[PPT] - Environmental forecasting on the base of online-integrated modeling PowerPoint Presentation

SLIDE 1

Environmental forecasting on the base of online-integrated modeling technology

Vladimir Penenko Elena Tsvetova

Institute of Computational Mathematics and Mathematical Geophysics SD RAS

SLIDE 2

Online-integrated modeling technology

Models of hydrodynamics
Models of atmospheric chemistry
Data of observations / assimilation
Technology of modeling
New algorithms
New mathematics/numerics

(tera- (12), peta-(15), exa- (18)

SLIDE 3

Model of atmospheric dynamics ¡

1

u

u p u fv kw F t x ρ ∂ ∂ + ⋅∇ − + = − + ∂ ∂ v 1

v

v p v fu F t y ρ ∂ ∂ + ⋅∇ + = − + ∂ ∂ v 1

w

w p w ku g F t z ρ ∂ ∂ + ⋅∇ − = − − + ∂ ∂ v ( ) 1

p p p p p v v

c c p p p c F f t c c ρ ⎛ ⎞ ∂ + ⋅∇ + ∇⋅ = − + ⎜ ⎟ ∂ ⎝ ⎠ v v ( (1 ) )

p d T T v v

c T R T T F f t c c α ∂ + ⋅∇ + + ∇⋅ = + ∂ v v

( )

1

0, (1 )

a d

p R T t ρ ρ ρ α

−

∂ + ∇⋅ = = + ∂ v

SLIDE 4

¡ Transport and transformation

f humidity

¡

( )

v

v v l f q

q q S S F t ∂ + ⋅∇ = − + + ∂ v 1

l

l l l lT l q

q q q v S F t z ρ ρ ∂ ∂ + ⋅∇ + = + ∂ ∂ v 1

f

f f f fT f q

q q q v S F t z ρ ρ ∂ ∂ + ⋅∇ + = + ∂ ∂ v

c

c c c q

q q S F t ∂ + ⋅∇ = + ∂ v

SLIDE 5

( )

i i i i i i i i

L div ( grad ) ((S ) f (x,t) r ) 0, t ∂ ≡ + − + − − = ∂ u πϕ πϕ ϕ π ϕ π ϕ µ ϕ π ϕ

Transport and transformation model

f gas pollutants and aerosols

Operators of transformation

{ } ( )

i r j

R U s (q ) g i i j i q 1 j 1

S ( ) P( ) ( ) k(q) s (q ) s (q )

−

− + = =

⎧ ⎫ = − ≡ − ⎨ ⎬ ⎩ ⎭

∑ ∏

ϕ ϕ ϕ ϕ Π ϕ ϕ ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1 1 2 1 1 1 2 1

1 , 2 , , 1,

M

M M a i ik k km k k m M i ik k i i i i k i i i

S K d K d e R Q i M

σ σ σ σ

ϕ σ γ ϕ σ α σ σ σ ϕ σ σ σ ϕ σ σ σ β ϕ σ σ ϕ σ ν ϕ σ σ σ ϕ σ σ

= = =

⎡ ⎤ ⎛ ⎞ = − − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ∂ ∂ ⎛ ⎞ − − + ⎡ ⎤ ⎡ ⎤ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ∂ ∂ ⎝ ⎠ − + =

∑ ∑ ∫ ∑ ∫

)

SLIDE 6

Goals

Development of methodology for

construction of integrated models of atmospheric dynamics and air quality in

n(off)-line regimes with account of all

accessible observational data

Design of adequate/open modeling system

SLIDE 7

Variational principle as a tool for model integration

It presents the equations and relations describing the development of multi-component and multi-scale processes in a simple invariant form; It holds the synthesis of continuum and discrete presentations of mathematical models and state functions;

SLIDE 8

Variational principle as a tool for model integration

It provides consistency for all technology stages of mathematical modeling;

Euler, Lagrange, Jacobi, and Hamilton principles

are interconnected among themselves: the stationary value of some definite integral (functional) is considered in each of them.

