Progress in the implementation of the adjoint of the Ocean model NEMO by using the YAO

software

M. Berrada, C. Deltel, M. Crépon, F. Badran, S. ThiriaWork supported by the SHOM (Hydrographic and Oceanographic Department of the French Navy)

under SINOBAD project

University of Paris VILOCEAN Laboratory

9th European AD Workshop INRIA Sophia-Antipolis, 26-27, November 2009

NEMO (Nucleus for European Modeling of the Ocean)is a platform for numerical modeling of ocean

The Ocean model NEMO

NEMO Ocean Model

Ocean Circulation

OPA

Model of sea ice

LIM

MarineBiogeochemistry

LOBSTER, PISCES

Idealized configuration of the physical part of NEMO

The domain is a limited area in the North of the Atlantic ocean (Gulf stream region)

The horizontal dimension 32x22 and 31 vertical levels

GYRE area localisation

GYRE configuration

The ocean trajectory depends on the initial state Accurate initial ocean environment (V0 ,ssh0 ,T0 ,S0)

V=(u,v) velocity ssh the sea surface height T Temperature S salinity

Variational assimilation (YAO software) x=T0 : Control vector y=M(x)=ssh at t=T : Observations (twin data)

Cost functionJ(x)=||y-yobs||2

A validation experiment with twin data

YAO

Semi- automatic generator of the adjoint code Based on a modular graph structure The modular graph is a data flow diagram which

describes the underlying physical model It consists of a set of modules, where the input of

each one is provided by the output of its predecessors

YAO: Modular Graph

M1

x11y12

y11

x31

x32

x33y32

y31

x21

x22y21

Forward model

dd

d

d

dd

d

d

d

d

d

1i

1j1 x

yF

)(xfy 111

)(xfy qqq

)(xfy 333

)(xfy 222

3i

3j3 x

yF

2i

2j2 x

yF

M3

M2

Backward model qTqq dydx F

M’2

1. Defined the modular graph structure of the model

2. Coding of the local functions fq

3. Coding of the Jacobian Fq

Modular graph in a point of the grid

Modular graph in a point of the grid

Modular graph of the space discretization

M1

M2

M3 M1

M2

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M3

Time evolution of the modular graph of the space

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M2

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t2t1 t3

x x

xx

t0

YAO: Modular Graph

Accomplished work

M1

x11

1i

1j1 x

yF

)(xfy 111 y11

y12

1. Defined the modular graph structure of the GYRE model under YAO

2. Coded the forward model

3. Implementation of the Jacobian of each module which is needed for the backpropagation (used for the computation of the cost function gradient )

d

d

d

Accomplished work

Comparison: GYRE-YAO vs GYRE-Fortran (accuracy 10-11)

Comparison of the ssh at t=100

(1 time step =2h)

Linear tangent test

α(k+1)=α(k)/2 1st order test 2nd order test : α (1): 2.500000e-02 : -1-> 7.914089e+00 -K-> 3.121720e+02 : α (2): 1.250000e-02 : -1-> 1.015254e+01 -K-> 8.051761e+02 : α (3): 6.250000e-03 : -1-> 3.269313e+00 -K-> 5.011360e+02 : α (4): 3.125000e-03 : -1-> 4.561872e+01 -K-> 1.460673e+04 : α (5): 1.562500e-03 : -1-> 8.877574e+01 -K-> 5.683726e+04 : α (6): 7.812500e-04 : -1-> 1.205476e+01 -K-> 1.542101e+04 : α (7): 3.906250e-04 : -1-> 1.799721e+02 -K-> 4.606209e+05 : α (8): 1.953125e-04 : -1-> 6.400455e+01 -K-> 3.278745e+05 : α (9): 9.765625e-05 : -1-> 3.628549e+03 -K-> 3.715594e+07 : α (10): 4.882813e-05 : -1-> 7.207983e+03 -K-> 1.476187e+08 : α (11): 2.441406e-05 : -1-> 9.999715e-01 -K-> 2.649493e+01 : α (12): 1.220703e-05 : -1-> 9.999856e-01 -K-> 2.674571e+01 : α (13): 6.103516e-06 : -1-> 9.999939e-01 -K-> 2.245280e+01 : α (14): 3.051758e-06 : -1-> 9.999962e-01 -K-> 2.876924e+01 : α (15): 1.525879e-06 : -1-> 9.999983e-01 -K-> 2.558008e+01

1).(

)()(lim

0

00

0

dXX

XMdXM

M

x KdXX

XMdXXM

1

).(

)()(1lim

0

00

0

M

1st order test 2nd order test

Adjoint test

dYdXdYdXdYdX TMM and ,,.,

13-4.198756eerror Relative

19-3179589e1.30390134MM

07-7329680e3.10544677M

07-7330984e3.10544677 M

T

T

dYdXdYdX

dYdX

dYdX

,,.

,

,.

Conclusion

Flexibility: Modifying the model at any time is straightforward due to modular graph structure

One can consider a more complex function as a module for the YAO graph and uses Tapenade (or other) to get the local adjoint

Encouraging results of the adjoint tests

Thank you!