October,25 2005Scripps Institution of Oceanography An Alternative Method to Building Adjoints Julia...
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October,25 2005 Scripps Institution of Oceanography
An Alternative Method to Building Adjoints
Julia LevinRutgers University
Andrew Bennett “Inverse Modeling of the Ocean and Atmosphere”
Carl Wunsch “Ocean Circulation Inverse Problem”
Julia Muccino’s Inverse Ocean Modeling Website http://216.216.95.110/index.cfm?fuseaction=home.main
Outline
• Background: basic ideas of building adjoint operators on a simple 1D advection equation.
• Continuous and discrete adjoint operators
• Comparison between “analytic” and “symbolic” approaches to building discrete adjoint operators.
• Examples
Advection equation
( , ) ( ) ( )
( ,0) ( ) ( )
( , )m m m m
c F ft xa t B t b t
x I x i x
x t y n
are errors in the model, boundary conditions, initial conditions and data respectively
mntitbtxf ),(),(),,(
Least squares minimization (weak constraint)
2 2 2
0
0
2
2 ( )
2 ( ( , ) ( ) ( ))
2 ( ( ,0) ( ) ( ))
( ( , ) )
T b
f b I
a
TB
bI
a
d m m mm
J W f dxdt W b dt W i dx
c F f dxdtt x
a t B t b t dt
x I x i x dx
W x t y
Minimize model errors, subject to
constraints from PDE
from boundary conditions
from initial conditions
and make solution closest to the data
Analytic Forward and Inverse model
( , ) ( ) ( , )
( ,0) ( ) ( ,0)
f
b
I
c F Ct xa t B t C c a t
x I x C x
( ) ( ) ( )
( , ) 0
( , ) 0
m m m m
m
d y x x t tc Wt xb t
x T
0,0f
JJ
0J
111 ,, bbIIff WCWCWC
Also sensitivity:
To forcing
To BC
To IC)0,(2)(
),(2)(
),(2),(
xxI
J
tactB
J
txtxF
J
m
mmmd
T
b
b
a
if ytxWdtbWdxiWdxdtfWJ 2
0
222 )),((
Strong constraint
2 2 2
0
0
2
2 ( )
2 ( ( , ) ( ) ( ))
2 ( ( ,0) ( ) ( ))
( ( , ) )
T b
f b I
a
TB
bI
a
d m m mm
J W f dxdt W b dt W i dx
c F f dxdtt x
a t B t b t dt
x I x i x dx
W x t y
No model errors
Constraints from PDE
from boundary conditions
from initial conditions
How far solution is from to the data
Forward and adjoint models
( , ) ( ) ( , )
( ,0) ( ) ( ,0)
f
b
I
c F Ct xa t B t C c a t
x I x C x
( ) ( ) ( )
( , ) 0
( , ) 0
m m m m
m
d y x x t tc Wt xb t
x T
,0J
0J
Discrete Equations
( , ) ( ) ( )
( ,0) ( ) ( )
( , )m m m m
c F ft xa t B t b t
x I x i x
x t y n
bA
nyE
}{)},,({ mk
i yytx
2 2 2
0
0
2
2 ( )
2 ( ( , ) ( ) ( ))
2 ( ( ,0) ( ) ( ))
( ( , ) )
T b
f b I
a
TB
bI
a
d m m mm
J W f dxdt W b dt W i dx
c F f dxdtt x
a t B t b t dt
x I x i x dx
W x t y
Discrete Least squares
nWn
bA
WJ
dT
T
T
)(2
bA
yE
“Symbolic” computation of adjoint
Since we don’t have forward operator in a closed form, we can not compute its transpose directly. Instead we break the operator into basic operations according to the code structure (loops, sums, products, etc).
nn AAAAA 121
A
Then we derive transpose for those elementary operators, and change the order, according to a formula
TTTn
Tn
T AAAAA 121
Continuous adjoint:2 2 2
0
0
2
2 ( )
2 ( ( , ) ( ) ( ))
2 ( ( ,0) ( ) ( ))
( ( , ) )
T b
f b I
a
TB
bI
a
d m m mm
J W f dxdt W b dt W i dx
c F f dxdtt x
a t B t b t dt
x I x i x dx
W x t y
Minimize model errors, subject to
constraints from PDE
from boundary conditions
from initial conditions
and make solution closest to the data
“Analytic” Discrete adjoint
mmmmd
ki
ki
ki
ki
ki
ki
kik
i
ytxWetc
txfFx
ct
J
2
1
,
11
)),((
)(2
)(0
0
111
1
11
1
EyWEx
ct
J
CFx
ct
J
dT
ki
ki
ki
ki
ki
ki
ki
ki
ki
ki
ki
Twin Experiment
2 year run
NL simulation
Init. Cond.,
Forcing
True run:
NL simulation
True
solution
Modify Init Cond and Forcing
Dif
fere
nce
is
mod
el e
rror
Prior run :
NL simulation
Prior
solution
Synthetic data:
True + noise
Intr
odu
ce
obse
rvat
ion
al e
rror
Test run:
assimilation
Optimal solution
Statistics of
errors
Dif
fere
nce
is
resi
du
al e
rror
Dif
fere
nce
is
pri
or e
rror
Twin experiment with SEOM Comparison of Prior and Residual Error (free surface).
Assimilation is over 10 hour period (40 time steps).
Synthetic data is free surface.
Experiment 1: inserted at time 9.5h on all grid points, variance 10;
Experiment 2: inserted at time 4.5 and 9.5h at all grid points, variance 10
Experiment 3: inserted at tiime 9.5 h at all grid points, variance 5.
data
data
data
One iteration in outer loop, 9 iterations in the inner loop
Twin experiment with SEOM Comparison of Prior and Residual Error (velocity).
data
data data
Assimilation is over 10 hour period (40 time steps).
Synthetic data is free surface.
Experiment 1: inserted at time 9.5h on all grid points, variance 10;
Experiment 2: inserted at time 4.5 and 9.5h at all grid points, variance 10
Experiment 3: inserted at tiime 9.5 h at all grid points, variance 5.
Conclusion
• “Symbolic” method is a powerful tool to build complex adjoint operators, but it produces a unintuitive code, and does not provide any insight on the resulting discrete adjoint equations, more difficult to debug and analize.
• Continuous adjoint equations are very important in understanding the adjoint processes. Usually, they are not difficult to derive.
• “Analytic”approach to building discrete adjoint equations provides good understanding of the numerics of an adjoint code; the adjoint code is more compact and more readable; but the equations can be difficult to derive, especially for complex difference schemes.