10. Ocean State Estimation Packages¶
This chapter describes packages that have been introduced for ocean state estimation purposes and in relation with automatic differentiation (see Automatic Differentiation).
10.1. ECCO: model-data comparisons using gridded data sets¶
Author: Gael Forget
The functionalities implemented in pkg/ecco
are: (1) output
time-averaged model fields to compare with gridded data sets; (2)
compute normalized model-data distances (i.e., cost functions); (3)
compute averages and transports (i.e., integrals). The former is
achieved as the model runs forwards in time whereas the others occur
after time-integration has completed. Following
[FCH+15] the total cost function is formulated
generically as
using symbols defined in Table 10.1. Per Equation (10.3) model counterparts (\(\vec{m}_i\)) to observational data (\(\vec{o}_i\)) derive from adjustable model parameters (\(\vec{v}\)) through model dynamics integration (\(\mathcal{M}\)), diagnostic calculations (\(\mathcal{D}\)), and averaging in space and time (\(\mathcal{S}\)). Alternatively \(\mathcal{S}\) stands for subsampling in space and time in the context of Section 10.2 (PROFILES: model-data comparisons at observed locations). Plain model-data misfits (\(\vec{m}_i-\vec{o}_i\)) can be penalized directly in Eq. (10.1) but penalized misfits (\(\vec{d}_i\)) more generally derive from \(\vec{m}_i-\vec{o}_i\) through the generic \(\mathcal{P}\) post-processor (Eq. (10.2)). Eqs. (10.4)-(10.5) pertain to model control parameter adjustment capabilities described in Section 10.3 (CTRL: Model Parameter Adjustment Capability).
symbol | definition |
---|---|
\(\vec{u}\) | vector of nondimensional control variables |
\(\vec{v}\) | vector of dimensional control variables |
\(\alpha_i, \beta_j\) | misfit and control cost function multipliers (1 by default) |
\(R_i\) | data error covariance matrix (\(R_i^{-1}\) are weights) |
\(\vec{d}_i\) | a set of model-data differences |
\(\vec{o}_i\) | observational data vector |
\(\vec{m}_i\) | model counterpart to \(\vec{o}_i\) |
\(\mathcal{P}\) | post-processing operator (e.g., a smoother) |
\(\mathcal{M}\) | forward model dynamics operator |
\(\mathcal{D}\) | diagnostic computation operator |
\(\mathcal{S}\) | averaging/subsampling operator |
\(\mathcal{Q}\) | Pre-processing operator |
\(\mathcal{R}\) | Pre-conditioning operator |
10.1.1. Generic Cost Function¶
The parameters available for configuring generic cost function terms in
data.ecco
are given in Table 10.2 and
examples of possible specifications are available in:
- MITgcm_contrib/verification_other/global_oce_cs32/input/data.ecco
- MITgcm_contrib/verification_other/global_oce_cs32/input_ad.sens/data.ecco
- MITgcm_contrib/gael/verification/global_oce_llc90/input.ecco_v4/data.ecco
The gridded observation file name is specified by gencost_datafile
.
Observational time series may be provided as on big file or split into
yearly files finishing in ‘_1992’, ‘_1993’, etc. The corresponding
\(\vec{m}_i\) physical variable is specified via the
gencost_barfile
root (see Table 10.3).
A file named as specified by gencost_barfile
gets created where
averaged fields are written progressively as the model steps forward in
time. After the final time step this file is re-read by
cost_generic.F
to compute the corresponding cost function term. If
gencost_outputlevel
= 1 and gencost_name
=‘foo’ then
cost_generic.F
outputs model-data misfit fields (i.e.,
\(\vec{d}_i\)) to a file named ‘misfit_foo.data’ for offline
analysis and visualization.
In the current implementation, model-data error covariance matrices
\(R_i\) omit non-diagonal terms. Specifying \(R_i\) thus boils
down to providing uncertainty fields (\(\sigma_i\) such that
\(R_i=\sigma_i^2\)) in a file specified via gencost_errfile
. By
default \(\sigma_i\) is assumed to be time-invariant but a
\(\sigma_i\) time series of the same length as the \(\vec{o}_i\)
time series can be provided using the variaweight
option
(Table 10.4). By
default cost functions are quadratic but
\(\vec{d}_i^T R_i^{-1} \vec{d}_i\) can be replaced with
\(R_i^{-1/2} \vec{d}_i\) using the nosumsq
option
(Table 10.4).
