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A gradient-projective basis of compactly supported wavelets in dimension n > 1

100%

Battle G.

Open Mathematics

2013

tom 11

nr 7

1317-1333

A given set W = {W X } of n-variable class C 1 functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm $\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W x } of compactly supported class C 2−ɛ functions on ℝn such that [...] where X has the structure $\chi = (\tilde \chi ,\nu )$ , ν ∈ ℤ. W is a wavelet set in the sense that the functions indexed by $\tilde \chi $ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde \chi $ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy - just anti-differentiate the Haar functions.)

Approximate multiplication in adaptive wavelet methods

80%

Černá D., Finěk V.

Open Mathematics

2013

tom 11

nr 5

972-983

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2-problem, finding the convergent iteration process for the l 2-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.

Ograniczanie wyników

2 Open Mathematics

1 Battle G.

1 Finěk V.

1 Černá D.

2 2013

Wyniki wyszukiwania

A gradient-projective basis of compactly supported wavelets in dimension n > 1

Approximate multiplication in adaptive wavelet methods