 -n 30 -eps_nev 5 -st_type sinvert -st_pc_type lu
>> It will report Detected zero pivot in LU factorization
>> ------------------------------------------------------------------
>> if I use direct pc with shift value
>> bobye@:tutorials$ ./ex11 -n 30 -eps_nev 5 -st_type sinvert -st_shift
>> 1e-9 -st_pc_type lu
>>
>> Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
>>
>> Number of iterations of the method: 1
>> Solution method: krylovschur
>>
>> Number of requested eigenvalues: 5
>> Stopping condition: tol=1e-08, maxit=100
>> Number of converged approximate eigenpairs: 5
>>
>> k ||Ax-kx||/||kx||
>> ----------------- ------------------
>> 0.010956 5.37411e-08
>> 0.010956 5.3741e-08
>> 0.021912 5.13947e-14
>> 0.043705 1.95903e-11
>> 0.054661 2.62529e-09
>>
>> ------------------------------------------------------------------
>> if I use iterative pc with shift-and-invert, the result seems to be wrong.
>> bobye@:tutorials$ ./ex11 -n 30 -eps_nev 5 -st_type sinvert -st_pc_type
>> jacobi -st_ksp_rtol 1e-12
>>
>> Fiedler vector of a 2-D regular mesh, N=900 (30x30 grid)
>>
>> Number of iterations of the method: 1
>> Solution method: krylovschur
>>
>> Number of requested eigenvalues: 5
>> Stopping condition: tol=1e-08, maxit=100
>> Number of converged approximate eigenpairs: 12
>>
>> k ||Ax-kx||/||kx||
>> ----------------- ------------------
>> 2.000000 0.707107
>> 2.000000 0.707107
>> 2.000000 0.707107
>> 2.004191 0.704215
>> 3.000000 0.579634
>> 3.000000 0.578826
>> 3.000000 0.580728
>> 3.000000 0.57262
>> 3.000000 0.597176
>> 3.000000 0.562004
>> 3.000000 0.587957
>> 3.096958 0.710225
>>
>>
>> -jianbo
>>
>> On Sat, May 26, 2012 at 5:47 PM, Jose E. Roman <jroman at dsic.upv.es> wrote:
>>>
>>> El 26/05/2012, a las 10:47, Ye Jianbo Bob escribió:
>>>
>>>> Hi, I have some problem when using SLEPc to compute eigenvalue
>>>> problems. I do not know how to implement my idea properly. Let me
>>>> explain in detail.
>>>>
>>>> I am interested in solving a set of eigenvalue problems (to find the
>>>> smallest magnitude eigenvalues)
>>>>
>>>> A x = \lambda B_i x
>>>> for i=1,2,3,...
>>>>
>>>> where A is the Laplacian of some regular grid, B_i is diagonal. It is
>>>> known that A is semi-positive with a null vector [1,1,...,1].
>>>>
>>>> I found SLEPc provides EPSSetDeflationSpace to set deflation space
>>>> when applies iterative eigen solver. But I am not sure what default
>>>> precondition is used when I defined my problem by setting
>>>> eps_smallest_magnitude through EPSSetWhichEigenpairs.
>>>>
>>>> It is possible to use shift-and-invert to explicitly address my
>>>> problem. If the shift value is 0, zero pivot will be reported during
>>>> LU precondition stage. But since my set of eigen problems has the
>>>> exact the same A, I hope I could somehow do the precondition (direct
>>>> solver) offline only associate with A and apply the iterative eigen
>>>> solver online with set of different B_i.
>>>>
>>>> The very initiative is the direct solver applied in precondition stage
>>>> is O(N^2) while the matrix-free eigen solver is O(N), thus I think
>>>> this would improve the efficiency of my situation. Is there any way to
>>>> realize it?
>>>>
>>>> Here is some rough idea:
>>>> To prevent the zero pivot during LU, I would dampshift A with a small
>>>> quantity A-\sigma I, and then compute its LU offline and store it. In
>>>> online stage, I could build an ST shell that read LU computed in
>>>> offline stage and solve. This is only an approximated approach,
>>>> hopefully not degrading the performance.
>>>>
>>>> Thank you!
>>>
>>> You can use STSHELL to perform any customized spectral transformation. But in your case I don't think it is necessary. [1,1,...,1] is also an eigenvector of the matrix pair (A,B), so you can pass it through EPSSetDeflationSpace, then use shift-and-invert with zero target - in that case the internal KSP will have A as the coefficient matrix and KSPSetNullSpace will be invoked with the [1,1,...,1] vector. The only thing is that you have to use a KSP solver that supports nullspaces. I am not sure if PETSc's direct solvers (or external solvers) support it, so you may have to use an iterative solver (e.g., -st_ksp_type bcgs -st_pc_type bjacobi -st_ksp_rtol 1e-9).
>>>
>>> Jose
>>>
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