In principle, any averaging frequency should be possible, but only
‘day’, ‘month’, ‘step’, and ‘const’ are implemented for
gencost_avgperiod
. If two different averaging frequencies are needed
for a variable used in multiple cost function terms (e.g., daily and
monthly) then an extension starting with ‘_’ should be added to
gencost_barfile
(such as ‘_day’ and ‘_mon’). [1] If two cost
function terms use the same variable and frequency, however, then using
a common gencost_barfile
saves disk space.
Climatologies of \(\vec{m}_i\) can be formed from the time series of
model averages in order to compare with climatologies of
\(\vec{o}_i\) by activating the ‘clim’ option via
gencost_preproc
and setting the corresponding gencost_preproc_i
integer parameter to the number of records (i.e., a # of months, days,
or time steps) per climatological cycle. The generic post-processor
(\(\mathcal{P}\) in Eq. (10.2)) also
allows model-data misfits to be, for example, smoothed in space by
setting gencost_posproc
to ‘smooth’ and specifying the smoother
parameters via gencost_posproc_c
and gencost_posproc_i
(see
Table 10.4).
Other options associated with the computation of
Eq. (10.1) are summarized in
Table 10.4 and
further discussed below. Multiple gencost_preproc
/
gencost_posproc
options may be specified per cost term.
In general the specification of gencost_name
is optional, has no
impact on the end-result, and only serves to distinguish between cost
function terms amongst the model output (STDOUT.0000, STDERR.0000,
costfunction000, misfit*.data). Exceptions listed in
Table 10.6 however
activate alternative cost function codes (in place of
cost_generic.F
) described in Section 10.1.3. In this
section and in Table 10.3
(unlike in other parts of the manual) ‘zonal’ / ‘meridional’ are to be
taken literally and these components are centered (i.e., not at the
staggered model velocity points). Preparing gridded velocity data sets
for use in cost functions thus boils down to interpolating them to XC /
YC.
parameter | type | function |
---|---|---|
gencost_name |
character(*) | Name of cost term |
gencost_barfile |
character(*) | File to receive model counterpart \(\vec{m}_i\) (See Table 10.3) |
gencost_datafile |
character(*) | File containing observational data \(\vec{o}_i\) |
gencost_avgperiod |
character(5) | Averaging period for \(\vec{o}_i\) and \(\vec{m}_i\) (see text) |
gencost_outputlevel |
integer | Greater than 0 will output misfit fields |
gencost_errfile |
character(*) | Uncertainty field name (not used in Section 10.1.2) |
gencost_mask |
character(*) | Mask file name root (used only in Section 10.1.2) |
mult_gencost |
real | Multiplier \(\alpha_i\) (default: 1) |
gencost_preproc |
character(*) | Preprocessor names |
gencost_preproc_c |
character(*) | Preprocessor character arguments |
gencost_preproc_i |
integer(*) | Preprocessor integer arguments |
gencost_preproc_r |
real(*) | Preprocessor real arguments |
gencost_posproc |
character(*) | Post-processor names |
gencost_posproc_c |
character(*) | Post-processor character arguments |
gencost_posproc_i |
integer(*) | Post-processor integer arguments |
gencost_posproc_r |
real(*) | Post-processor real arguments |
gencost_spmin |
real | Data less than this value will be omitted |
gencost_spmax |
real | Data greater than this value will be omitted |
gencost_spzero |
real | Data points equal to this value will be omitted |
gencost_startdate1 |
integer | Start date of observations (YYYMMDD) |
gencost_startdate2 |
integer | Start date of observations (HHMMSS) |
gencost_is3d |
logical | Needs to be true for 3D fields |
gencost_enddate1 |
integer | Not fully implemented (used only in Section 10.1.3) |
gencost_enddate2 |
integer | Not fully implemented (used only in Section 10.1.3) |
variable name | description | remarks |
---|---|---|
m_eta |
sea surface height | free surface + ice + global steric correction |
m_sst |
sea surface temperature | first level potential temperature |
m_sss |
sea surface salinity | first level salinity |
m_bp |
bottom pressure | phiHydLow |
m_siarea |
sea-ice area | from pkg/seaice |
m_siheff |
sea-ice effective thickness | from pkg/seaice |
m_sihsnow |
snow effective thickness | from pkg/seaice |