SLIDE 9

variational principles
combined use of models and observed data,
forward and inverse modeling procedures,
methodology for description of links between

regional and global processes ( including climatic changes) by means of orthogonal decomposition

f functional spaces for analysis of data bases and

phase spaces of dynamical systems Theoretical background

control theory,
sensitivity theory,
risk and vulnerability theory

A CONCEPT OF ENVIRONMENTAL MODELING

SLIDE 10

Basic elements for concept implementation:

models of processes
data and models of measurements
adjoint problems
constraints on parameters and state functions
functionals: objective, quality, control, restrictions

etc.

sensitivity relations for target functionals and

constraints

feedback equations for inverse problems

SLIDE 11

Desired!

Statement of the problem

Mathematical model

) , ( = − − + ∂ ∂ r f Y G t B      ϕ ϕ

,

ς ξ ϕ ϕ       + = + = , Y Y

;

) (

t

D ℑ ∈ ϕ 

is the state function ,

) (

t

D Y ℜ ∈ 

is the parameter vector.

G is the “space” operator of the model

A set of measured data

m

ϕ  ,

m

Ψ 

n

m t

D η ϕ    + = Ψ

m m

H )] ( [

,

m

H )] ( [ ϕ 

is the model of observations.

η

ς ξ     , , , r are the terms describing uncertainties and errors of the corresponding objects.

SLIDE 12

General form of functionals

( )

( ) ( ) ( , ) , , 1,...,

t

k k k k k D

F x t dDdt F k K ϕ ϕ χ χ Φ = ≡ =

∫

r r r

k

F are the Lipschitz's functions of the given form, differentiable, bounded dDdt

k

χ

are Radon’s or Dirac’s measures on

t

D , ) (

* t k

D ℑ ∈ χ

.

Quality functionals

( ) ( ( )) ( ( )) ( , ) ,

t

T k m m k D

H M H x t dDdt Φ ϕ = Ψ− ϕ Ψ− ϕ χ

∫

r r r r

“Measurement” functionals

[ ]

1

( ) ( ) ( ) ,

t

K m mk mk t mk k D

H x x dDdt x D

=

Φ ϕ = ϕ δ − ∈

∑ ∫

r r r r r

Restriction functionals ) , ( ( , ) , ( ≤ ≤ t x N t x

k

    ϕ ϑ ϕ

distributive constraints

dDdt t x

k k D k k

t

) , ( ) ) ( ) ( ( ) (     χ ϕ ϑ ϕ ϑ ϕ + = Φ

∫

Differentiability in extended sense

SLIDE 13

Variational form of model’s set: hydrodynamics+ chemistry+ hydrological cycle

( ) ( )

t

n i i i i i 1 D

I , ((S ) f (x,t) r ) dDdt

∗ ∗ ∗ ∗ ∗

ϕ, ϕ Λϕ,ϕ π ϕ ϕ

=

⎧ ⎫ ⎪ ⎪ ≡ + − − + ⎨ ⎬ ⎪ ⎪ ⎩ ⎭

∑ ∫

Y

( ) ( )

{ }

t

* * * * p T D

pp div pdiv pp TT div dDdt + α + α − +

∫

u u u

*

t

n

pu d dt

Ω

Ω =

∫

( ) ( )

{ }

t

T a a a D

W dDdt

ρ

α ρ α ρ ρ ρ ρ − − +

∫

SLIDE 14

Variational form

f convection-diffusion operators

( ) ( )

t

i D

1 div div 2 t t

∗ ∗ ∗ ∗ ∗ ∗ ∗

πϕ πϕ πϕ πϕ Λϕ,ϕ ϕ ϕ + ϕ πϕ ϕ πϕ ⎛ ⎧ ⎡ ⎤ ⎛ ⎞ ∂ ∂ ⎪ ≡ − − ⎜ ⎨ ⎢ ⎥ ⎜ ⎟ ⎜ ∂ ∂ ⎝ ⎠ ⎪ ⎣ ⎦ ⎩ ⎝ ∫ u u

