[petsc-users] solving system with 2x2 block size
Barry Smith
bsmith at mcs.anl.gov
Tue Nov 15 15:34:38 CST 2016
> On Nov 15, 2016, at 3:23 PM, Manav Bhatia <bhatiamanav at gmail.com> wrote:
>
> I have a complex system, (A + i B) (x + i y) = (f + ig), that I am trying to solve using real matrices:
>
> [A -B; B A ] [x; y] = [f; g]
>
> So, the 2x2 block is made of the real and imaginary component of each entry in the complex matrix.
>
> I am following the discussion in the following paper:
>
> DAY D. \& HEROUX M.A. 2001. Solving complex-valued linear systems via equivalent real formulations. \textit{SIAM Journal on Scientific Computing} 23: 480-498.
>
> Following is an excerpt.
>
> **********************************************************************************
>
> The matrix K in the K formulation has a natural 2-by-2 block structure that can be exploited by using block entry data structures. Using the block entry features of these packages has the following benefits.
>
> • Applying 2-by-2 block Jacobi scaling to K corresponds exactly to applying point Jacobi scaling to C.
>
> • The block sparsity pattern of K exactly matches the point sparsity pattern of C. Thus any pattern-based preconditioners such as block ILU(l) applied to K correspond exactly to ILU(l) applied to C. See section 4 for definitions of block ILU(l) and ILU(l).
>
> • Any drop tolerance-based complex preconditioner has a straightforward K formulation since the absolute value of a complex entry equals the scaled Frobenius norm of the corresponding block entry in K.
>
> **********************************************************************************
>
> The paper additional outlines the challenges of the poor spectral properties of the equivalent real system.
>
> So, I am assembling the system with a 2x2 block, but am not sure how to best pick the right solver options in Petsc.
>
> I agree that I am getting confused by the “block” nomenclature. Particularly, I am not sure how to reconcile the different notions with points 1 and 2 from the paper (noted above).
In PETSc we call this 2x2 block Jacobi "point-block Jacobi" you can use the option -pc_type pbjacobi. The ILU() in PETSc can also be "point block", this is obtained with the usual -pc_type ilu (that is there is no different preconditioner name for ILU point block). To use all these things you need to make your matrix a BAIJ matrix (not an AIJ) and set its block size to 2.
Have you tried solving the matrices as complex? Is there a reason you wish to reformulate them as real?
The convergence of iterative methods (either with real or complex numbers) depends on the properties of the A and B (i.e. C) matrix. Where does the C matrix come from? There are many applications that result in complex matrices that are poorly conditioned for iterative methods.
Barry
>
> Any guidance would be appreciated!
>
> Thanks,
> Manav
>
>
>> On Nov 15, 2016, at 3:12 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
>>
>>
>> We can help you if you provide more information about what the blocks represent in your problem.
>>
>> Do you have two degrees of freedom at each grid point? What physically are the two degrees of freedom. What equations are you solving?
>>
>> I think you may be mixing up the "matrix block size" of 2 with the blocks in "block Jacobi". Though both are called "block" they really don't have anything to do with each other.
>>
>> Barry
>>
>>> On Nov 15, 2016, at 3:03 PM, Manav Bhatia <bhatiamanav at gmail.com> wrote:
>>>
>>> Hi,
>>>
>>> I am setting up a matrix with the following calls. The intent is to solve the system with a 2x2 block size.
>>>
>>> What combinations of KSP/PC will effectively translate to solving this block matrix system?