m_theta |
potential temperature | three-dimensional |
m_salt |
salinity | three-dimensional |
m_UE |
zonal velocity | three-dimensional |
m_VN |
meridional velocity | three-dimensional |
m_ustress |
zonal wind stress | from pkg/exf |
m_vstress |
meridional wind stress | from pkg/exf |
m_uwind |
zonal wind | from pkg/exf |
m_vwind |
meridional wind | from pkg/exf |
m_atemp |
atmospheric temperature | from pkg/exf |
m_aqh |
atmospheric specific humidity | from pkg/exf |
m_precip |
precipitation | from pkg/exf |
m_swdown |
downward shortwave | from pkg/exf |
m_lwdown |
downward longwave | from pkg/exf |
m_wspeed |
wind speed | from pkg/exf |
m_diffkr |
vertical/diapycnal diffusivity | three-dimensional, constant |
m_kapgm |
GM diffusivity | three-dimensional, constant |
m_kapredi |
isopycnal diffusivity | three-dimensional, constant |
m_geothermalflux |
geothermal heat flux | constant |
m_bottomdrag |
bottom drag | constant |
name | description | gencost_preproc_i
, _r , or _c |
---|---|---|
gencost_preproc |
||
clim |
Use climatological misfits | integer: no. of records per climatological cycle |
mean |
Use time mean of misfits | — |
anom |
Use anomalies from time mean | — |
variaweight |
Use time-varying weight \(W_i\) | — |
nosumsq |
Use linear misfits | — |
factor |
Multiply \(\vec{m}_i\) by a scaling factor | real: the scaling factor |
gencost_posproc |
||
smooth |
Smooth misfits | character: smoothing scale file |
integer: smoother # of time steps |
10.1.2. Generic Integral Function¶
The functionality described in this section is operated by
cost_gencost_boxmean.F
. It is primarily aimed at obtaining a
mechanistic understanding of a chosen physical variable via adjoint
sensitivity computations (see Automatic Differentiation) as done for example in
[MGZ+99][HWP+11][FWL+15]. Thus the
quadratic term in Eq. (10.1)
(\(\vec{d}_i^T R_i^{-1} \vec{d}_i\)) is by default replaced with a
\(d_i\) scalar [2] that derives from model fields through a generic
integral formula (Eq. (10.3)). The
specification of gencost_barfile
again selects the physical variable
type. Current valid options to use cost_gencost_boxmean.F
are
reported in Table 10.5. A
suffix starting with ‘_’
can again be appended to
gencost_barfile
.
The integral formula is defined by masks provided via binary files which
names are specified via gencost_mask
. There are two cases: (1) if
gencost_mask = ‘foo_mask’
and gencost_barfile
is of the
‘m_boxmean*’ type then the model will search for horizontal, vertical,
and temporal mask files named foo_maskC
, foo_maskK
, and
foo_maskT
; (2) if instead gencost_barfile
is of the
‘m_horflux_’ type then the model will search for foo_maskW
,
foo_maskS
, foo_maskK
, and foo_maskT
.
The ‘C’ mask or the ‘W’ / ‘S’ masks are expected to be two-dimensional
fields. The ‘K’ and ‘T’ masks (both optional; all 1 by default) are
expected to be one-dimensional vectors. The ‘K’ vector length should
match Nr. The ‘T’ vector length should match the # of records that the
specification of gencost_avgperiod
implies but there is no
restriction on its values. In case #1 (‘m_boxmean*’) the ‘C’ and ‘K’
masks should consists of +1 and 0 values and a volume average will be
computed accordingly. In case #2 (‘m_horflux*’) the ‘W’, ‘S’, and ‘K’
masks should consists of +1, -1, and 0 values and an integrated
horizontal transport (or overturn) will be computed accordingly.
variable name | description | remarks |
---|---|---|
m_boxmean_theta |
mean of theta over box | specify box |
m_boxmean_salt |
mean of salt over box | specify box |
m_boxmean_eta |
mean of SSH over box | specify box |
m_horflux_vol |
volume transport through section | specify transect |
10.1.3. Custom Cost Functions¶
This section (very much a work in progress…) pertains to the special
cases of cost_gencost_bpv4.F
, cost_gencost_seaicev4.F
,
cost_gencost_sshv4.F
, cost_gencost_sstv4.F
, and
cost_gencost_transp.F
. The cost_gencost_transp.F function can be
used to compute a transport of volume, heat, or salt through a specified
section (non quadratic cost function). To this end one sets
gencost_name = ‘transp*’
, where *
is an optional suffix starting
with ‘_’
, and set gencost_barfile
to one of m_trVol
,
m_trHeat
, and m_trSalt
.
