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Essay On Multiscale Basis

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2.2 Modeling the solution using multiscale basis
Selecting the dominant scale corresponding to the small eigen values gives rise to a fixed basis sets and using fixed basis to solve the weak form produces the fixed solution un,fixed(x,t) = ﰄ n n,ωj n H n,ωj i,j βi,j φi (x,t), where βi,j’s are defined in each computational time interval and φi (x,t) are fixed basis functions. Fixed solution at n + 1 th time point is computed by solving equation
(3) by setting un as the fixed solution at n th time point and writing un+1 in the space of HH
5
1 0.9
0.8ωE K 0.7
0.6
0.5
0.4
0.3
KK 1ω2 i 0.2 KK
0.1
0
0 0.2 0.4 0.6 0.8 1
Figure 3: Illustration of fine grid, coarse grid, coarse neighborhood and oversampled domain. fixed basis at n + 1 th time
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The true solution is assumed to be normal around the fixed solution with small variance. Finally, this structure enables us to compute the posterior or conditional distribution of the basis selection probability and conditional solution of the system given the observation and the pde model.
Residual and selection probability on the subregion and basis
From equation (3), the residual is defined as
Rn(un+1,un+1 ,un
) =

Ω fn+1v − ﰆ
ﰆ un+1+un+1−un
+ fixed fixed v v + fixed fixed
Ω n+1
∆t
+ ufixed) · ∇v.
(4)
+ κ∇(u+ n+1
Ω
6
For any fixed basis φn,ωj ’s this equation is zero as the fixed solution is constructed by setting the k n,ωj equation zero for each fixed basis. Using φk,+ ’s ∀k, j in the residual function one can compute the residual for additional basis and writing down the residual as a long vector over subregions and basis the following quantity is defined. Let αωk = ∥Rnωk ∥/∥Rn∥, where Rn is the global residual vector and Rnωk is the local residual vector in ωk (as mentioned earlier) and L1 norm is used .
Let Nω be the average number of subregions where additional basis will added. Furthermore, αωj αﰇ j = ﰄj αωj Nω, (5) ω With probability proportional to αﰇ j ∧ 1, the region ωj , is selected and Jj = 1 if the region is ω selected and zero otherwise. Given subregion j is selected the k th extra basis is
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