}

1 grad grad 2

t D

dDdt dD

∗ ∗ ∗ ∗ ϕ

πµ ϕ ϕ ϕϕ π + + +

∫

( )

1 2

t

n b b b i

u R q d dt n

∗ Ω

ϕ ϕ µ α ϕ − ϕ π Ω ⎞ ∂ ⎡ ⎤ ⎛ ⎞ − + ⎟ ⎜ ⎟ ⎢ ⎥ ⎟ ∂ ⎝ ⎠ ⎣ ⎦ ⎠

∫

b b

R q = ϕ − ϕ −

boundary conditions on

t

Ω

SLIDE 15

{ 1

1 2 2

( ) ( ) 0.5 ( ) ( )

m h t t

h h T T k D D

M r M r ϕ ϕ α η η α Φ = Φ + + r r r r r r %

}

3 3 4 4 ( )

( ) ( )

h h h t

T T h D R D

M M α ξ ξ α ζ ζ + + r r r r

( , , )

h t

h D

Y ϕ ϕ

∗

⎡ ⎤ +⎣ ⎦ I r r r

) 4 , 1 ( , = i M i

are the weight matrices, 1 ,

4 1

= ≥

∑

= i i i

α α are the weight coefficients,

∗

ϕ ϕ  , are the solutions of the direct and adjoint problems generated from

) , , ( =

∗

ϕ ϕ    Y

h

I

Augmented functional for construction

f optimal algorithms and uncertainty assessment

Additive aggregation of the functionals for decomposition

SLIDE 16

* *

( ) ( ) ( ) , ,

h h h k k k

δ ϕ δ ϕ δ ϕ δϕ δϕ ϕ ϕ ⎛ ⎞ ⎛ ⎞ Φ Φ Φ = + + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ r r % % r r r % r r ( ) ( ) ( ) , , ,

h h h k k k

r Y r Y δ ϕ δ ϕ δ ϕ δ δξ δ ξ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Φ Φ Φ + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ r r r % % %

Variation of augmented cost functional. Consistency and optimization of algorithms

SLIDE 17

( , )

h h k t

B G Y f r ϕ ϕ ϕ ∗ ∂Φ ≡ Λ + − − = ∂ % r r r r r r

( ) ( , ) 0,

h T T k t k k k

B A Y d ϕ ϕ ϕ ϕ

∗ ∗

∂Φ ≡ Λ + + = ∂ % r r r r r

)), ( 5 . ) ( (

1 1

η η α ϕ ϕ    M d

T h k k

+ Φ ∂ ∂ =

( )

k t t

x ϕ

∗ =

= r r

), , (

1 3

= + =

∗ −

t x M

k a

    ϕ ϕ ϕ

), , ( ) , (

* 1 2

t x M t x r

k

   ϕ

−

=

k a

M Y Y Γ − =

− 

 

1 4

) , , (

∗

∂ ∂ = Γ

k h k

Y I Y ϕ ϕ      [ ]

) , ( ) , (

=

+ ∂ ∂ ≡

α

ϕ δ α ϕ α ϕ δ ϕ Y G Y A

h

     

ϕ

t

Λ is the approximation of time derivatives Initial guess:

a a

Y Y r      = = =

) ( ) ( ) (

, , ϕ ϕ

The universal algorithm

f forward & inverse modeling

SLIDE 18

1 2 * k

r( x,t ) M ( x,t ),

−

= ϕ r r r

1 3 k

M ( x, ), t

− ∗

ξ = ϕ = r r r

1 1 4 4 h k k

M M I ( ,Y , ) Y

− − ∗

∂ ζ = Γ = ϕ ϕ ∂ r r r r r r

Algorithms of uncertainty calculation based on sensitivity analysis and data assimilation:

in model in initial state in model parameters and sources

2 4 ,( , )

i

M i =

are the weight matrices

SLIDE 19

Fundamental role

f uncertainty functions
integration of all technology components
bringing control into the system
regularization of inverse methods
targeting of adaptive monitoring
cost effective data assimilation