name | description | remarks |
---|---|---|
sshv4-mdt |
sea surface height | mean dynamic topography (SSH - geod) |
sshv4-tp |
sea surface height | Along-Track Topex/Jason SLA (level 3) |
sshv4-ers |
sea surface height | Along-Track ERS/Envisat SLA (level 3) |
sshv4-gfo |
sea surface height | Along-Track GFO class SLA (level 3) |
sshv4-lsc |
sea surface height | Large-Scale SLA (from the above) |
sshv4-gmsl |
sea surface height | Global-Mean SLA (from the above) |
bpv4-grace |
bottom pressure | GRACE maps (level 4) |
sstv4-amsre |
sea surface temperature | Along-Swath SST (level 3) |
sstv4-amsre-lsc |
sea surface temperature | Large-Scale SST (from the above) |
si4-cons |
sea ice concentration | needs sea-ice adjoint (level 4) |
si4-deconc |
model sea ice deficiency | proxy penalty (from the above) |
si4-exconc |
model sea ice excess | proxy penalty (from the above) |
transp_trVol |
volume transport | specify masks (Section 10.1.2) |
transp_trHeat |
heat transport | specify masks (Section 10.1.2) |
transp_trSalt |
salt transport | specify masks (Section 10.1.2) |
10.1.4. Key Routines¶
TBA… ecco_readparms.F
, ecco_check.F
, ecco_summary.F
, …
cost_generic.F
, cost_gencost_boxmean.F
, ecco_toolbox.F
, …
ecco_phys.F
, cost_gencost_customize.F
,
cost_averagesfields.F
, …
10.1.5. Compile Options¶
TBA… ALLOW_GENCOST_CONTRIBUTION, ALLOW_GENCOST3D, … ALLOW_PSBAR_STERIC, ALLOW_SHALLOW_ALTIMETRY, ALLOW_HIGHLAT_ALTIMETRY, … ALLOW_PROFILES_CONTRIBUTION, … ALLOW_ECCO_OLD_FC_PRINT, … ECCO_CTRL_DEPRECATED, … packages required for some functionalities: smooth, profiles, ctrl
10.2. PROFILES: model-data comparisons at observed locations¶
Author: Gael Forget
The purpose of pkg/profiles is to allow sampling of MITgcm runs according to a chosen pathway (after a ship or a drifter, along altimeter tracks, etc.), typically leading to easy model-data comparisons. Given input files that contain positions and dates, pkg/profiles will interpolate the model trajectory at the observed location. In particular, pkg/profiles can be used to do model-data comparison online and formulate a least-squares problem (ECCO application).
The pkg/profiles namelist is called data.profiles. In the example below, it includes two input netcdf file names (ARGOifremer_r8.nc and XBT_v5.nc) that should be linked to the run directory and cost function multipliers that only matter in the context of automatic differentiation (see Automatic Differentiation). The first index is a file number and the second index (in mult* only) is a variable number. By convention, the variable number is an integer ranging 1 to 6: temperature, salinity, zonal velocity, meridional velocity, sea surface height anomaly, and passive tracer.
The netcdf input file structure is illustrated in the case of XBT_v5.nc To create such files, one can use the MITprof matlab toolbox obtained from https://github.com/gaelforget/MITprof . At run time, each file is scanned to determine which variables are included; these will be interpolated. The (final) output file structure is similar but with interpolated model values in prof_T etc., and it contains model mask variables (e.g. prof_Tmask). The very model output consists of one binary (or netcdf) file per processor. The final netcdf output is to be built from those using netcdf_ecco_recompose.m (offline).
When the k2 option is used (e.g. for cubed sphere runs), the input file is to be completed with interpolation grid points and coefficients computed offline using netcdf_ecco_GenericgridMain.m. Typically, you would first provide the standard namelist and files. After detecting that interpolation information is missing, the model will generate special grid files (profilesXCincl1PointOverlap* etc.) and then stop. You then want to run netcdf_ecco_GenericgridMain.m using the special grid files. This operation could eventually be inlined.
