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Next we will need to discuss how to reduce the infinite systems to computable finite versions. 1 Finite Systems on Uniform Grids Let us first consider finite-dimensional trial spaces with respect to uniform discretizations. , IIJ,H := {λ ∈ IIH : |λ| ≤ J} ⊂ IIH satisfying NJ,H := #IIJ,H < ∞. 3 with respect to this truncated index set which corresponds to deleting all rows and columns that refer to indices λ such that |λ| > J, and correspondingly for functions. There is by construction also a coarsest level of resolution denoted by j0 .

Numerical Example For illustration of the choice of different norms for the Dirichlet boundary control problem, consider the following example taken from [P]. 2 while a prescribed state y∗ ≡ 1 on the observation boundary Γy opposite the control boundary is to be achieved. The right hand side is chosen as constant f ≡ 1, and ω = 1. 35) contains a term y − y∗ 2H s (Γy ) for s = 0, 1/10, 2/10, 3/10, 4/10, 5/10, 7/10, 9/10 from bottom to top. We see that as the smoothness index s for the observation increases, the state moves towards the target state at the observation boundary.

42) results in the reconstruction identity Φj+1 = GTj Φj Ψj = GTj,0 Φj + GTj,1 Ψj . 36) so that SJ = S(ΦJ ) can be written in terms of the functions from the coarsest space supplied with the complement functions from all intermediate levels, J−1 S(ΦJ ) = S(Φj0 ) ⊕ S(Ψj ). 46) with respect to the multiscale or wavelet basis J−1 ΨJ := Φj0 ∪ J−1 Ψj =: j=j0 Ψj . 47) j=j0 −1 Often the single–scale representation of a function may be easier to compute and evaluate while the multiscale representation allows one to separate features of the underlying function characterized by different length scales.

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A Handbook for Designing Mathematics Laboratory in Schools by Hukum Singh

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