SLIDE 20

( ) ( , Y) ( ,Y Y, )

h h k k k

I

α

δ δ α δ α

∗ =

∂ Φ ≡ ≡ + ∂ ϕ Γ ϕ Γ ϕ ϕ

( ,Y Y, ) Y

h k k

I

α

α δ δ α

∗ =

∂ ∂ ⎛ ⎞ = + ⎜ ⎟ ∂ ∂ ⎝ ⎠ Γ ϕ Γ ϕ ϕ

The main sensitivity relations Algorithm for calculation

f sensitivity functions

} {

ki k

Γ = Γ

are the sensitivity functions

} { Y

i

Y δ δ =

are the parameter variations

N i K k , 1 , , 1 = = N N N dt dY

k

≤ = Γ − =

α α α α α

α η , , 1 , The feed-back relations

Some elements

f optimal forecasting and design

SLIDE 21

Realization

f main operator equations

Convection & diffusion

SLIDE 22

Idea and basic approximations

( )

1 1 1 1

* * * * *

, ( ) ( )

i i i i i i i i

x x x x x x x x

L f dx L dx A f x x dx ϕ ϕ ϕ ϕ ϕ ϕ ϕ

− − − −

= − = + − =

∫ ∫ ∫

Differential operators of common type in the models

* *

0, Lϕ =

( )

1 1

* *

, ( ) ( ) 0, 2,

i i i i

x x x x x

A f x x dx i n ϕ ϕ ϕ

− −

− = =

∫

If then

*( ) 1

( ), , 1,2

i i

x x x x

α

ϕ α

− ≤

≤ =

integrating multipliers

{ } { }

*(1) *(1) *(2) *(2) 1 1

1, 0 , 0, 1 , 1, 1

i i i i x

i n ϕ ϕ ϕ ϕ

+ +

= = = = = −

Fundamental analytical solutions of local adjoint problems

SLIDE 23

Additive form of integral identity

( )

* * 1

( , )

t

r D

I L f dDdt t

α α α

ϕ ϕ ϕ ϕ ϕ

=

⎛ ⎞ ∂ = + − = ⎜ ⎟ ∂ ⎝ ⎠

∑ ∫

( )

{ }

* 1 * 1 * 2 1

( , , ) ( ) 1

h h t t J r j j j j j j j j j j j j j D

I Y L f t

α α α α α α α α α α α α

ϕ ϕ ϕ ϕ τ σ ϕ σ ϕ ϕ δ

− − = =

+ Φ = ⎧ ⎡ ⎤ ⎡ ⎤ Ψ − + Δ Ψ − Ψ + Ψ − − − ⎨ ⎣ ⎦ ⎣ ⎦ ⎩

∑ ∑ ∫

* 1 2

1 ( )

r J j j j j j j

t dD r

α α

ϕ ϕ ϕ δ ϕ

= =

⎫ ⎛ ⎞ ⎪ + − + Φ ⎬ ⎜ ⎟ ⎪ ⎝ ⎠ ⎭

∑ ∑

Variational construction of additive schemes

SLIDE 24

Decomposition of integral identity

* *

( , ) 0;

b j j a

I u d W dx x x x

α ϕ ϕ

τ µ ϕ ⎛ ⎞ ∂Ψ ∂ ∂Ψ ⎛ ⎞ ≡ Ψ + Δ − + Ψ − = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠

∫

%

1 j j j

W f ϕ τ

−

= + Δ

( )

1 1

1 , , , 0.5 1

j j j j j j j

t t t t σϕ σ ϕ τ σ σ

− −

Ψ = + − Δ = Δ Δ = − ≤ ≤

( )

