topic Re: Re:Solver for special structured linear system of equations in Intel® oneAPI Math Kernel Library

Solver for special structured linear system of equations

NilsA — Wed, 08 Jun 2022 07:44:44 GMT

I have to solve a matrix of the following special structure:

| | | | | | |
| | | | | | |
| A | B | | | | |
| N1 x N1 |N1xN2| |X | = |F |
|_____________|______| | | | |
| C | | | | |
| N2 x NEQ | | | | |

whereas N1 >> N2 and A is a band matrix. The complete matrix is not symmetric and not necessarily positiv definite. Is there linear solver which is more suitable for this problem then dgesv from LAPACK, which solves a general NxN-matrix. LAPACK: dgesv (netlib.org)

Re:Solver for special structured linear system of equations

VidyalathaB_Intel — Wed, 08 Jun 2022 11:57:03 GMT

Hi Nils,

Thanks for reaching out to us.

Since in your case A is a band matrix could you please let us know if using LAPACKE_dgbsv from Intel oneMKL matches your use case?

You can refer to the below link for more details (maybe you can also refer to other sections as well in the manual)

https://www.intel.com/content/www/us/en/develop/documentation/onemkl-developer-reference-c/top/lapack-routines/lapack-linear-equation-routines/lapack-linear-equation-driver-routines/gbsv.html

You can also take help from the oneMKL LAPACK Function Finding Advisor which helps in recommending the routines for your particular use case.

https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl-function-finding-advisor.html#gs.32hv75

Kindly let us know if that doesn't match your use case.

Regards,

Vidya.

Re: Re:Solver for special structured linear system of equations

NilsA — Wed, 08 Jun 2022 13:45:25 GMT

Thanks for your reply, but unfortanetly gbsv does not fit, since in my "A", which is banded, is only a submatrix of the global matrix. I still have B and C, which destroy the banded structure. You obtain a matrix structure like this for example using Lagrangian multipliers connected to multiple nodes in a finite element simulation.

But thanks for your link to the function finding advisor. This might be helpful in future situations.

Re: Solver for special structured linear system of equations

mecej4 — Wed, 08 Jun 2022 14:23:43 GMT

There is not enough to indicate that there is some property of your matrix that enables a divide-and-conquer approach. If, however, you break your C block into a N2 X N1 block C1 and a N2 X N2 block D, i.e., [C] = [C1 D], and D is diagonal or tridiagonal, then it may be worthwhile to see if you can use the Schur-complement approach.

The Schur-complement approach does have at least one disadvantage, in that the Schur Complement may be more dense than the block A. Some PDE solvers, such as the HST3D of USGS, use the Schur Complement in combination with an iterative solver such as GMRES, QMR, etc. It can be error-prone and tricky to infer the sparse matrix representation of the Schur Complement from the sparse matrix representation of the original matrix.

Re:Solver for special structured linear system of equations

VidyalathaB_Intel — Thu, 16 Jun 2022 06:08:34 GMT

Hi Nils,

Reminder:

Could you please let us know if the information provided by mecej4 helps? If yes, please let us know if we could close this thread from our end.

Regards,

Vidya.

Re:Solver for special structured linear system of equations

VidyalathaB_Intel — Thu, 23 Jun 2022 04:29:46 GMT

Hi Nils,

As we haven't heard back from you we are closing this thread. Please post a new question if you need any additional assistance from Intel as this thread will no longer be monitored.

Regards,

Vidya.