Example: data.profiles
#
# \*****************\*
# PROFILES cost function
# \*****************\*
&PROFILES_NML
#
profilesfiles(1)= ’ARGOifremer_r8’,
mult_profiles(1,1) = 1.,
mult_profiles(1,2) = 1.,
profilesfiles(2)= ’XBT_v5’,
mult_profiles(2,1) = 1.,
#
/
Example: XBT_v5.nc
netcdf XBT_v5 {
dimensions:
īPROF = 278026 ;
iDEPTH = 55 ;
lTXT = 30 ;
variables:
double depth(iDEPTH) ;
depth:units = "meters" ;
double prof_YYYYMMDD(iPROF) ;
prof_YYYYMMDD:missing_value = -9999. ;
prof_YYYYMMDD:long_name = "year (4 digits), month (2 digits), day (2 digits)" ;
double prof_HHMMSS(iPROF) ;
prof_HHMMSS:missing_value = -9999. ;
prof_HHMMSS:long_name = "hour (2 digits), minute (2 digits), second (2 digits)" ;
double prof_lon(iPROF) ;
prof_lon:units = "(degree E)" ;
prof_lon:missing_value = -9999. ;
double prof_lat(iPROF) ;
prof_lat:units = "(degree N)" ;
prof_lat:missing_value = -9999. ;
char prof_descr(iPROF, lTXT) ;
prof_descr:long_name = "profile description" ;
double prof_T(iPROF, iDEPTH) ;
prof_T:long_name = "potential temperature" ;
prof_T:units = "degree Celsius" ;
prof_T:missing_value = -9999. ;
double prof_Tweight(iPROF, iDEPTH) ;
prof_Tweight:long_name = "weights" ;
prof_Tweight:units = "(degree Celsius)-2" ;
prof_Tweight:missing_value = -9999. ;
}
10.3. CTRL: Model Parameter Adjustment Capability¶
Author: Gael Forget
The parameters available for configuring generic cost terms in
data.ctrl
are given in Table 10.7.
parameter | type | function |
---|---|---|
xx_gen*_file |
character(*) | Control Name: prefix from Table 10.8 + suffix. |
xx_gen*_weight |
character(*) | Weights in the form of \(\sigma_{\vec{u }_j}^{-2}\) |
xx_gen*_bounds |
real(5) | Apply bounds |
xx_gen*_preproc |
character(*) | Control preprocessor(s) (see Table 10.9 ) |
xx_gen*_preproc_c |
character(*) | Preprocessor character arguments |
xx_gen*_preproc_i |
integer(*) | Preprocessor integer arguments |
xx_gen*_preproc_r |
real(*) | Preprocessor real arguments |
gen*Precond |
real | Preconditioning factor (\(=1\) by default) |
mult_gen* |
real | Cost function multiplier \(\beta_j\) (\(= 1\) by default) |
xx_gentim2d_period |
real | Frequency of adjustments (in seconds) |
xx_gentim2d_startda
te1 |
integer | Adjustment start date |
xx_gentim2d_startda
te2 |
integer | Default: model start date |
xx_gentim2d_cumsum |
logical | Accumulate control adjustments |
xx_gentim2d_glosum |
logical | Global sum of adjustment (output is still 2D) |
name | description | |
---|---|---|
2D, time-invariant controls | genarr2d |
|
xx_etan |
initial sea surface height | |
xx_bottomdrag |
bottom drag | |
xx_geothermal |
geothermal heat flux | |
3D, time-invariant controls | genarr3d |
|
xx_theta |
initial potential temperature | |
xx_salt |
initial salinity | |
xx_kapgm |
GM coefficient | |
xx_kapredi |
isopycnal diffusivity | |
xx_diffkr |
diapycnal diffusivity | |
2D, time-varying controls | gentim2D |
|
xx_atemp |
atmospheric temperature | |
xx_aqh |
atmospheric specific humidity | |
xx_swdown |
downward shortwave | |
xx_lwdown |
downward longwave | |
xx_precip |
precipitation | |
xx_uwind |
zonal wind | |
xx_vwind |
meridional wind | |
xx_tauu |
zonal wind stress | |
xx_tauv |
meridional wind stress | |
xx_gen_precip |
globally averaged precipitation? |
name | description | arguments |
---|---|---|
WC01 |
Correlation modeling | integer: operator type (default: 1) |
smooth |
Smoothing without normalization | integer: operator type (default: 1) |
docycle |
Average period replication | integer: cycle length |
replicate |
Alias for docycle |
(units of
xx_gentim2d_period ) |
rmcycle |
Periodic average subtraction | integer: cycle length |
variaweight |
Use time-varying weight | — |
noscaling :math:
^{a} |
Do not scale with
xx_gen*_weight |
— |
documul |
Sets
xx_gentim2d_cumsum |
— |
doglomean |
Sets
xx_gentim2d_glosum |
— |
The control problem is non-dimensional by default, as reflected in the
omission of weights in control penalties [(\(\vec{u}_j^T\vec{u}_j\)
in (10.1)]. Non-dimensional controls
(\(\vec{u}_j\)) are scaled to physical units (\(\vec{v}_j\))
through multiplication by the respective uncertainty fields
