[ecco-support] Change-of-variable formulas for adjoint sensitivities

[Christopher Wolfe]

It looks like my pasted graphics don’t show up well in Gmail, so I’ve attached the whole thing as a PDF.



> On Jul 7, 2020, at 10:43 AM, Christopher Wolfe <christopher.wolfe at stonybrook.edu> wrote:
> 
> Hi Matt and Hong:
> 
> Thanks for the pointers. I had initially thought something similar: the sensitivities are linear, so you should just be able to do calculus operations on them. On further reflection, though, the units don’t work out. Going back to the wind stress curl example, if the cost function has units C, the sensitivities to wind stress are ∂J/∂τ and have units of C per N m^–2 = C m^2 N^–1. The sensitivities to wind stress curl, w, are ∂J/∂w and should have units of C per N m^–3 = C m^3 N^–1. However, if you just take the curl of the sensitivities, you get units of C m N^–1, which are off by a factor of m^2. 
> 
> It’s straightforward to work out the transformation rule for the derivative of τ in 1D using finite differences. The sensitivity of J to τ at the ith grid point is
> 
> 	
> <pdd_mathcal_J_ta.pdf>
> 
> On a N+1 point grid, we can use the chain rule to write the sensitivity to the derivative of the stress at a point j, τ’_j, as
> 
> 	
> <PastedGraphic-1.pdf>
> 
> On a C-grid,
> 
> 	
> <PastedGraphic-2.pdf>
> 
> from which it follows that
> 
> 	
> <PastedGraphic-3.pdf>
> 
> The derivative of τ with respect to τ’ is therefore
> 
> 	
> <PastedGraphic-4.pdf>
> 
> and the transformed sensitivity is
> 
> 	
> <PastedGraphic-5.pdf>
> 
> This has the correct units, but is effectively an integral of the original sensitivities rather than a derivative. I ran into trouble in 2D because writing the stress in terms of the curl requires solving an elliptic problem and things got a little hairy. 
> 
> Christopher
> 
> 
>> On Jul 6, 2020, at 6:47 PM, Zhang, Hong (US 398K) <hong.zhang at jpl.nasa.gov <mailto:hong.zhang at jpl.nasa.gov>> wrote:
>> 
>> Hi Chris,
>> You might check this paper about transformed gradient:
>> https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/jgrc.20240 <https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/jgrc.20240>paragraph [26] and figure6, figure7.
>> hope it helps
>> Hong
> 
> 
>> On Jul 7, 2020, at 12:27 AM, Matthew Mazloff <mmazloff at ucsd.edu <mailto:mmazloff at ucsd.edu>> wrote:
>> 
>> Ariane V, Bruce C, and I worked this out some time ago (~2013), so the details are blurry. But I am fairly certain that it works out fine to just take the curl of the sensitivity of J to the wind stress. That should give you the sensitivity to the curl. The operation is linear - we should be able to work this out.... Happy to discuss, though like I said, last time I thought about this was ~2013.
>> 
>> Matt
>>  
>> 
>>> On Jul 6, 2020, at 3:20 PM, Christopher Wolfe <christopher.wolfe at stonybrook.edu <mailto:christopher.wolfe at stonybrook.edu>> wrote:
>>> 
>>> Hi all:
>>> 
>>> Does any know of a simple formula (or reference) for a changing the dependent variables of adjoint sensitivities? For example, suppose you have the sensitivities of a cost function, J, to zonal and meridional wind stress. Is there a straightforward way to use these to calculate the sensitivity of J to the wind stress curl? I figured that there ought to be, but I got buried under a mountain of functional analysis and worried I was overthinking it.
>>> 
>>> Not sure if this is the right forum. If not, I’m happy to ask the wider MITgcm-support list.
>>> 
>>> Thanks in advance for any pointers!
>>> 
>>> Cheers,
>>> Christopher
>>> 
>>> 
>>> ----------------------------------------------------------------------------
>>> Christopher L. Pitt Wolfe
>>> Associate Professor (Physical Oceanography)
>>> School of Marine and Atmospheric Sciences
>>> Stony Brook University
>>> christopher.wolfe at stonybrook.edu <mailto:christopher.wolfe at stonybrook.edu>             631-632-3152
>>> ----------------------------------------------------------------------------
>>> 
>>> 
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