1

1 1

j j

ϕ σ ϕ σ

−

= Ψ − −

SLIDE 25

Solutions of the local adjoint problems

for convection-diffusion operators

* ( ) 1

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

i i i i

L x x x x i n

α

ω α

+

= = ≤ ≤ = −

(1) 1 1 * * (2) 1 (2) (1) 1 1 1 2 1 1

( ) ( ) ( ), ( ) , 1, ( ) ( ) ( ), ; ( ) ( ),

i i i i i i i i i i n n n n

x x x x x x x i n x x x x x x x x x x x x x x ω ϕ ϕ ω ω ω ω ω ω ω

− − + − −

⎧ ≤ ≤ ⎪ = = = ⎨ ≤ ≤ ⎪ ⎩ = ≤ ≤ = ≤ ≤

( )

2 1 1

1 (1)( ) x x i i

x A e eν

ν ν

ω

ʹ″ − − ʹ″ −

= −

( )

2 2 1

1 (2)( ) x x i i

x A e e

ν ν ν

ω

ʹ″ − ʹ″ −

= −

1

( )/

i i

x x x x

+

ʹ″ = − Δ

( )

2 1

( )

1/(1 )

i i

A e ν

ν −

= −

{ }

1 1 2 2 1 2

, , 0, 0 i x x ν λ ν λ λ λ = Δ = Δ ≥ ≤

2

1

j i

u d τ λ λ µ µ τ ⎧ ⎫ + Δ + − = ⎨ ⎬ Δ ⎩ ⎭

SLIDE 26

approximation, stability, monotonicity, transportivity, differentiability with respect to parameters and state functions , uniform schemes without flux-correction

Properties of DA numerical schemes:

Three-point numerical scheme

0, 0, , , , при

i i i i i i i i i i

a c b b b b a b c µ

+ − + −

≥ ≥ = + ≥ ≥ ≥

2

2 max 1,

i

u CFL d x x τ µ τ τ Δ ⎧ ⎫ Δ = + + Δ < ⎨ ⎬ Δ Δ ⎩ ⎭

( ) (

)

{ }

1 2

max , / 10

i i

u x ν ν µ − Δ ≤

1 1 i i i i i i i

a b c f ϕ ϕ ϕ

+ −

− + − =

SLIDE 27

Singular case

µ =

( )

1 1

(1) (2) 1/ 2 1 1 1 1/ 2 1/ 2 1/ 2 1 (1) (2) 1

exp( ) exp( ) ( )exp ( )exp .

i i i i

i i i i i i i i i i i x x i i i i x x

u x u u u x f x x x dx f x x x dx

+ −

+ + − − − − − − − + + + −

− −λ Δ ϕ + + ϕ − −λ Δ ϕ = −λ − + −λ −

∫ ∫

j j

u d f x ∂ϕ + ϕ = ∂ Numerical scheme

1 1; i i i

x x x

− −

Δ = −

1

;

i i i

x x x

+

Δ = −

( )

(1) 1 1/2

/ ;

i i

d u+

− −

λ =

( )

(2) 1/2

/ ;

i i

d u−

+

λ =

( )

0,5 0; u u u

+ =

+ >

( )

0,5 0. u u u

− =

− >

SLIDE 28

Godunov’s test problem

(С.К.Годунов Математический сборник, т. 47(89):3, 1959, с.271-306.) 0, ( ),

t

u x u const t x

=

∂ϕ ∂ϕ + = ϕ =ϕ = ∂ ∂

2

( , ) 0,5 0,25 x ut x t x − ⎛ ⎞ ϕ = − − ⎜ ⎟ Δ ⎝ ⎠ Analytic solution

(G1) (G2)

SLIDE 29

Difference-analytic scheme

1 2

3 4 2

j j j j

u t x

− −

ϕ − ϕ + ϕ ∂ϕ + = Δ ∂

1 2

3 4 2 ( , ) 0,5 2

j

j j j j t t

u x ut x t t x x t

− − =

⎛ ⎞ ϕ − ϕ + ϕ − ∂ϕ = − − = ⎜ ⎟ Δ Δ Δ ∂ ⎝ ⎠

(G3) (G4)

