Thank you very much for the input!
I've spent a lot of time to compress the linear system to a quarter of
its size. This resulted in a form, though, which cannot be represented
by Kronecker products. Maybe I should return to the original form..
The new structure of the linear system is as follows:
+---+---+--------+
| A | 0 | -2*B^T |
+---+---+--------+
| 0 | D | -C^T |
+---+---+--------+
| B | C | D |
+---+---+--------+
The matrices D and C are huge and should therefore be accountable for
most of the computational cost. Looking at the visual structure of C, I
assume that it can maybe be written as a sum of (skew-)symmetric
matrices. A, B, and D are not symmetric at all, though.
Can the block substructure alone be helpful for solving the linear
system in a smarter manner?
On 1/31/24 18:45, Barry Smith wrote:
For large problems, preconditioners have to take advantage of some
underlying mathematical structure of the operator to perform well (require few
iterations). Just black-boxing the system with simple preconditioners will not
be effective.
So, one needs to look at the Liouvillian Superoperator's structure to see
what one can take advantage of. I first noticed that it can be represented as a
Kronecker product: A x I or a combination of Kronecker products? In theory,
one can take advantage of Kronecker structure to solve such systems much more
efficiently than just directly solving the huge system naively as a huge
system. In addition it may be possible to use the Kronecker structure of the
operator to perform matrix-vector products with the operator much more
efficiently than by first explicitly forming the huge matrix representation and
doing the multiplies with that. I suggest some googling with linear solver,
preconditioning, Kronecker product.
On Jan 31, 2024, at 6:51 AM, Niclas Götting <[email protected]> wrote:
Hi all,
I've been trying for the last couple of days to solve a linear system using
iterative methods. The system size itself scales exponentially (64^N) with the
number of components, so I receive sizes of
* (64, 64) for one component
* (4096, 4096) for two components
* (262144, 262144) for three components
I can solve the first two cases with direct solvers and don't run into any
problems; however, the last case is the first nontrivial and it's too large for
a direct solution, which is why I believe that I need an iterative solver.
As I know the solution for the first two cases, I tried to reproduce them using GMRES and
failed on the second, because GMRES didn't converge and seems to have been going in the
wrong direction (the vector to which it "tries" to converge is a totally
different one than the correct solution). I went as far as -ksp_max_it 1000000, which
takes orders of magnitude longer than the LU solution and I'd intuitively think that
GMRES should not take *that* much longer than LU. Here is the information I have about
this (4096, 4096) system:
* not symmetric (which is why I went for GMRES)
* not singular (SVD: condition number 1.427743623238e+06, 0 of 4096 singular
values are (nearly) zero)
* solving without preconditioning does not converge (DIVERGED_ITS)
* solving with iLU and natural ordering fails due to zeros on the diagonal
* solving with iLU and RCM ordering does not converge (DIVERGED_ITS)
After some searching I also found [this](http://arxiv.org/abs/1504.06768)
paper, which mentions the use of ILUTP, which I believe in PETSc should be used
via hypre, which, however, threw a SEGV for me, and I'm not sure if it's worth
debugging at this point in time, because I might be missing something entirely
different.
Does anybody have an idea how this system could be solved in finite time, such
that the method also scales to the three component problem?
Thank you all very much in advance!
Best regards
Niclas
