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

A new variational technique for direct and inverse problems of 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 operator

Model of transport and transformation of pollutants ∂ π c ϕ ≡ + π − µ + ϕ − − = i ( u ) ( ) (x, ) L div c grad c S f t r 0 ∂ i c i i i i t i ϕ = = - state function { c i i 1 , m } , f - source term i ( ϕ - operator of transformation, S ) i r = ≤ ≤ ∈ . D t { 0 t t ; x D } Boundary and initial conditions ( ) ϕ = ∈Ω x R q , ( , ) t i t i ϕ = ϕ 0 ( ,0) x ( ) x

Splitting stage: transformation ∂ ϕ r r + ϕ − ϕ = i L ( ) P ( ) Q ∂ i i i t r r r ˆ ϕ = ϕ ≤ ≤ ϕ ∈ j ( ) t , t t t , Q D ( ) + j j j 1 t { } r ϕ = ϕ = ≥ i i , 1, , n n 1 = + τ ≤ τ ≤ ∆ t t , 0 t j r L ϕ ( ) loss operators i r P ϕ ( ) production operators i Q sources and sinks of pollutants i r ∈ x D The space co-ordinates are parametrically taken t

Properties of operators and functions r r r ϕ ≥ ϕ ≥ ϕ ≥ 0, L ( ) 0, P ( ) 0, i i r r r r r ϕ = ϕ ϕ α ϕ = ϕ α ϕ ≥ % % L ( ) L ( ) , ( ) L ( ), ( ) 0 i i i i i i Monotony is qualifying standard: Q ≥ 0 the state functions have to be positive if . i

S-stage Runge-Kutta method s ∑ + ϕ = ϕ + ∆ = + ∆ j 1 j 0 j j t b k , T t c t , j j j = j 1 s ∑ Ψ = ϕ + ∆ = Ψ % j j 0 0 j j t a k , k F ( , , , Y x T ) jq q j = q 1 a , b c , parameters jq j j

S-stage Rosenbrock methods s ∑ + ϕ = ϕ + ∆ = + ∆ j 1 j 0 j j t b k , T t c t , j j j = 1 j − j 1 ∑ Ψ = ϕ + ∆ j j 0 t a k , jq q = q 1   1 − ϕ = Ψ + % j j 0 j j   E J t ( , , , ) Y x k F ( , , , Y x T ) ∆ γ j t   jj − ∂ j 1 c ∑ + ∆ γ ϕ jq % 0 j x k t F ( , , , ), Y T ∆ ∂ q j t t = 1 q γ γ − b a , , c , c , , parameters j jq jq j jj j % Jacobian matrix of derivative functions of F J

Variational form of the model ∂ ϕ r + α ϕ ϕ = ϕ ( ) F t ( , , Q ) ∂ t ∆ ∂ ϕ t   r ∫ + α ϕ ϕ − ϕ τ = *   ( ) F d ∂   t 0 ∆     ∂ ϕ t * ( ) ∆ r t ∫ − + α ϕ ϕ ϕ − ϕ τ + ϕϕ * * *     ( ) F d ∂   t   0 0

Local adjoint problem ϕ τ is chosen from the conditions: * ( ) The function ∂ ϕ * r + − + α ϕ ϕ = ϕ ≡ ϕ = * * * j 1 ( ) 0, 1 ∂ =∆ t t t + e α − ∆ − τ ϕ τ = ϕ * * j 1 ( t ) ( ) ≤ τ ≤ ∆ 0 t

Balance relations ∆ t ( ) t ∫ + ϕ ϕ τ τ + ϕϕ = j 1 * * F t ( , , Q ) ( ) d 0 t j 0 ∆ t + = + ∫ ϕ ϕ ϕ ϕ ϕ τ τ j 1 j * j * F t ( , , ) ( ) Q d 0 τ ∫ − ατ − α τ τ − ′ ′ ′ ϕ τ = ϕ + τ ϕ τ ≤ τ ≤ ∆ j ( ) ( ) e e F ( , , Q d ) , 0 t 0

The second order monotonic scheme − ∆ α − ∆ / 2 t ( ) 1 e t + − ∆ α − ∆ α ϕ = ϕ + + j 1 j t t / 2 e f e f α ∆ 1 2 t / 2 2 + = ϕ = ϕ j j j 1 % ( , , ), ( , ) f F t Q f F t + 1 j 2 j 1 − ∆ α − t / 2 1 e + − ∆ α ϕ = ϕ + ∆ 1 j j t % e f t α ∆ 1 t / 2 α ≥ 0 Монотонный характер схемы при непосредственно следует из свойств неотрицательности всех сомножителей слагаемых . α ≡ ∆ 2 0 o ( t ) При получается аналог формулы Рунге - Кутта .

