An Investigation on Minimization Algorithm for Nhm-4dvar

1. An Investigation on Minimization Algorithm for Nhm-4dvar 1 2 KURODA Tohru, KAWABATA Takuya 1)JST Cooperative System for Supporting Priority Research 2)2nd Lab, Forecast Research Department, MRI

2. 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.

3. Deformation of Cost Functioninduced by physical process. Non- differentiable L-BFGS

4. Choices • Smoothing NHM • Global Algorithm (GA, Sim.Anealing, etc) Large development Cost! Is there any choice else still using ∇F(x) and adjoint codes?

5. Motivation To reduce non-differentiable cost function Using descent algorism, Non Smooth Optimization (NSO) • Bundle algorithm (ex. Zhang,et al. 2000) • Random Gradient Sampling • ....

6. Zhang(2000), L-BFGS Optimization

7. Zhang(2000), Bundle Optimization

8. Convex Analysis ・Classification Bundle Algorithm L-BFGS Non- differentiable ------ Convex func. ------- Non-convex Convex algorithm with Non-convex Setting

9. Subgradient(劣勾配） Example. At non-differentiable point x=0, “Subdifferential consists of 2 subgradients.” “Not differentiable, But Directional differentiable”

10. Example: Descent Direction at non-differentiable point Directional derivative ,and do line-search in the direction of d. Solve Several Gradients may give some useful information.

11. 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; • We only have to change the optimization procedure, • without changing the main framework. • 2) We can examine the existing optimization solvers.

12. Reduce f(y) ! <=Bundle Information 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

13. 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.

14. Nhm-4dvar Test1-2 • Moisture: Vapor-advection • Grids: horizontal 22x22, vertical 40 (N1CV2)

15. 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

16. Nhm-4dvar Test2-1 • Window : 10min 　・ Grids: horizontal 122x122, vertical 40 (FunctionCalls/Iteration) Function Call Per Iteration(Descent) L-BFGS1.5 BUNDLE 2.4

17. 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.