By James H. Bramble, Albert Cohen, Wolfgang Dahmen, Claudio G Canuto

This quantity goals to disseminate a few new principles that experience emerged within the previous couple of years within the box of numerical simulation, all bearing the typical denominator of the

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Each index λ ∈ J 40 Wolfgang Dahmen encodes diﬀerent types of information, namely the scale j = j(λ) = |λ|, the spatial location k = k(λ) and the type e = e(λ) of the wavelet. g. for tensor product constructions one has 2d − 1 diﬀerent types of wavelets associated with each spatial index k. For d = 2 one has, for instance, ψλ (x, y) = 2j ψ 1,0 (2j (x, y) − (k, l)) = 2j/2 ψ(2j x − k)2j/2 φ(2j y − l). We will explain later what exactly qualiﬁes Ψ as a wavelet basis in our context. 2 Notational Conventions As before it will be convenient to view a collection Ψ as an (inﬁnite) vector (with respect to some ﬁxed but unspeciﬁed order of the indices in J ).

Although in practice one would not apply an iterative scheme for the solution of the particular system (9), it serves well to explain what will be relevant for more realistic multidimensional problems. The performance of an iterative scheme for a symmetric positive system is known to depend on the condition number of that system which in this case is the quotient of the maximal and minimal eigenvalue. 3), it should suﬃce for the moment to note that the condition numbers grow like h−2 (here 22J for h = 2−J ) for a given mesh size h, which indeed adversely aﬀects the performance of the iteration.

Since the support of low level wavelets is comparable to the domain, a suﬃciently accurate quadrature would be quite expensive. A remedy is oﬀered by the following strategy which can be used when the wavelet coeﬃcients of interest have at most some highest level J say. The accurate computation of the scaling function coeﬃcients cJ,k := f, φJ,k , k = 0, . . , 2J −1, is much less expensive, due to their uniformly small support. The transformation from the array cJ into the array of wavelet coeﬃcients Multiscale and Wavelet Methods for Operator Equations 35 dJ = (c0 , d0 , .