You may try MINRES (-ksp_type minres) See http://web.stanford.edu/group/SOL/reports/SOL-2011-2R.pdf
Hong On Fri, Jul 18, 2014 at 1:22 PM, Matthew Knepley <[email protected]> wrote: > On Fri, Jul 18, 2014 at 1:19 PM, Chetan Jhurani <[email protected]> > wrote: >> >> > From: Jed Brown <[email protected]> >> > >> > Jozsef Bakosi <[email protected]> writes: >> > >> > > Hi folks, >> > > >> > > I have a matrix as a result of a finite element discretization of the >> > > Poisson >> > > operator and associated right hand side. As it turns out, the matrix >> > > is >> > > symmetric but not positive definite since it has at least two negative >> > > small >> > > eigenvalues. I have been solving this system without problem using the >> > > conjugate >> > > gradients (CG) algorithm with ML as a preconditioner, but I'm >> > > wondering why it >> > > works. >> > >> > Is the preconditioned matrix >> > >> > P^{-1/2} A P^{-T/2} >> > >> > positive definite? >> >> This does not answer the original question regarding CG/ML, but > > > It is possible for CG to convergence with an indefinite matrix, but it is > also > possible for it to fail. > > Matt > >> >> will reduce some debugging work. P^{-1/2} A P^{-T/2} cannot be >> positive definite since A is not positive definite. This is due >> to Sylvester's inertia theorem. >> >> http://books.google.com/books?id=P3bPAgAAQBAJ&pg=PA202 >> Applied Numerical Linear Algebra By James W. Demmel, p 202 >> >> Chetan >> >> > > Shouldn't CG fail for a non-positive-definite matrix? Does PETSc do >> > > some >> > > additional magic if it detects that the dot-product in the CG >> > > algorithm is >> > > negative? Does it solve the system using the normal equations, A'A, >> > > instead? >> > >> > CG will report divergence in case a direction of negative curvature is >> > found. >> > > > > -- > What most experimenters take for granted before they begin their experiments > is infinitely more interesting than any results to which their experiments > lead. > -- Norbert Wiener
