[petsc-dev] How do you get RIchardson?

[Jed Brown]

On Fri, Sep 16, 2011 at 23:54, Matthew Knepley <knepley at gmail.com> wrote:

> On Fri, Sep 16, 2011 at 4:38 PM, Jed Brown <jedbrown at mcs.anl.gov> wrote:
>
>> On Fri, Sep 16, 2011 at 23:21, Matthew Knepley <knepley at gmail.com> wrote:
>>
>>> it still converges, conditionally to the same solution as exact
>>>> newton. Variations for A yield different rates of convergence. When
>>>> A=1, you get the classical Picard iteration that Matt mentioned (?).
>>>>
>>>
>>> Not even close.
>>>
>>
>> From Barry's description at the top of this thread:
>>
>>  x^{n+1}   = x^{n}  - lambda F(x^{n})
>>
>>
>> This looks oddly similar to
>>
>>  x^{n+1}   = x^{n}  - J(x^n)^{-1} F(x^{n})
>>
>> I wonder where I've seen that before.
>>
>
> So you are saying that you agree with me, what is coded is in fact the
> Picard iteration, and we are done arguing?


I was objecting to your "not even close". I agree with Vijay's terminology.

I agree that your choice is _a_ Picard iteration, I do not agree that it is
the One True Picard iteration. In particular, it is not the one that is most
commonly used in practice (e.g. SISC or JCP publications).

Since Newton is also a fixed point iteration, I do not think the name
"Picard" is a useful way to distinguish. Furthermore, since "Picard with
solve" as you like to call it can easily be formulated in defect-correction
mode as a Newton-like update with a different "Jacobian", it doesn't require
any special support from PETSc.
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