(\(\sigma_{\vec{u}_j}\)), as part of the generic preprocessor
\(\mathcal{Q}\) in (10.4). Besides the
scaling of \(\vec{u}_j\) to physical units, the preprocessor
\(\mathcal{Q}\) can include, for example, spatial correlation
modeling (using an implementation of Weaver and Coutier, 2001) by
setting xx_gen*_preproc = ’WC01’
. Alternatively, setting
xx_gen*_preproc = ’smooth’
activates the smoothing part of WC01
,
but omits the normalization. Additionally, bounds for the controls can
be specified by setting xx_gen*_bounds
. In forward mode, adjustments
to the \(i^\text{th}\) control are clipped so that they remain
between xx_gen*_bounds(i,1)
and xx_gen*_bounds(i,4)
. If
xx_gen*_bounds(i,1)
\(<\) xx_gen*_bounds(i+1,1)
for
\(i = 1, 2, 3\), then the bounds will “emulate a local
minimum;” otherwise, the bounds have no effect in adjoint mode.
For the case of time-varying controls, the frequency is specified by
xx_gentim2d_period
. The generic control package interprets special
values of xx_gentim2d_period
in the same way as the exf
package:
a value of \(-12\) implies cycling monthly fields while a value of
\(0\) means that the field is steady. Time varying weights can be
provided by specifying the preprocessor variaweight
, in which case
the xx_gentim2d_weight
file must contain as many records as the
control parameter time series itself (approximately the run length
divided by xx_gentim2d_period
).
The parameter mult_gen*
sets the multiplier for the corresponding
cost function penalty [\(\beta_j\) in (10.1);
\(\beta_j = 1\) by default). The preconditioner, \(\cal{R}\),
does not directly appear in the estimation problem, but only serves to
push the optimization process in a certain direction in control space;
this operator is specified by gen*Precond
(\(=1\) by default).
10.5. The line search optimisation algorithm¶
Author: Patrick Heimbach
10.5.1. General features¶
The line search algorithm is based on a quasi-Newton variable storage method which was implemented by [GL89].
TO BE CONTINUED…
10.5.2. The online vs. offline version¶
- Online versionEvery call to simul refers to an execution of the forward and adjoint model. Several iterations of optimization may thus be performed within a single run of the main program (lsopt_top). The following cases may occur:
- cold start only (no optimization)
- cold start, followed by one or several iterations of optimization
- warm start from previous cold start with one or several iterations
- warm start from previous warm start with one or several iterations
- Offline versionEvery call to simul refers to a read procedure which reads the result of a forward and adjoint run Therefore, only one call to simul is allowed, itmax = 0, for cold start itmax = 1, for warm start Also, at the end, x(i+1) needs to be computed and saved to be available for the offline model and adjoint run
In order to achieve minimum difference between the online and offline code xdiff(i+1) is stored to file at the end of an (offline) iteration, but recomputed identically at the beginning of the next iteration.
10.5.3. Number of iterations vs. number of simulations¶
10.5.3.1. Summary¶
10.5.3.2. Description¶
\[\tt xdiff(i,1) = xx(i-1) + tact(i-1,1)*dd(i-1)\]serves as input for a forward and adjoint model run yielding a new gg(i,1). In general, the new solution passes the 1st and 2nd Wolfe tests so xdiff(i,1) represents the solution sought:
\[{\tt xx(i) = xdiff(i,1)}\]If one of the two tests fails, an inter- or extrapolation is invoked to determine a new step size tact(i-1,2). If more than one function call is permitted, the new step size is used together with the “old” descent direction dd(i-1) (i.e. dd is not updated using the new gg(i)), to compute a new
\[{\tt xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1)}\]that serves as input in a new forward and adjoint run, yielding gg(i,2). If now, both Wolfe tests are successful, the updated solution is given by
\[\tt xx(i) = xdiff(i,2) = xx(i-1) + tact(i-1,2)*dd(i-1)\]
In order to save memory both the fields dd and xdiff have a double usage.
