On Applications of State-of-the-Art
Mathematical-Computer Models in OceanicAtmospheric & Environmental Sciences
&Technologies.
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/~lely
Outline
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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
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.
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)!
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.
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
uij = fij + ∑ wkdk
Єij = δfij + ∑k wkdk
Gandin finds wk so that the mean-square error of the estimate is
minimized.
• To minimize , av(Єij .Єij), a necessary condition is that
derev of av(Єij .Єij) = 0 for each wk.
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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
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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.
• 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!
Kalman Filter
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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:
Xn+1 = Φn Xn + Weigh . [ Θn+1
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- Φn X n ]
Again in the Spirit of Gandin, we can subtract the True Value from both sides:
Єn+1 = fn + W . [ δΘn+1 - δfn+1 ]
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Need to find W so that: Ave(Єn+1 x Єn+1) minimized!
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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
V - Supercomputing/HPC with
Numerical Grid Generation Technique
• Traditional grids: Rectangular, orthogonal grids
• Non-traditional grids: Curvilinear, nearly-orthogonal grids
• Vertical grids:
NUMERICAL GRID GENERATION
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We need more complicated grid systems than traditional single block grids
Such as: a) Nesting grids
b) Curvilinear-coastal following (orthogonal & nearly orthogonal) multiblock 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
IV - Supercomputing/HPC with
Multi-Block and 2-D Domain Decomposition (Traditional App.) on
Parallel Platforms for Coastal Ocean Modeling
Thank You!