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Le Ngoc Ly (1,2,3) (1) Institute of Oceanic, Atmos. & Env. Technology (IOAET) Hanoi, Vietnam (2) Applied Science International (APSIN), Hanoi, Vietnam (3) Naval Postgraduate School, Monterey, CA USA Emails: le_ASI@yahoo.com ; lely@ioaet.org Webs: www.ioaet.org;www.apsin.net;www.oc.nps.edu/~lelywww.ioaet.orgwww.apsin.netwww.oc.nps.edu/~lely On Applications of State-of-the-Art Mathematical-Computer Models in Oceanic- Atmospheric & Environmental Sciences &Technologies.

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Outline I - Introduction II - Supercomputing/HPC in Vietnam III - Basic of Atmospheric and Oceanic Models IV - Supercomputing/HPC with Data Assimilation Technique Numerical Grid Generation Technique Multi-block Grid & 2-D Domain Decomposition Technique on Parallel Platforms V - Some Results of Hurricane Prediction Model With SC/HPC Technology VI - Conclusion

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I-Introduction Most complicated problems for Atm., Ocean & Env. Modeling: Complicated Physics, Numerical techniques, Dealing with “big” Data sets, Needing very High Resolutions (especially for Marine Biology modeling), very “big” Computing domain! No SC is “too big, too fast” for these problems! Typical Problems of Atmospheric, Oceanic & Env. Sciences: Weather & Climate of various scales, Large Rain Forecasts, Hurricane & Storms, Floods, Tide, Waves, Strom Surges, Ocean Circulation, Physical-Biological Coupling, Tsunami, Air/Water Pollutions, ….Forecasts & especially, Climate Modeling & Climate Change! They are among #1 customers of Supercomputing (especially Climate Modeling, see next!).

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HPC/SU IN VIETNAM IOAET and now APSIN is the sole representative for Cray CX1 Supercomputer in Vietnam. We will have Cray CX1 “desk” supercomputer with HPC window2008 & Red Hat Linux ROCKS+ by 12/2009 in Hanoi.

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Some Advantages of Cray CX1 Architecture “vSMP” using “ScaleMP” allows CX1 effective as a “big supercomputer” (global memory), while CX1 is “low cost” supercomputer as a cluster “Desk” supercomputer Low level of Noise and Heat to be a “desk/personal” supercomputer Do not need a big room for CX1 Ideal supercomputer for a group of researchers/production or a U. Dept., Institutions … or even CX1 can be a “personal supercomputer” Low cost of maintenance, operating and software writing Highest level of ratio (performance/total cost)!

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III - Basic of Atmospheric/Oceanic & Evm Models Hydrodynamic Model with Primitive Equations. Full Air/Sea Boundary Layer Physics (Simplified BL Physics for Climate models!): Full and simplified Turbulence Closure. Numerical schemes/Methods: Fine-differential methods with Popular “Progonki”/”Pivotal-Condensation”. Some most popular forecast model:WRF (atm. meso-scale); POM (ocean); GFDL Hurricane Model; NCAR Climate model; Physical-Biological Coupling Model; NAM (Coupled wave-circulation model). Most Importance: a) Micro-Physics Parameterizations b) SC/HPC Technology: Resolutions problems and Fast with Large Memory to handle Forecasts/Prediction problems (including Economy/Financial Services) with Large Data Sets and Data Assimilation Techniques.

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IV - Supercomputing/HPC with DATA ASSIMILATION 1808? C. Gauss found solution for “forecast” Astronomy problem of appearing “Heaven body” based on observations. He formulated the problem as a minimization (or optimization) problem. Follow Gauss need to formulation the optimization problem as least square-fit problem. Almost all data assimilation schemes are based on this idea! Prof. Lev Gandin (LHMU, USSR, 1985? USA) formulated: Increment (d) = Observation - Forecast u ij = f ij + ∑ w k d k Є ij = δf ij + ∑ k w k d k Gandin finds w k so that the mean-square error of the estimate is minimized. To minimize, av(Є ij.Є ij ), a necessary condition is that derev of av(Є ij.Є ij ) = 0 for each w k.

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We have: T = F + W(Θ – F) where T: Temperature estimate; F: Forecast Temp; Θ: Observation of Temp; and W: Weight. Then the Principle : Estimate at Grd pnts = Forecast /Guess at Grd pnts + W Sum of d We have fields on grid points. With some model/dynamical consistency/condition adjustments, we have objectively analyze the data to gridded fields. Adjust the control vector (initial conditions, boundary conditions, other parameters) so that the difference between the forecast and “observations” (now in the form of estimates from the objective analysis) is minimized! We need to set up “cost function(al)” ℑ of difference of obs and forecast. Find min by taking derivative of ℑ set it to zero. Solution will be the optimum initial state.

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In general, neither one of these methods are practical for typical problems in geophysical sciences. These methods require too much Computer Time!!! There are various strategies for finding Grad ℑ but the most efficient is associated with the mathematical methodology called adjoint model. The adjoint model basically achieves the back substitution in a most efficient manner – equivalent to 1 “forward” model integration. One of popular adjoint model of Thacker (Oceanographer at Atlantic Oceanographic & Marine Lab in Miami, FL) is based on The Method of Lagrange Multiplier from Math Physics by modifying ℑ then L (Lagrangian) is expressed in terms of ℑ and forecast equations. Here, we would like to find Initial (guess) fields such that the forecasts minimize the squared (discrepancies: d). That means to find minimum we need to take derivatives of L and then set to zero. Name adjoint cones from the matrix algebra!

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Kalman Filter In Kalman Filter we need to have Tangent Linear Model (TLM) for the model forecast equation. This can handle nonlinear dynamics though linearization. We formulating problem as: X n+1 = Φ n X n + Weigh. [ Θ n+1 - Φ n X n ] Again in the Spirit of Gandin, we can subtract the True Value from both sides: Є n+1 = f n + W. [ δΘ n+1 - δf n+1 ] Need to find W so that: Ave(Є n+1 x Є n+1 ) minimized! Kalman permits a step by step update of the weight W based on previous history of the estimation process. Advantage of Kalman Filter: a) model and observational errors are simultaneously accounted. b) weight matrix is automatically updated each time step. Disadvantage: weighting matrix can be a big problem - a lot of matrix operations: Computing time problem!

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Nudging Technique of DA for an Ocean Forecast System PHYSICS - W(Mod – Obs) Obs: Surface Current

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V - Supercomputing/HPC with Numerical Grid Generation Technique Traditional grids: Rectangular, orthogonal grids Non-traditional grids: Curvilinear, nearly-orthogonal grids Vertical grids:

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NUMERICAL GRID GENERATION We need more complicated grid systems than traditional single block grids Such as: a) Nesting grids b) Curvilinear-coastal following (orthogonal & nearly orthogonal) multi- block grids c) Moving grids d) Adapted grids Numerical Grid Generation (Jose Thompson & Soni) & International Grid Associate Software developed by a group of Applied Math people. They used properties of Elliptic & Parabolic Equations to generate numerical grids. These Grid Packages are very popular in the world. Numerical Grid Generation Technique to Coastal Ocean Modeling. “A New Advance in Coastal Ocean Modeling: Application of the Grid Generation Technique.” High Performance Computing Contributions to DoD Mission Success 1998

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IV - Supercomputing/HPC with Multi-Block and 2-D Domain Decomposition (Traditional App.) on Parallel Platforms for Coastal Ocean Modeling

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Thank You!

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