- - in lsopt_top: used as x(i) - x(i-1) for Hessian update- in lsline: intermediate result for control update x = x + tact*dd
- - in lsopt_top, lsline: descent vector, dd = -gg and hessupd- in dgscale: intermediate result to compute new preconditioner
10.5.3.3. The parameter file lsopt.par¶
- NUPDATE max. no. of update pairs (gg(i)-gg(i-1), xx(i)-xx(i-1)) to be stored in OPWARMD to estimate Hessian [pair of current iter. is stored in (2*jmax+2, 2*jmax+3) jmax must be > 0 to access these entries] Presently NUPDATE must be > 0 (i.e. iteration without reference to previous iterations through OPWARMD has not been tested)
- EPSX relative precision on xx bellow which xx should not be improved
- EPSG relative precision on gg below which optimization is considered successful
- IPRINT controls verbose (>=1) or non-verbose output
- NUMITER max. number of iterations of optimisation; NUMTER = 0: cold start only, no optimization
- ITER_NUM index of new restart file to be created (not necessarily = NUMITER!)
- NFUNC max. no. of simulations per iteration (must be > 0); is used if step size tact is inter-/extrapolated; in this case, if NFUNC > 1, a new simulation is performed with same gradient but “improved” step size
- FMIN first guess cost function value (only used as long as first iteration not completed, i.e. for jmax <= 0)
10.5.3.4. OPWARMI, OPWARMD files¶
Two files retain values of previous iterations which are used in latest iteration to update Hessian:
OPWARMI: contains index settings and scalar variables
n = nn no. of control variables fc = ff cost value of last iteration isize no. of bytes per record in OPWARMD m = nupdate max. no. of updates for Hessian jmin, jmax pointer indices for OPWARMD file (cf. below) gnorm0 norm of first (cold start) gradient gg iabsiter total number of iterations with respect to cold start OPWARMD: contains vectors (control and gradient)
entry name description 1 xx(i) control vector of latest iteration 2 gg(i) gradient of latest iteration 3 xdiff(i),diag preconditioning vector; (1,…,1) for cold start 2*jmax+2 gold=g(i)-g(i-1) for last update (jmax) 2*jmax+3 xdiff=tact*d=xx(i)-xx (i-1) for last update (jmax)
Example 1: jmin = 1, jmax = 3, mupd = 5
1 2 3 | 4 5 6 7 8 9 empty empty
|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___|
0 | 1 2 3
Example 2: jmin = 3, jmax = 7, mupd = 5 ---> jmax = 2
1 2 3 |
|___|___|___| | |___|___| |___|___| |___|___| |___|___| |___|___|
| 6 7 3 4 5
10.5.3.5. Error handling¶
lsopt_top
|
|---- check arguments
|---- CALL INSTORE
| |
| |---- determine whether OPWARMI available:
| * if no: cold start: create OPWARMI
| * if yes: warm start: read from OPWARMI
| create or open OPWARMD
|
|---- check consistency between OPWARMI and model parameters
|
|---- >>> if COLD start: <<<
| | first simulation with f.g. xx_0; output: first ff_0, gg_0
| | set first preconditioner value xdiff_0 to 1
| | store xx(0), gg(0), xdiff(0) to OPWARMD (first 3 entries)
| |
| >>> else: WARM start: <<<
| read xx(i), gg(i) from OPWARMD (first 2 entries)
| for first warm start after cold start, i=0
|
|
|
|---- /// if ITMAX > 0: perform optimization (increment loop index i)
| (
| )---- save current values of gg(i-1) -> gold(i-1), ff -> fold(i-1)
| (---- CALL LSUPDXX
| ) |
| ( |---- >>> if jmax=0 <<<
| ) | | first optimization after cold start:
| ( | | preconditioner estimated via ff_0 - ff_(first guess)
| ) | | dd(i-1) = -gg(i-1)*preco
| ( | |
| ) | >>> if jmax > 0 <<<
| ( | dd(i-1) = -gg(i-1)
| ) | CALL HESSUPD
| ( | |
| ) | |---- dd(i-1) modified via Hessian approx.
| ( |
| ) |---- >>> if <dd,gg> >= 0 <<<
| ( | ifail = 4
| ) |
| ( |---- compute step size: tact(i-1)
| ) |---- compute update: xdiff(i) = xx(i-1) + tact(i-1)*dd(i-1)
| (
| )---- >>> if ifail = 4 <<<
| ( goto 1000
| )
| (---- CALL OPTLINE / LSLINE
| ) |
... ... ...
