<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class=""><br class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Jul 7, 2020, at 10:43 AM, Christopher Wolfe <<a href="mailto:christopher.wolfe@stonybrook.edu" class="">christopher.wolfe@stonybrook.edu</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta http-equiv="Content-Type" content="text/html; charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class="">Hi Matt and Hong:</div><div class=""><br class=""></div><div class="">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 <span style="caret-color: rgb(0, 0, 0);" class="">∂J/∂w</span><span style="caret-color: rgb(0, 0, 0);" class=""> and </span>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. </div><div class=""><br class=""></div><div class="">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</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:7DE1FDE9-1B48-4DCD-AF67-D77B42D31822@lan"><pdd_mathcal_J_ta.pdf></span><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><meta http-equiv="Content-Type" content="text/html; charset=utf-8" class=""><div class=""></div><div class=""><br class=""></div><div class="">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</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:5CFFD504-A410-4B59-B72E-3A50A2DB979D@lan"><PastedGraphic-1.pdf></span><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><meta http-equiv="Content-Type" content="text/html; charset=us-ascii" class=""><div class=""></div><div class=""><br class=""></div><div class="">On a C-grid,</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:2CA7A297-F5B2-4160-9988-B5D905B41CE4@lan"><PastedGraphic-2.pdf></span><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><meta http-equiv="Content-Type" content="text/html; charset=us-ascii" class=""><div class=""></div><div class=""><br class=""></div><div class="">from which it follows that</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:A7E3981D-064C-4FBB-864A-13661787B51B@lan"><PastedGraphic-3.pdf></span><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><meta http-equiv="Content-Type" content="text/html; charset=utf-8" class=""><div class=""></div><div class=""><br class=""></div><div class="">The derivative of τ with respect to τ’ is therefore</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:00503687-59FB-4474-BB91-7F6BA149DC99@lan"><PastedGraphic-4.pdf></span><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><meta http-equiv="Content-Type" content="text/html; charset=us-ascii" class=""><div class=""></div><div class=""><br class=""></div><div class="">and the transformed sensitivity is</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span></div></div><span id="cid:A8905A98-D5A8-491C-A22D-4263F314A522@lan"><PastedGraphic-5.pdf></span><meta http-equiv="Content-Type" content="text/html; charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class=""></div><div class=""><br class=""></div><div class="">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. </div><div class=""><br class=""></div><div class="">Christopher</div><div class=""><br class=""></div><div class=""><br class=""></div><blockquote type="cite" class=""><div class="">On Jul 6, 2020, at 6:47 PM, Zhang, Hong (US 398K) <<a href="mailto:hong.zhang@jpl.nasa.gov" class="">hong.zhang@jpl.nasa.gov</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div class="" style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Hi Chris,<br class=""><div class="">You might check this paper about transformed gradient:</div><div class=""><a href="https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/jgrc.20240" class="">https://agupubs.onlinelibrary.wiley.com/doi/full/10.1002/jgrc.20240</a></div>paragraph [26] and figure6, figure7.<div class="">hope it helps</div><div class="">Hong</div></div></div></blockquote><div class=""><div class=""><div class="" style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div class=""><br class=""></div></div></div></div><div class=""><br class=""><blockquote type="cite" class=""><div class="">On Jul 7, 2020, at 12:27 AM, Matthew Mazloff <<a href="mailto:mmazloff@ucsd.edu" class="">mmazloff@ucsd.edu</a>> wrote:</div><div class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class=""><br class=""></div><div class="">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.</div><div class=""><br class=""></div><div class="">Matt</div><div class=""> </div><div class=""><br class=""></div><div class=""><div class=""><blockquote type="cite" class=""><div class="">On Jul 6, 2020, at 3:20 PM, Christopher Wolfe <<a href="mailto:christopher.wolfe@stonybrook.edu" class="">christopher.wolfe@stonybrook.edu</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta http-equiv="Content-Type" content="text/html; charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">Hi all:<div class=""><br class=""></div><div class="">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.</div><div class=""><br class=""></div><div class="">Not sure if this is the right forum. If not, I’m happy to ask the wider MITgcm-support list.</div><div class=""><br class=""></div><div class="">Thanks in advance for any pointers!</div><div class=""><br class=""></div><div class="">Cheers,</div><div class="">Christopher</div><div class=""><br class=""></div><div class=""><br class=""></div><div class="">
<div dir="auto" style="caret-color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div dir="auto" style="caret-color: rgb(0, 0, 0); letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class="">----------------------------------------------------------------------------<br class="">Christopher L. Pitt Wolfe<br class="">Associate Professor (Physical Oceanography)<br class="">School of Marine and Atmospheric Sciences<br class="">Stony Brook University<br class=""><a href="mailto:christopher.wolfe@stonybrook.edu" class="">christopher.wolfe@stonybrook.edu</a> 631-632-3152<br class="">----------------------------------------------------------------------------<br class=""><br class=""></div></div></div>
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