A new variational technique for direct and inverse problems of - - PowerPoint PPT Presentation

A new variational technique for direct and inverse problems

• f atmospheric chemistry

V.V. Penenko

Institute of Computational Mathematics and Mathematical Geophysics SB RAS 630090, Novosibirsk

Goal

to derive effective and concordant algorithms for realization of both the direct and adjoint problems for analysis of comprehensive chemical mechanisms, parameter estimation techniques, data assimilation, etc.

Flux Corrected Transport (FCT) Schemes/ Monotony algorithms

• The mesh refinement schemes
• The monotone interpolation routines
• Overlapping and moving grids
• Richardson extrapolation
• Romberg’s method
• “Mother” domain- “daughter” domain interactions
• Averager procedures
• “Smoother-dismoother”
• Non-linear renormalization
• “Lagrangian-type” monotonization

Flux Corrected Transport (FCT) Schemes

• Richardson (1910)
• Romberg (1955)
• Godunov S.K. (1959)
• Gol’din V.Ya, Kalitkin, Shishova (1965)
• Van Leer (1974-79) self-limiting diffusion, Taylor’s series

expansion

• A.Harten et al (1978-87) TVD, ENO
• Tremback et al (1987) Non-linear renormalization
• A.Bott ( 1989) Non-linear renormalization
• Smolarkiewicz and Grell (1992)

Critique of FCT schemes

• non-linear with respect to the state function;
• explicit ( in most cases);
• non-differential in classical sense ( flux

correction, logical operators)

• self-diffusive;
• each component of the state function

defines own character of approximating

• perator

grad div = − − + − + ∂ ∂ ≡

i i i i c i i

r t f S c c t c L

i

) (x, ) ( ) u ( ϕ ϕ µ π π

i

f - source term

i

S ) ( ϕ - operator of transformation, } ; { D x t t Dt ∈ ≤ ≤ = r

.

} , {

,

m i ci 1 = = ϕ

• state function

Boundary and initial conditions

Model of transport and transformation of pollutants

( )

, ( , ) ( ,0) ( )

i t i

R q t = ∈Ω = x x x ϕ ϕ ϕ

( ) ( )

i i i i

L P Q t ϕ ϕ ϕ ∂ + − = ∂ r r

1

ˆ ( ) , , ( )

j j j j t

t t t t Q D ϕ ϕ ϕ

+

= ≤ ≤ ∈ r r r

{ }

, 1, , 1

i i

n n ϕ ϕ = = ≥ r , 0

j

t t t τ τ = + ≤ ≤ ∆ ( )

i

L ϕ r

loss operators

( )

i

P ϕ r

production operators

i

Q sources and sinks of pollutants

The space co-ordinates are parametrically taken

t

x D ∈ r

Splitting stage: transformation

Properties of operators and functions

0, ( ) 0, ( ) 0, ( ) ( ) , ( ) ( ), ( )

i i i i i i i i

L P L L L ϕ ϕ ϕ ϕ ϕ ϕ α ϕ ϕ α ϕ ≥ ≥ ≥ = = ≥ r r r r r r r r % %

Monotony is qualifying standard: the state functions have to be positive if

i

Q ≥ .

S-stage Runge-Kutta method

1 1 1

, , , ( , , , )

s j j j j j j j j s j j j j jq q j q

t b k T t c t t a k k F Y T ϕ ϕ ϕ

+ = =

= + ∆ = + ∆ Ψ = + ∆ = Ψ

∑ ∑

x %

, ,

jq j j

a b c

parameters

S-stage Rosenbrock methods

1 1 1 1 1 1

, , , 1 ( , , , ) ( , , , ) ( , , , ), , , , , , parameters

s j j j j j j j j j j j jq q q j j j j j jj j jq j q j q j jq jq j jj j

t b k T t c t t a k E J t Y k F Y T t c k t F Y T t t b a c c ϕ ϕ ϕ ϕ γ γ ϕ γ γ

+ = − = − =

= + ∆ = + ∆ Ψ = + ∆   − = Ψ +   ∆   ∂ + ∆ ∆ ∂ −

∑ ∑ ∑

x x x % %

J

Jacobian matrix of derivative functions of F

%

Variational form of the model

( ) ( , , ) F t Q t ϕ α ϕ ϕ ϕ ∂ + = ∂ r

( )

* * * * *

( ) ( )

t t t

F d t F d t ϕ α ϕ ϕ ϕ τ ϕ α ϕ ϕ ϕ ϕ τ ϕϕ

∆ ∆ ∆

∂   + − =   ∂       ∂ − + − +     ∂    

∫ ∫

r r

Local adjoint problem

* * * * 1

( ) 0, 1

j t t

t ϕ α ϕ ϕ ϕ ϕ

+ =∆

∂ − + = ≡ = ∂ r

The function

*( )

ϕ τ is chosen from the conditions:

* * 1 ( )

( )

j t

e α

τ

ϕ τ ϕ

+ − ∆ −

=

t τ ≤ ≤ ∆

Balance relations

( )

1

* *

( , , ) ( )

j j

t t t

F t Q d ϕ ϕ τ τ ϕϕ

+

∆

+ =

∫

1 * *

( , , ) ( )

t j j j

F t Q d ϕ ϕ ϕ ϕ ϕ τ τ

∆ + =

+∫

( )

( ) ( , , ) , 0

j

e e F Q d t

τ ατ α τ τ

ϕ τ ϕ τ ϕ τ τ

′ − − −

′ ′ = + ≤ ≤ ∆

∫

The second order monotonic scheme

( )

/ 2 1 / 2 1 2 / 2 1 1 2 1 / 2 1 1 / 2

1 2 ( , , ), ( , ) 1

t j j t t t j j j j j t j j t t

e t e f e f f F t Q f F t e e f t

α α α α α α α

ϕ ϕ ϕ ϕ ϕ ϕ

− ∆ + − ∆ − ∆ ∆ + + − ∆ + − ∆ ∆

− ∆ = + + = = − = + ∆ % %

Монотонный характер схемы при

α ≥

непосредственно следует из свойств неотрицательности всех сомножителей слагаемых.

