[petsc-users] Problems about Picard and NolinearGS

[Matthew Knepley]

On Thu, Jan 3, 2019 at 7:36 AM Yingjie Wu via petsc-users <
petsc-users at mcs.anl.gov> wrote:

> 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?
>

It is solving the equations, but we know the analytical answer. KSP is our
abstraction for linear equation solver,
so we also use it in this case.


> 2. Where and how to  use preconditioner in quasi-Newton method?
>

You do not need a PC here. Note that this is only going to work well if you
have a good initial
guess for the inverse of your Jacobian. The optimization people who
invented have that (it is
the identity). In Jed's paper, they use a V-cycle as the good initial guess.

  Thanks,

    Matt


> 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.
>>
>

-- 
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

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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