petsc-3.8.4 2018-03-24

KSPQCG

Code to run conjugate gradient method subject to a constraint on the solution norm. This is used in Trust Region methods for nonlinear equations, SNESNEWTONTR

Options Database Keys

-ksp_qcg_trustregionradius <r> -Trust Region Radius

Notes: This is rarely used directly

Notes: Use preconditioned conjugate gradient to compute an approximate minimizer of the quadratic function

q(s) = g^T * s + .5 * s^T * H * s

subject to the Euclidean norm trust region constraint

|| D * s || <= delta,

where

delta is the trust region radius, g is the gradient vector, and H is Hessian matrix, D is a scaling matrix.

KSPConvergedReason may be

 KSP_CONVERGED_CG_NEG_CURVE if convergence is reached along a negative curvature direction,

 KSP_CONVERGED_CG_CONSTRAINED if convergence is reached along a constrained step,

 other KSP converged/diverged reasons

Notes

Currently we allow symmetric preconditioning with the following scaling matrices

PCNONE: D = Identity matrix PCJACOBI: D = diag [d_1, d_2, ...., d_n], where d_i = sqrt(H[i,i]) PCICC: D = L^T, implemented with forward and backward solves. Here L is an incomplete Cholesky factor of H.

References

1. -Trond Steihaug, The Conjugate Gradient Method and Trust Regions in Large Scale Optimization, SIAM Journal on Numerical Analysis, Vol. 20, No. 3 (Jun., 1983).

KSPQCG

Options Database Keys

Notes

Currently we allow symmetric preconditioning with the following scaling matrices

References

See Also