2 0,5

j

t t

x ut x x x

=

∂ϕ − ⎛ ⎞ = − ⎜ ⎟ ∂ Δ Δ ⎝ ⎠

(G5)

SLIDE 30

Chemical operators

SLIDE 31

Stiff differential systems

1 1 1 1

1000 0; (0) 1, ( ) x x x x t + = = > &

2 2 1 2 2

0,999 ; (0) 0,999, ( ) x x x x x t + = = > &

1( )

exp( 1000 ) x t t = −

2( )

0,001exp( 1000 ) exp( ) x t t t = − − + −

Analytical solution

( )

1 1

( ) 1 1000 ( ) x t t t x t + Δ = − Δ

( )

2 2 1

( ) 1 ( ) 0,999 ( ) x t t t x t tx t + Δ = − Δ + Δ

Euler explicit scheme Euler implicit scheme

( )

1 1

( ) ( )/ 1 1000 x t t x t t + Δ = + Δ

( ) ( )

2 2 1

( ) ( ) 0,999 ( ) / 1 x t t x t tx t t + Δ = + Δ + Δ

SLIDE 32

Direct and adjoint operators for kinetics of chemical transformations

{ } ( )

r i j

U R s (q ) g i i j i q 1 j 1

S ( ) P( ) ( ) k(q) s (q ) s (q )

−

− + = =

⎧ ⎫ = − ≡ − ⎨ ⎬ ⎩ ⎭

∑ ∏

r r r ϕ ϕ ϕ ϕ ϕ Π ϕ ϕ

{ } ( )

q q i i

U U R s (q ) * * * i g j i q 1 j 1 1 i i

s (q ) S ( ) k(q) s (q ) s (q )

−

− − + = = =

⎧ ⎫ ≡ − ⎨ ⎬ ⎩ ⎭

∑ ∏ ∑

r

α α α α α α

ϕ ϕ ϕ ϕ ϕ ϕ

SLIDE 33

Relations for reaction operators. Monotonicity and positivity properties

( ) ( )

1

*

P( ) ( )

j j

t t

f dt t

α α α α α α

ϕ ϕ ϕ ϕ ϕ

−

∂ ⎛ ⎞ + − Π − = ⎜ ⎟ ∂ ⎝ ⎠

∫

r r

{ }

*

, 1, , ( ) 1, 2, ,

h j t

n t j J x D

α α α

ϕ ϕ α ϕ = = = = ∈ r r

( ) ( )

0, P( ) 0, ( )

α α α

ϕ ϕ ϕ ≥ ≥ Π ≥ r r

SLIDE 34

Structure of aerosol operators

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1 1 1 1 1 1 1 1 2 2

, 1 , 2 , , 1,

M

i M M ik k km k k m M i ik k i i k i i i i i

t t K d K d R e Q i M

σ σ σ σ

ϕ σ γ ϕ σ α σ σ σ ϕ σ σ σ ϕ σ σ σ β ϕ σ σ ϕ σ ϕ σ ν ϕ σ σ σ σ

= = =

∂ = ∂ ⎡ ⎤ ⎛ ⎞ − − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠ ∂ ∂ − + ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ∂ ∂ + =

∑ ∑ ∫ ∑ ∫

)

production destruction condensation / evaporation diffusion sources / sinks

SLIDE 35

Variational principles for transformation models

( ) ( )

1

* 1

( ) ( )

j j

n t t

P f dt t

α

α α α α α α

ϕ ϕ ϕ ϕ

−

=

⎧ ⎫ ∂ ⎛ ⎞ + − Π − = ⎨ ⎬ ⎜ ⎟ ∂ ⎝ ⎠ ⎩ ⎭

∑ ∫

r r

{ }

*

, 1, , ( ) 1, 2, ,

h j t

n t j J x D

α α α

ϕ ϕ α ϕ = = = = ∈ r r

( ) ( )

0, ( ) 0, ( ) P

α α α

ϕ ϕ ϕ ≥ ≥ Π ≥ r r

( ) ( )