При

α ≡

получается аналог формулы Рунге-Кутта

2

( )

• t

∆

.

The forth order monotonic scheme

( )

( )

1 / 2 1 2 3 4 / 2 1 1 1 / 2 2 1 2 2 / 2 3 2 3 3 4 3

2 6 ( , ), 2 ( , ), 2 2 ( , ), 2 ( , )

j j t t t j j t j j t j j t t j j

t e f e e f f f t f F t f e t t f F t e f t f F t e tf e f F t t

α α α α α α α

ϕ ϕ ϕ η ϕ η η ϕ η η ϕ η

+ − ∆ − ∆ − ∆ − ∆ − ∆ − ∆ − ∆

∆ = + + + + ∆   = = +     ∆ ∆ = + = + ∆ = + = + ∆ = + ∆

Variational form of transformation model for multi-stage algorithms ( the 2nd order)

( )

{

( )

( )

( )

( )

}

1 * 1 * 1 1 2 1 1 * 1 * 1 1 2 1 2 * 1 * 1 / 2 / 2 / 2

( , , ) ( , ) ( , ) ( , ) 0, 1 , , 2 1

i i i i i

n J j j j tr i i i i i i i i i j j j i i j i i i j i j j i i p i i i i i i i t t i i i t t i

I Y H U f W f f F t f f F t f L H V f t e t W U W e e V

α α α α α

ϕ ϕ ϕ ϕ ϕ ϕ ϕ α ϕ κ α ϕ ϕ ϕ

− + + = = + + + − ∆ − ∆ ∆ − ∆

  ≡ − − +   + − + − + − + − − ∆ = − ∆ = = − =

∑∑

r r r r r % r r r % % % ,

i t

i t

t H e t

α − ∆ ∆

∆ = ∆

Adjoint transformation problem

( )

* * 1 2 1 * 1 * 2 1 1 * * 1 * 1 1 * * 1 * 1 1 1 * * 1 * 1 * 1

, ( ) ( ) (

j i i i j n j k i k j k i j j i i i i i j j j j i i i i i i i i i i i i i i i i i j j j j j k k i i i i k j i

f W F f f U V H U W H V f f F L H f ϕ ϕ ϕ ϕ ϕ ϕ α ϕ ϕ ϕ ϕ α α α α α ϕ ϕ ϕ ϕ ϕ ϕ

+ + + + = + + + + + +

= ∂ = ∂ = +       ∂ ∂ ∂ ∂ ∂ = + + + +       ∂ ∂ ∂ ∂ ∂       ∂ ∂ = + + + ∂

∑

r % % % % % r r % %

* 1

) 1,

n k j k i

i n α ϕ

=

    ∂   =

∑

Sensitivity relations for transformation problem

( )

* Y 1

( , , ) ( ) grad ( ),

N tr k k k

I Y Y Y Y ϕ ϕ δ ϕ δ ϕ δ

=

∂ Φ = = Φ ∂

∑

r

r r r r %

{ } ( )

{ }

, 1, , , , 1, , 1, , 1,

j k i i q q

Y Y k N Q q n i n j J ϕ κ = = = = = = r

Feedback equation

k

Y

grad ( ), 1,

k

dY k N dt ϕ = Φ = r

Sensitivity relations

}

}

1 * 1 1 1 * * 1 2 1 * * *0 1 2

( ) ( ) ( )

k k

q n J j i q i i j q q j j q i i q i q i q q q i i i i i i

L F F f f Q f Q f t δ ϕ δκ α κ ϕ ϕ δκ δκ κ κ δ δ δϕ δϕ

− = = = =

  ∂   Φ = − +   ∂       ∂ ∂ + + +     ∂ ∂   + ∆ +

∑ ∑ ∑ ∑

% r r r %

HCOOH O HOCH O HOCH OH CH NO O CH O NO HO O CH HO NO CO CO HCO CO H CH O CH CH CH OH O h OH O NO NO HO O OH ONO CH OOH H C

→ → ↓ ↓ ↓ → → → → → → → ↑ ↑

2 2 2 3 2 2 3 2 2 2 2 2 2 2 3 2 3 3 4 2 2 2 2 2 2 3 3

, , ν

Methane transformation in the atmosphere

Daily behavior of transformation products in methane-nitrogen-sulfur cycle

54 substances and 170 equations

Conclusion

• The universal technique for construction of algorithms

for direct and adjoint atmospheric chemistry problems is proposed.

• The new schemes hold the properties of unconditional monotony

and are efficiently realized.

• A new way for construction of variational principles

for multi-stage recursive algorithms is proposed. It gives the possibility of simple realization of adjoint problems and sensitivity studies.

Acknowledgements

The work is supported by

• RFBR

Grant 07-05-00673

• Presidium of the Russian Academy of Sciences

Program 16

• Department of Mathematical Science of RAS

Program 1.3.

• European Commission

contract No 013427