Steps needed to invert model output

[cspencerjones]

This is my first attempt at writing down the steps required to get to a place where we can invert model output to get the transport matrix. Comments welcome!

• [x] Replace cost function: tracer plus tracer control vector should equal model output of tracer (22 in the notes)
• [x] Check that with noisy input we get back the model output of tracer
• [x] Tracer control vector should be minimized (first term of 18)
  • [x] The vector f (and not the lu factorized version of A) should be an input to the cost function
• [x] It then probably makes sense to add tracer conservation (17 in the notes) to the cost function
• [x] Find the Lagrange multiplier for the tracer (26)
• [ ] Test to see if tracer is conserved more accurately after optimization (Edit: there is actually no reason for this to be true at this point)
• [ ] A second control vector u_f should also be an input to the cost function
• [ ] Changes in u_f should be minimized (second term of 18)
• [ ] Add constraint that f cannot become negative (21)
• [ ] Add Lagrange multiplier from first term of (27)
• [ ] Add mass conservation to cost function (19) and to Lagrange multiplier
• [ ] Test to see if mass is conserved more accurately after optimization
• [ ] Add geostrophy to cost function (20) and to Lagrange multiplier
• [ ] Test to see if geostrophic balance is conserved more accurately

[cspencerjones]

I have been playing around here: https://github.com/ggebbie/TMI.jl/blob/d0971825c9f60552400dd83e50aa194748577840/src/TMI.jl#L2219-L2295 (script to run this is here: https://github.com/ggebbie/TMI.jl/blob/invert-model/scripts/invert_model_TS.jl)

If I optimize right now the cost function grows, so I'm definitely doing something wrong. It was working ok until I implemented the Lagrange multipliers, so I probably have an error there. I'm mostly just posting this so that we can look at the code and talk about it at some point - see if I'm going in a good direction or if I need to refocus/rethink.

[ggebbie]

I'll take a look. I am currently working on issue #123 which has led me to refactor many things. I can help merge your work with the other changes.

[cspencerjones]

Updated list given that we are not learning the cross-face flux, but the water-mass matrix

• [x] Replace cost function: tracer plus tracer control vector should equal model output of tracer (22 in the notes)
• [x] Check that with noisy input we get back the model output of tracer
• [x] Tracer control vector should be minimized (first term of 18)
  • [x] The vector f (and not the lu factorized version of A) should be an input to the cost function
• [x] It then probably makes sense to add tracer conservation (17 in the notes) to the cost function
• [x] Find the Lagrange multiplier for the tracer (26)
• [x] A second control vector u_f should also be an input to the cost function
• [x] Changes in u_f should be minimized (second term of 18)
• [x] Add mass conservation to cost function (19) and to Lagrange multiplier
• [x] Test to see if mass is conserved more accurately after optimization

[cspencerjones]

I ended up getting rid of some of the Lagrange multiplier code but I now think I'm optimizing successfully here: https://github.com/ggebbie/TMI.jl/tree/invert-model.

The next step is to try this on actual model output.