[petsc-users] Problems about Picard and NolinearGS

[Yingjie Wu]

Thanks for your reply.
I read the article you provided. This is my first contact with the
quasi-Newton method.
I have some problem:
1. From the point of view of algorithm, the quasi-Newton method does not
need to be solved by linear equations, so why would KSP be used?
2. Where and how to  use preconditioner in quasi-Newton method?

Thanks,
Yingjie

Jed Brown <jed at jedbrown.org> 于2018年12月27日周四 上午10:11写道：

> Yingjie Wu via petsc-users <petsc-users at mcs.anl.gov> writes:
>
> > I my opinion, the difficulty in constructing my Jacobian matrix is
> complex
> > coefficient.(eg, thermal conductivity* λ* , density )
> > For example, in temperature equation(T):
> >                 ∇(*λ*∇T) - ∇(ρ* Cp* u ) + Q =  0
> > *λ* is thermal conductivity , ρ* is density  Cp* is specific heat , u is
> > velocity, Q is source.
> > *λ = *1.9*(1.0e-3)*pow(T+273.0-150.0,1.29)            function of T
> > ρ=
> >
> (0.4814*P/1.0e3/(T+273.15))/(1.0+0.446*(1.0e-2)*P/1.0e3/(pow(T+273.15,1.2)))
> > function of T and P
> >
> > In theory, the coefficient contain variable. So it is complicated to
> > calculate the element of Jacobian.
> > In my snes_mf_operator method, I treat λ,ρ as constant. In every
> nonlinear
> > step, I use the solution update the λ,ρ  and thus update the
> > preconditioning matrix. At each residual function call(in
> > SNESFormFunction), I also update the coefficient to ensure the correction
> > of the residual function.
>
> If those Jacobian entries are really that hard to get right, you can try
> using quasi-Newton as an alternative to -snes_mf_operator; similar to
>
>
> https://jedbrown.org/files/BrownBrune-LowRankQuasiNewtonRobustJacobianLagging-2013.pdf
>
>
> In general, I would start with an exact Jacobian (computed by coloring,
> AD, or just in one specific material/configuration).  Test Newton using
> direct solvers to see your "best case" convergence (globalization may be
> needed).  Test fieldsplit using direct solvers on the blocks so you know
> how much error is attributable to that approximation.  Only then look at
> backing off on the approximation of the Jacobian and the preconditioner.
>
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