Nhm-4dvarの評価関数最小化の検討

An Investigation on
Minimization Algorithm
for Nhm-4dvar
1
KURODA Tohru,
KAWABATA Takuya 2
1)JST Cooperative System for Supporting Priority Research
2)2nd Lab, Forecast Research Department, MRI
1
Introduction
• Nhm-4dvar is the 4-dimensional variational data
assimilation system being developed by Forecast
Research Division, based on JMANHM, aiming for the
cloud-resolution data assimilation.
• Nhm-4dvar searches the optimal value of control
variable x s.t. x minimizes the cost function F(x),
by use of the information of gradient ∇F(x).
• After calculating ∇F(x) with adjoint model, L-BFGS
optimization algorithm is applied in present version.
• By introducing the physical process, non-differentiable
functions may appear. Then, the optimization assuming
smooth function may suffer from the irregularity.
Acutually, Nhm-4dvar is facing with the difficulty.
2
Deformation of Cost Function
induced by physical process.
L-BFGS
F (x)
Nondifferentiable
3
Choices
• Smoothing NHM
• Global Algorithm (GA, Sim.Anealing, etc)
Large development Cost!
Is there any choice else
still using ∇F(x) and adjoint codes?
4
Motivation
To reduce non-differentiable cost function
Using descent algorism,
Non Smooth Optimization
(NSO)
• Bundle algorithm (ex. Zhang,et al. 2000)
• Random Gradient Sampling
• ....
5
Zhang(2000), L-BFGS Optimization
6
Zhang(2000), Bundle Optimization
7
Convex Analysis
・Classification
L-BFGS
Bundle Algorithm
F (x)
Nondifferentiable
------ Convex func. -------
Non-convex
Convex algorithm with Non-convex Setting
8
Subgradient(劣勾配）
Example.
for x  0
x
f ( x)   2
 2 x  3 x for x  0
for x  0
1
f ( x)  
4 x  3 for x  0
f (0)  {1, 3}
At non-differentiable point x=0,
“Subdifferential  f 0  consists of 2 subgradients.”
“Not differentiable, But Directional differentiable”
9
Example: Descent Direction at
non-differentiable point
Directional derivative
min max g d  max min g d
T
|d |1 gf ( x )
T
gf ( x ) |d |1
T
min g d  d  g / | g |
|d |1
Solve min | g | , and do line-search
gf ( x )
in the direction of d.
Several Gradients may give some useful information.
10
Background of Bundle Algorithm
• Since 1976
• Several Implementations
( N1CV2, N2FC1,PBUN, CPROX,ETC )
and the benchmarking reports exist.
• Given by C.Lemarechal, we can test N1CV2.
Summary of MERIT introducing bundle solvers;
1) We only have to change the optimization procedure,
without changing the main framework.
2) We can examine the existing optimization solvers.
11
Bundle: { f1, f 2 ,.., f  ; g1 , g 2 ,.., g  ; y1 , y2 ,.., y }

Reduce f(y) !
 g
i
i
i
<=Bundle Information
1
f( y) |y  y|2
t
Serious Step
Null Step
Descent Failure is considered to be
because of insufficiency of Bundle
A.Frangioni, Ph.D. Thesis, the University of Pisa,1997 12
Nhm-4dvar Test1-1
• Moisture: Vapor-advection, no physical process.
• Grids: horizontal 22x22, vertical 40
(N1CV2)
Functioncalls of Bundle is more than twice of L-BFGS.
13
Nhm-4dvar Test1-2
• Moisture: Vapor-advection
• Grids: horizontal 22x22, vertical 40
(N1CV2)
14
Nhm-4dvar Test2-1
• Moisture: Vapor-advection ・Window : 10min
• Grids: horizontal 122x122, vertical 40, Nerima Heavy Rain Case.
• DT=10sec, itend=60. Analysis at it=60 is depicted below.
L-BFGS
BUNDLE
15
Nhm-4dvar Test2-1
• Window : 10min ・ Grids: horizontal 122x122, vertical 40
(FunctionCalls/Iteration)
Function Call Per
Iteration(Descent)
L-BFGS 1.5
BUNDLE 2.4
16
Summary and Future Work
• Nhm-4dvar with Bundle Algorithm is under investigation, using
N1CV2, in order to deal with the assimilation of physical process.
• Demonstration on the vapor advection assimilation seems to be
valid. Examination of availability for larger window case and
Parameter tuning for N1CV2 are the present tasks.
• Although bundle Algorithm showed the expensive calculation cost,
we hope that the resulting analysis is so optimal as to correspond to
the cost.
• Demonstration for the Warm Rain assimilation is the coming task.
• Other NSO solvers including non-convex bundle algorithms can be
tested in the future.
17

Download Report