The forth order monotonic scheme ∆ ( ) t ( ) + − ∆ α − ∆ α − ∆ α ϕ = ϕ + + + + j 1 j t t t / 2 e f e 2 e f f f 1 2 3 4 6 ∆   t − ∆ α = ϕ η = ϕ + j j t / 2   f F t ( , ), f e 1 j 1 1   2 ∆ ∆ t t − ∆ α = + η η = ϕ + j t / 2 f F t ( , ), e f 2 j 1 2 2 2 2 ∆ t − ∆ α − ∆ α = + η η = ϕ + ∆ j t t / 2 f F t ( , ), e tf e 3 j 2 3 3 2 = + ∆ η f F t ( t , ) 4 j 3

Variational form of transformation model for multi-stage algorithms ( the 2nd order) − { r n J 1 r r ∑∑ ( )  +  + ϕ ϕ ≡ ϕ − ϕ − + ϕ * j 1 j * j 1 I ( , , ) Y H U f W f   tr i i i i 1 i i 2 i i = = i 1 j 1 ( r ) ( ) r + + − ϕ + − ϕ j * j 1 * % f F t ( , ) f f F t ( , ) f + 1 i i j 1 i 2 i i j 1 2 i } ( r ) r ( ) r + + α − ϕ κ α + ϕ − ϕ − ϕ ∆ = % * j 1 j * % % L ( , ) H V f t 0, i i p i i i i i 1 i i − α ∆ − ∆ / 2 t 1 e t i − α ∆ = = t / 2 W , U W e , i α ∆ i i i t / 2 2 i − α ∆ − t 1 e i − α ∆ = ∆ = ∆ i t V t , H e t α ∆ i i t i

Adjoint transformation problem + = ϕ * * j 1 f W , 2 i i i r + ∂ ϕ j 1 n % F ( ) ∑ + ϕ = * j 1 * k % f + ∂ ϕ i 2 k j 1 % = k 1 i + + = ϕ + ϕ * * j 1 * j 1 % f U V 1 i i i i i       ∂ ∂ ∂ ∂ ∂ H U W H V + + α = ϕ + + ϕ + ϕ + ϕ * j * j 1 j * j 1   i i i i i     % f f ∂ α ∂ α ∂ α ∂ α ∂ α i i 1 i i i 1 i i       i i i i i r r   ∂ ϕ ∂ ϕ % ( ) j j n ∑ F ( ) L ( ) + + ϕ = ϕ + ϕ + + α * j * j 1 * j 1 * * k k %   H f ∂ ϕ ∂ ϕ i i i i 1 k k j j   = k 1 i i = i 1, n

Sensitivity relations for transformation problem r r r ∂ ϕ ϕ * ( r ) N I ( , , ) Y ∑ δ Φ ϕ = δ = Φ ϕ δ tr % ( ) Y grad ( ), Y r ∂ k Y Y = k 1 k { } { } ( ) r = = = ϕ κ = = = 0 j Y Y , k 1, N , Q , , q 1, n , i 1, , n j 1, J k i i q q Feedback equation dY r = Φ ϕ = k grad ( ), k 1, N Y dt k

Sensitivity relations   − ∂   % q n J 1 r L ∑ ∑ ∑ k δ Φ ϕ = − δκ α + * j   i ( ) ∂ κ q i     = = = i 1 j 1 q 1 q r r   ∂ ϕ ∂ ϕ q j j % F ( ) F ( ) ∑ k + δκ + δκ +  * *  i i f f   ∂ κ ∂ κ q 1 i q 2 i   = 1 q q q } } δ + δ ∆ + δϕ δϕ * * 0 *0 Q f Q f t i 1 i i 2 i i i

Methane transformation in the atmosphere C H OOH CH ONO 3 3 2 ↑ ↑ , ν OH O HO NO NO O OH h O OH → → 2 2 → 2 → 2 → 2 → → CH CH CH O CH 0 H CO HCO CO CO 4 3 3 2 3 2 2 ↓ ↓ ↓ , NO HO CH O HO NO O → → 2 2 2 2 2 CH O NO CH OH HOCH O HOCH O HCOOH 3 2 2 3 2 2 2

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.