Computations in Higher Types by J. Moldestad
By J. Moldestad
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Extra resources for Computations in Higher Types
One should then of course address the questions of “what are the natural deterministic classes which model real signals” and “what can one say about the sparsity of wavelet representations for these classes”. An interesting example is given by real images which are often modelized by the space BV of functions with bounded variation. This function space represents functions which have one order of smoothness Nonlinear Approximation 21 in L1 in the sense that their gradient is a ﬁnite measure. This includes in particular functions of the type χΩ for domains Ω with boundaries of ﬁnite length.
Verf¨ urth, R. (1994) A-posteriori error estimation and adaptive mesh reﬁnement techniques, Jour. Comp. Appl. Math. 50, 67–83. de Summary. These notes are to bring out some basic mechanisms governing wavelet methods for the numerical treatment of diﬀerential and integral equations. Some introductory examples illustrate the quasi–sparsity of wavelet representations of functions and operators. This leads us to identify the key features of general wavelet bases in the present context, namely locality, cancellation properties and norm equivalences.
2j − 1}. Figure 1 displays such approximations for several levels of resolution. Multiscale and Wavelet Methods for Operator Equations 0 1 0 1 0 33 1 Fig. 1. Piecewise constant approximations to f If a chosen level of resolution turns out to be insuﬃcient, a naive approach would be to recompute for a smaller mesh size. More cleverly one can avoid wasting prior eﬀorts by monitoring the diﬀerence between successive levels of resolution as indicated by Figure 2 a = + b = Feinstruktur Mittelung + Detail Fig.