>>>
>>> I saw a discussion about bjacobi in the manual with the following calls (I omitted the prefixes from my actual command):
>>> -pc_type bjacobi -pc_bjacobi_blocks 2 -sub_ksp_type preonly -sub_pc_type lu -ksp_view
>>>
>>> which provides the following output:
>>> KSP Object:(fluid_complex_) 1 MPI processes
>>> type: gmres
>>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
>>> GMRES: happy breakdown tolerance 1e-30
>>> maximum iterations=10000, initial guess is zero
>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
>>> left preconditioning
>>> using PRECONDITIONED norm type for convergence test
>>> PC Object:(fluid_complex_) 1 MPI processes
>>> type: bjacobi
>>> block Jacobi: number of blocks = 2
>>> Local solve is same for all blocks, in the following KSP and PC objects:
>>> KSP Object: (fluid_complex_sub_) 1 MPI processes
>>> type: preonly
>>> maximum iterations=10000, initial guess is zero
>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
>>> left preconditioning
>>> using NONE norm type for convergence test
>>> PC Object: (fluid_complex_sub_) 1 MPI processes
>>> type: lu
>>> LU: out-of-place factorization
>>> tolerance for zero pivot 2.22045e-14
>>> matrix ordering: nd
>>> factor fill ratio given 5., needed 5.70941
>>> Factored matrix follows:
>>> Mat Object: 1 MPI processes
>>> type: seqaij
>>> rows=36844, cols=36844
>>> package used to perform factorization: petsc
>>> total: nonzeros=14748816, allocated nonzeros=14748816
>>> total number of mallocs used during MatSetValues calls =0
>>> using I-node routines: found 9211 nodes, limit used is 5
>>> linear system matrix = precond matrix:
>>> Mat Object: (fluid_complex_) 1 MPI processes
>>> type: seqaij
>>> rows=36844, cols=36844
>>> total: nonzeros=2583248, allocated nonzeros=2583248
>>> total number of mallocs used during MatSetValues calls =0
>>> using I-node routines: found 9211 nodes, limit used is 5
>>> linear system matrix = precond matrix:
>>> Mat Object: (fluid_complex_) 1 MPI processes
>>> type: seqaij
>>> rows=73688, cols=73688, bs=2
>>> total: nonzeros=5224384, allocated nonzeros=5224384
>>> total number of mallocs used during MatSetValues calls =0
>>> using I-node routines: found 18422 nodes, limit used is 5
>>>
>>>
>>> Likewise, I tried to use a more generic option:
>>> -mat_set_block_size 2 -ksp_type gmres -pc_type ilu -sub_ksp_type preonly -sub_pc_type lu -ksp_view
>>>
>>> with the following output:
>>> Linear fluid_complex_ solve converged due to CONVERGED_RTOL iterations 38
>>> KSP Object:(fluid_complex_) 1 MPI processes
>>> type: gmres
>>> GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
>>> GMRES: happy breakdown tolerance 1e-30
>>> maximum iterations=10000, initial guess is zero
>>> tolerances: relative=1e-05, absolute=1e-50, divergence=10000.
>>> left preconditioning
>>> using PRECONDITIONED norm type for convergence test
>>> PC Object:(fluid_complex_) 1 MPI processes
>>> type: ilu
>>> ILU: out-of-place factorization
>>> 0 levels of fill
>>> tolerance for zero pivot 2.22045e-14
>>> matrix ordering: natural
>>> factor fill ratio given 1., needed 1.
>>> Factored matrix follows:
>>> Mat Object: 1 MPI processes
>>> type: seqaij
>>> rows=73688, cols=73688, bs=2
>>> package used to perform factorization: petsc
>>> total: nonzeros=5224384, allocated nonzeros=5224384
>>> total number of mallocs used during MatSetValues calls =0
>>> using I-node routines: found 18422 nodes, limit used is 5
>>> linear system matrix = precond matrix:
>>> Mat Object: (fluid_complex_) 1 MPI processes
>>> type: seqaij
>>> rows=73688, cols=73688, bs=2
>>> total: nonzeros=5224384, allocated nonzeros=5224384
>>> total number of mallocs used during MatSetValues calls =0
>>> using I-node routines: found 18422 nodes, limit used is 5
>>>
>>> Are other PC types expected to translate to the block matrices?
>>>
>>> I would appreciate any guidance.
>>>
>>> Thanks,
>>> Manav
>>>
>>
>
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