Tamtam Proceedings - lamsin

Reduction methods and uncertainty propagation 311POD methods is based on a particular choice of the orthonormal basis Ψ such that c(t j )can be optimally approached by successive truncated sums. Ψ is then such that for all1 ≤ k ≤ d, (Ψ 1 . . . Ψ k ) minimizes the function:J k (Φ 1 , . . . , Φ k ) =N∑ ∑i=k‖c(t j ) − 〈Φ i , c(t j )〉 Φ i ‖ 2 (2)j=1Ψ can be computed from the correlation matrix of the snapshots ([11]). The choice ofa truncated basis of dimension p (the number of degrees of freedom we keep) is relatedto the decreasing rate of the eigenvalues of the correlation matrix. We now search for areduced solution z(t) = ∑ pi=1 z i(t) Ψ i , z i ∈ R such that for 1 ≤ i ≤ p:〈〉dz ip∑dt = f( z i (t) Ψ i ), Ψ i , z i (0) = 〈c(0), Ψ i 〉 (3)i=1Application and result: We have applied this technique for the reduction of the chemicalmechanism CBM IV ([12]) with 68 reactions and 31 species. The time period forcomputing the POD basis (t N −t 1 ) is 4 hours. In figure 1, we compare the results obtainedwith the reduced model and an exact solution. We can see that the reduced model is validduring more than 500 minutes (to be compared with 240 minutes). The reduced modelis then valid locally. Notice that p = 3 degrees of freedom describe the 31-dimensionalsystem.i=1O3 ConcentrationNO Concentration30040POD (Snap=4h)POLAIRPOD(Snap=4h)POLAIR30Concentration (mol/cm3)200100Concentration (mol/cm3)201000 500 1000 1500Time(min)00 500 1000 1500Time(min)Figure 1. Profiles of O 3 and NO: exact (line) versus POD (- - -) models.3. HDMR techniqueWe now consider one integration of the chemical model (1) from time t s to timet s+1 as a model with n input variables x = (x 1 , x 2 ..., x n ) ∈ Ω and an output c =TAMTAM –Tunis– 2005