1 An Investigation on Minimization Algorithm for Nhm-4dvar KURODA Tohru, KAWABATA Takuya 1)JST Cooperative System for Supporting Priority Research 2)2nd Lab, Forecast Research Department, MRI 12 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 Function induced by physical process. L-BFGS Non- differentiable 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 Non Smooth Optimization (NSO) Bundle algorithm (ex. Zhang,et al. 2000) Random Gradient Sampling.... To reduce non-differentiable cost function Using descent algorism, 6 Zhang(2000), L-BFGS Optimization 7 Zhang(2000), Bundle Optimization 8 Convex Analysis Convex func Non-convex Classification L-BFGS Non- differentiable Convex algorithm with Non-convex Setting Bundle Algorithm 9 Subgradient( Subdifferentialconsists of 2 subgradients. At non-differentiable point x=0, Example. Not differentiable, But Directional differentiable 10 Example: Descent Direction at non-differentiable point Several Gradients may give some useful information. Directional derivative Solve, and do line-search in the direction of d. 11 Background of Bundle Algorithm Since 1976 Several Implementations ( N1CV2, N2FC1,PBUN, CP ROX,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. 12 A.Frangioni, Ph.D. Thesis, the University of Pisa,1997