1

*

( ) ( )

j j

t t

P f dt t

α α α α α

ϕ ϕ ϕ ϕ

−

∂ ⎛ ⎞ + − Π − = ⎜ ⎟ ∂ ⎝ ⎠

∫

r r

Decomposition on reaction mechanisms

1,nα α =

SLIDE 36

Variational principles for transformation models (2)

( ) 1 1

( ) ( ) ( ) ,

ri i

U R S r i i i r

P k r s r ϕ

α

− α = α=

⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

∑ ∏

ϕ

( ) 1 1

( ) ( ) ( ) .

ri i

U R S r i i i

P k r s r ϕ

α

− α α=

⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦

∑ ∏

ϕ

( ) ( )

i i i

P L ϕ ≡ ϕ ϕ ϕ ϕ ( ) ( ) ( )

i i i i

F F q = = Π + q ϕ ϕ ϕ ϕ, κ, ϕ

( )

1,..., T n

k k

α

κ =

( )

1,..., T n

q q

α

q =

SLIDE 37

Variational principles for transformation models (3)

( ) ( )

1 1

* 1 * * * *

( ) ( ) ( )

j j j j

t t j j i i i i i i i i i i t t

L t dt F dt t ψ ψ ϕ ψ ϕψ ϕψ

+ +

+

⎛ ⎞ ∂ − + − + − = ⎜ ⎟ ∂ ⎝ ⎠

∫ ∫

ϕ ϕ ϕ ϕ

* *(

)

i t

Q D ψ ∈

1

* * * 1

( ) 0, , 1.

j

i i i j j i t

L t t t t ψ ψ ψ

+

∂ − = ≤ ≤ = ∂ ϕ Adjoint functions are chosen as solutions of equations :

* i

ψ

SLIDE 38

Variational principles for transformation models (4)

( )

1

1 * *

( ) ( )

j j

t j j i i i i i t

F t t dt ϕ ϕψ ψ

+

+ =

+ ∫ ϕ,

( ) 1

( )

j i j i j

t b t b t j j i i i

e F e d

τ

ϕ ϕ τ τ

Δ − Δ − Δ − + =

+ ∫ ϕ, Using the idea of Euler integrating multipliers (factors), we get the system of integral equations Finaly, ( )

j j i i

b L = ϕ

SLIDE 39

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

SLIDE 40

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

54 substances and 170 equations

SLIDE 41

Sensitivity relations for the problems

f atmospheric chemistry

( )

* 1 1

, ,

k

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

F q q f

γ γ γ γ

δ δ δ

= = =

⎧ ⎧ ⎡ ⎤ ⎛ ⎞ ∂ ⎪ ⎪ ⎢ ⎥ ⎜ ⎟ Φ = + + ⎨ ⎨ ⎜ ⎟ ∂ ⎢ ⎥ ⎪ ⎪ ⎝ ⎠ ⎣ ⎦ ⎩ ⎩

∑ ∑ ∑

k k η , κ ϕ

( ) ( )

* *0 1

,

j j j n i j j i j ai i j i

L b t

γ

δ δϕ ϕ

=

⎫ ⎫ ⎛ ⎞ ∂ ⎪ ⎪ ⎜ ⎟ Δ − ⎬ ⎬ ⎜ ⎟ ∂ ⎪ ⎪ ⎝ ⎠ ⎭ ⎭

∑

k k η , κ

SLIDE 42

Stability and convergence of DA schemes

Let us consider an example:

1 , , ( , ) ( )

t t t

t D ϕ ϕ ϕ ϕ ϕ Δ

=

− = = ∈ x 2 , , ( , ) ( )

h h h t t t

t D ϕ ϕ ϕ ϕ ϕ Δ

=

− = = ∈ x

For one-step schemes

1 2 3 ( ) ; , ,...., ( )

j h j

S t j ϕ ϕ Δ Δ Δ Δ = = ( ) S

Transition operator (TO), its spectral form

( ) S tλ Δ

corresponds to TO of the model problem

4 , , ( )

t t

ψ λψ ψ ψ λ

=

+ = = ∈£

SLIDE 43

L-stability

Definition. The method is called L-stable,

if

1 S z for all z t c z и S λ ≤ = ∈ ≥ ∞ = £ ( ) , Re( ) ( ) Δ

Typical cases:

