Multiscale problems and methods in numerical simulations by James H. Bramble, Albert Cohen, Wolfgang Dahmen, Claudio G

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 , .

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