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Min
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Assimilation Algorithms:Minimisation Techniques
Yannick TrémoletECMWF
Data Assimilation Training CourseMarch 2006
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4D Variational Data Assimilation
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Incremental 4D-Var
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Incremental 4D-Var
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Incremental 4D-Var
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The Outer Iterations
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The Inner Iterations
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Minimisation: Newton method
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Minimisation: Newton method
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Minimisation: Quasi-Newton method
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Minimisation: Quasi-Newton method
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Limited Memory Quasi-Newton
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Minimisation: Steepest Descent
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Minimisation: Steepest Descent
The first step fully minimizes the function in the descent direction, but this is undone by subsequent steps. We want to avoid this.
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Minimisation: Conjugate Gradient
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Conjugate Gradient Convergence
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Preconditioning
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4D-Var Preconditioning
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A case of poor convergence
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Theoretical example
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Theoretical example
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A case of poor convergence
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Hessian Preconditioning
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Hessian Eigenvectors Preconditioning
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4D-Var Eigenvalues
Eig
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N=1
1=3105.4
26=492.75
Preconditioning reduces the condition number k=1/N from 3105.4 to 492.75
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Conjugate Gradient and Lanczos Algorithm
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Lanczos Algorithm
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Lanczos Algorithm
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Conjugate Gradient and Lanczos Algorithm
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Superlinear Convergence
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Rounding Error
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CG Cost function reduction
Quasi-Newton with inexact line searches
Quasi-Newton with exact line searches
Conjugate Gradient without orthogonalisation
Conjugate Gradient with orthogonalisation
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CG Gradient norm reduction
Quasi-Newton with inexact line searches
Quasi-Newton with exact line searches
Conjugate Gradient without orthogonalisation
Conjugate Gradient with orthogonalisation
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4D-Var Cost function reduction
Variational Quality Control
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4D-Var gradient norm reduction
Convergence is roughly twice as fast with Hessian preconditioning.
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CG reduction of norm of gradient
0.05
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Spectrum of preconditioned Hessian
0
200
400
600
800
1000
1200
1 3 5 7 9 11 13 15 17 19 21 23
Min_42
Min_95
Min_255
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RMS of T analysis increments
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4D-Var Convergence
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4D-Var Convergence
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Summary
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