1. Implicit scheme

1

1 ( ) ( ) ( ) S z z S

−

= + ⇒ ∞ =

2. Explicit scheme

1 ( ) ( ) ( ) S z z S = − ⇒ ∞ = −∞

3.Weighted scheme

1 1 1 1 1 1

1 1 1 1 1 ( ) ( )( ( ) ) , ( ) ( ) S z z z S α α α α α

−

= − + − ≤ ≤ ⇒ ∞ = − −

4. Implicit two-step Euler scheme

( )

1 2

3 4 2 ( )

j j j j

t S ψ ψ ψ λψ Δ

− −

− + + = ⇒ ∞ =

( )

1 1 2 1 2

2 1 2 1

j j

z ψ ξ ξ ξ

−

= = ± + <

, ,

; ;

SLIDE 44

Scenario forecasts of climate- caused risks by inverse methods

SLIDE 45

Winter pattern of the global 500-hPa geopotential height (the 1st main factor) 1950-2005

5 7 7 9 9 9 1 1 1 1 1 1 11 11 1 1 13 1 3 13 1 3 13 13 13 13 13 1 3 13 13 13 15 1 5 15 1 5 15 15 15 1 7 1 7 17 19 21 21

Longitude y, deg 30 60 90 120 150 180 210 240 270 300 330 30 60 90 120 150 180 Level value: 1

0.85

3

0.71

5

0.56

7

0.42

9

0.28

11

0.13

13 0.01 15 0.16 17 0.30 19 0.45 21 0.59

January 15

SLIDE 46

Winter pattern of global circulation (the 1st main factor) 1950-2005

x y

100 200 300 50 100 150 January 15

SLIDE 47

1 3 3 3 5 5 5 5 7 7 7 7 7 7 7 7 7 7 9 9 9 9 9 9 9 9 9 9 9 9 1 1 1 1 11 1 1 11 11 11 1 1 11 11 11 11 11 11 13 13 1 3 13 13 15 1 7

Longitude y 30 60 90 120 150 180 210 240 270 300 330 30 60 90 120 150 180 Level value: 1

0.64

3

0.49

5

0.35

7

0.20

9

0.06

11 0.09 13 0.23 15 0.38 17 0.52 19 0.66 21 0.81

July 15

Summer pattern of the global 500-hPa geopotential height (the 1st main factor) 1950-2005

SLIDE 48

Summer pattern of global circulation (the 1st main factor) 1950-2005

x y

100 200 300 50 100 150 June 15 June 15

SLIDE 49

East Siberia Region

90-140 E, 45-65 N June 1950-2005

SLIDE 50

500-hPa

SLIDE 51

Monthly risk functions for Lake Baikal

July December

SLIDE 52

Optimal forecasting and design

Optimality is meant in the sense that estimations of the goal functionals do not depend on the variations :

of the sought functions in the phase spaces of the

dynamics of the physical system under study

of the solutions of corresponding adjoint

problems that generated by variational principles

of the uncertainty functions of different kinds

which explicitly included into the extended functionals

SLIDE 53

Advantage of the approach

Consistency of all technology elements
Optimality of numerical schemes based on

discrete-analytical approximations(without flux-correction procedures )

Cost-effectiveness of computational

technology

SLIDE 54

Conclusion

Variational principle is the universal tool for construction of numerical models, algorithms, and integrated modeling technology

SLIDE 55

Acknowledgements

The work is supported by

RFBR

Grant 11-01-00187

Presidium of the Russian Academy of Sciences,

Program 4

Department of Mathematical Science of RAS

Program 1.3

SLIDE 56

Thank you for your time!