... ...
| )
| (---- CALL OPTLINE / LSLINE
| ) |
| ( |---- /// loop over simulations
| ) (
| ( )---- CALL SIMUL
| ) ( |
| ( ) |---- input: xdiff(i)
| ) ( |---- output: ff(i), gg(i)
| ( ) |---- >>> if ONLINE <<<
| ) ( runs model and adjoint
| ( ) >>> if OFFLINE <<<
| ) ( reads those values from file
| ( )
| ) (---- 1st Wolfe test:
| ( ) ff(i) <= tact*xpara1*<gg(i-1),dd(i-1)>
| ) (
| ( )---- 2nd Wolfe test:
| ) ( <gg(i),dd(i-1)> >= xpara2*<gg(i-1),dd(i-1)>
| ( )
| ) (---- >>> if 1st and 2nd Wolfe tests ok <<<
| ( ) | 320: update xx: xx(i) = xdiff(i)
| ) ( |
| ( ) >>> else if 1st Wolfe test not ok <<<
| ) ( | 500: INTERpolate new tact:
| ( ) | barr*tact < tact < (1-barr)*tact
| ) ( | CALL CUBIC
| ( ) |
| ) ( >>> else if 2nd Wolfe test not ok <<<
| ( ) 350: EXTRApolate new tact:
| ) ( (1+barmin)*tact < tact < 10*tact
| ( ) CALL CUBIC
| ) (
| ( )---- >>> if new tact > tmax <<<
| ) ( | ifail = 7
| ( ) |
| ) (---- >>> if new tact < tmin OR tact*dd < machine precision <<<
| ( ) | ifail = 8
| ) ( |
| ( )---- >>> else <<<
| ) ( update xdiff for new simulation
| ( )
| ) \\\ if nfunc > 1: use inter-/extrapolated tact and xdiff
| ( for new simulation
| ) N.B.: new xx is thus not based on new gg, but
| ( rather on new step size tact
| )
| (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries)
| )---- >>> if ifail = 7,8,9 <<<
| ( goto 1000
| )
... ...
... ...
| )
| (---- store new values xx(i), gg(i) to OPWARMD (first 2 entries)
| )---- >>> if ifail = 7,8,9 <<<
| ( goto 1000
| )
| (---- compute new pointers jmin, jmax to include latest values
| ) gg(i)-gg(i-1), xx(i)-xx(i-1) to Hessian matrix estimate
| (---- store gg(i)-gg(i-1), xx(i)-xx(i-1) to OPWARMD
| ) (entries 2*jmax+2, 2*jmax+3)
| (
| )---- CALL DGSCALE
| ( |
| ) |---- call dostore
| ( | |
| ) | |---- read preconditioner of previous iteration diag(i-1)
| ( | from OPWARMD (3rd entry)
| ) |
| ( |---- compute new preconditioner diag(i), based upon diag(i-1),
| ) | gg(i)-gg(i-1), xx(i)-xx(i-1)
| ( |
| ) |---- call dostore
| ( |
| ) |---- write new preconditioner diag(i) to OPWARMD (3rd entry)
| (
|---- \\\ end of optimization iteration loop
|
|
|
|---- CALL OUTSTORE
| |
| |---- store gnorm0, ff(i), current pointers jmin, jmax, iterabs to OPWARMI
|
|---- >>> if OFFLINE version <<<
| xx(i+1) needs to be computed as input for offline optimization
| |
| |---- CALL LSUPDXX
| | |
| | |---- compute dd(i), tact(i) -> xdiff(i+1) = x(i) + tact(i)*dd(i)
| |
| |---- CALL WRITE_CONTROL
| | |
| | |---- write xdiff(i+1) to special file for offline optim.
|
|---- print final information
|
O
[1] | ecco_check may be missing a test for conflicting names… |
[2] | The quadratic option in fact does not yet exist in
cost_gencost_boxmean.F … |