[petsc-users] MatPtAP

[Hong]

Samuel,
I have following questions:
1) Why solving \lambda*Pt*B*P*y=Pt*A*P*y is better than solving
original \lambda*B*x
= A*x?

2) Does your eigen solver require matrix factorization? If not, i.e., only
uses mat-vec multiplication, then you may implement
z = (Pt*(A*(P*y))) in your eigensolver instead of using mat-mat-mat multi
plication.

3) petsc PtAP() was implemented for multigrid applications, in which the
product C = PtAP is a denser but a much smaller matrix.
I have not seen the use of your case, that P is square with same size as A.
If C is much denser than A, then PtAP consumes a large portion of time is
anticipated.

Hong

On Fri, Jun 1, 2018 at 12:35 PM, Smith, Barry F. <bsmith at mcs.anl.gov> wrote:

>
>   Could you save the P matrix with MatView() using a binary viewer and the
> A matrix with MatView() and the binary viewer and email them to
> petsc-maint at mcs.anl.gov ? Then we can run the code in the profiler with
> your matrices and see if there is any way to speed up the computation.
>
>    Barry
>
>
> > On Jun 1, 2018, at 11:07 AM, Samuel Lanthaler <s.lanthaler at gmail.com>
> wrote:
> >
> > On 06/01/2018 03:42 PM, Matthew Knepley wrote:
> >> On Fri, Jun 1, 2018 at 9:21 AM, Samuel Lanthaler <s.lanthaler at gmail.com>
> wrote:
> >> Hi,
> >>
> >> I was wondering what the most efficient way to use MatPtAP would be in
> the following situation: I am discretizing a PDE system. The discretization
> yields a matrix A that has a band structure (with k upper and lower bands,
> say). In order to implement the boundary conditions, I use a transformation
> matrix P which is essentially the unit matrix, except for the entries
> P_{ij} where i,j<k and n-i,n-j<k, so
> >>
> >> P =  [ B, 0, 0, 0, ..., 0, 0 ]
> >>        [  0, 1, 0, 0, ..., 0, 0 ]
> >>        [                              ]
> >>        [                              ]
> >>        [                  ..., 1, 0 ]
> >>        [  0, 0, 0, 0, ..., 0, C ]
> >>
> >> with B,C are (k-by-k) matrices.
> >> Right now, I'm simply constructing A, P and calling
> >>
> >>     CALL MatPtAP(petsc_matA,petsc_matP,MAT_INITIAL_MATRIX,PETSC_
> DEFAULT_REAL,petsc_matPtAP,ierr)
> >>
> >> where I haven't done anything to pestc_matPtAP, prior to this call. Is
> this the way to do it?
> >>
> >> I'm asking because, currently, setting up the matrices A and P takes
> very little time, whereas the operation MatPtAP is taking quite long, which
> seems very odd... The matrices are of type MPIAIJ. In my problem, the total
> matrix dimension is around 10'000 and the matrix blocks (B,C) are of size
> ~100.
> >>
> >> Are you sure this is what you want to do? Usually BC are local, since
> by definition PDE are local, and
> >> are applied pointwise. What kind of BC do you have here?
> >>
> >
> > The boundary conditions are a mixture of Dirichlet and Neumann; in my
> case, the PDE is a system involving 8 variables on a disk, where the
> periodic direction is discretized using a Fourier series expansion, the
> radial direction uses B-splines.
> >
> > In reality, I have two matrices A,B, and want to solve the eigenvalue
> problem \lambda*B*x = A*x.
> > I found it quite convenient to use a transformation P to a different set
> of variables y, such that x=P*y and x satisfies the BC iff certain
> components of y are 0. The latter is enforced by inserting spurious
> eigenvalues at the relevant components of y in the transformed eigenvalue
> problem \lambda*Pt*B*P*y=Pt*A*P*y. After solving the EVP in terms of y, I
> get back x=P*y.
> > Is this an inherently bad/inefficient way of enforcing BC's? Thanks.
> >
> >
> >
> >
> >>   Thanks,
> >>
> >>     Matt
> >>
> >> Thanks in advance for any ideas.
> >>
> >> Cheers,
> >> Samuel
> >>
> >>
> >>
> >> --
> >> 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://www.cse.buffalo.edu/~knepley/
> >
>
>
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