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# KSPTSIRM

Implements the two-stage iteration with least-squares residual minimization method [CKG16]

## Options Database Keys

- -ksp_ksp_type solver - the type of the inner solver (GMRES or any of its variants for instance)

- -ksp_pc_type preconditioner - the type of the preconditioner applied to the inner solver

- -ksp_ksp_max_it maxits - the maximum number of inner iterations (iterations of the inner solver)

- -ksp_ksp_rtol tol - sets the relative convergence tolerance of the inner solver

- -ksp_tsirm_cgls number - if 1 use CGLS solver in the minimization step, otherwise use LSQR solver

- -ksp_tsirm_max_it_ls maxits - the maximum number of iterations for the least-squares minimization solver

- -ksp_tsirm_tol_ls tol - sets the convergence tolerance of the least-squares minimization solver

- -ksp_tsirm_size_ls size - the number of residuals for the least-squares minimization step

## Notes

`KSPTSIRM` is a two-stage iteration method for solving large sparse linear systems of the form \(Ax=b\). The main idea behind this new method is the use a least-squares residual minimization to improve the convergence of Krylov based iterative methods, typically those of GMRES variants. The principle of `TSIRM` algorithm is to build an outer iteration over a Krylov method, called the inner solver, and to frequently store the current residual computed by the given Krylov method in a matrix of residuals S. After a few outer iterations, a least-squares minimization step is applied on the matrix composed by the saved residuals, in order to compute a better solution and to make new iterations if required. The minimization step consists in solving the least-squares problem \(\min||b-ASa||\) to find ‘a’ which minimizes the residuals \((b-AS)\). The minimization step is performed using two solvers of linear least-squares problems: `KSPCGLS` or `KSPLSQR`. A new solution x with a minimal residual is computed with \(x=Sa\).

Defaults to 30 iterations for the inner solve, use option `-ksp_ksp_max_it <it>` to change it.

## Contributed by

Lilia Ziane Khodja

## References

[CKG16]

Raphaël Couturier, Lilia Ziane Khodja, and Christophe Guyeux. TSIRM: a two-stage iteration with least-squares residual minimization algorithm to solve large sparse linear and nonlinear systems. Journal of Computational Science, 17:535–546, 2016.

## See Also

KSP: Linear System Solvers, `KSPCreate()`, `KSPSetType()`, `KSPType`, `KSP`, `KSPFGMRES`, `KSPLGMRES`, `KSPGMRESSetRestart()`, `KSPGMRESSetHapTol()`, `KSPGMRESSetPreAllocateVectors()`, `KSPGMRESSetOrthogonalization()`, `KSPGMRESGetOrthogonalization()`, `KSPGMRESClassicalGramSchmidtOrthogonalization()`, `KSPGMRESModifiedGramSchmidtOrthogonalization()`, `KSPGMRESCGSRefinementType`, `KSPGMRESSetCGSRefinementType()`, `KSPGMRESGetCGSRefinementType()`, `KSPGMRESMonitorKrylov()`, `KSPSetPCSide()`

## Level

advanced

## Location

src/ksp/ksp/impls/tsirm/tsirm.c

Index of all KSP routines
Table of Contents for all manual pages
Index of all manual pages