[petsc-users] Preconditioning of Liouvillian Superoperator

[Niclas Götting]

The mathematical structure of D (and also A, B, and C for that matter) 
is hard to recover for me without lots of extra work, as I have pages of 
equations for each index due to the transformation. I will try to 
reformulate the equations tomorrow and see, if I can maybe at least find 
that C=X+Y, with X, Y (skew-)symmetric.

What I of course have are plots of the matrix and maybe this is enough 
to see, if certain methods are more fitting than others. I have attached 
the nonzero and block structure of the matrices for the one and two 
component systems (hoping that attachments are allowed in this mailing 
list). I believe that D could strongly benefit from appropriate 
reordering, but I think I'll need to learn how to setup PCFIELDSPLIT to 
test this.

For the three component system, A is as large as the whole two component 
system, so I should be able to solve it with LU decomposition without 
problems, which is great. You mentioned that (only?) A and D are 
important for the solution of this system, which has me wondering why B 
and C are less relevant?

Thank you very much for your time and patience!

On 2/1/24 16:21, Barry Smith wrote:
>
>> On Feb 1, 2024, at 6:57 AM, Niclas Götting <ngoetting at itp.uni-bremen.de> wrote:
>>
>> 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?
>    Possibly, it depends on details about A and D. If good preconditioners are available for A and D separately then PCFIELDSPLIT can be used to construct a preconditioner for the entire matrix. If A is small then in the fieldsplit process LU can be used for A so it does not need a good preconditioner. So what is the structure of D?
>> 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 <ngoetting at itp.uni-bremen.de> 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
>>>>
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