Tree Algorithms in Wavelet Approximations by Helmholtz Potential Operators

Freeden, Willi; Mayer, Carsten; Schreiner, Michael

doi:10.1081/nfa-120026374

Cited by 20 publications

(13 citation statements)

References 10 publications

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“…[74]). Diesbezüglich wird in [30] eine Multiskalenmethode vorgeschlagen und in [45] im Rahmen der Seismik als Postprocessing auf Basis von Helmholtz-Wavelets vorgestellt. Im Folgenden werden diese Betrachtungen genauer vorgestellt.…”

Section: Seismischeunclassified

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Mathematische Methoden in der Geothermie

et al. 2011

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Section: Seismischeunclassified

“…In dieser Arbeit werden -basierend auf den Betrachtungen in [3,28,30,33,35,37,45,46,59,68,69] -zuerst kurz der geothermische Hintergrund und insbesondere die relevanten Reservoirtypen dargestellt. Im Anschluss werden für diese Reservoirtypen die den einzelnen Säulen zugrundeliegenden Differential-und Integralgleichungen beschrieben.…”

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Mathematische Methoden in der Geothermie

et al. 2011

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“…For the case the case of the Laplace equation this approach presented above for the construction of scaling functions and wavelets has been discussed in [1] and for the Helmholtz equation is has been studied in [2].…”

Section: Construction Of Waveletsmentioning

confidence: 99%

Wavelets on Regular Surfaces Generated by Stokes Potentials

Mayer

2006

Proc Appl Math and Mech

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By means of the limit and jump relations of potential theory with respect to the Stokes equations the framework of a tensorial wavelet approach on a regular (Lyapunov-) surface is established. The setup of a multiresolution analysis is defined by interpreting the kernel functions of the limit and jump integral operators as scaling functions on the regular surfaces. The distance of the parallel surface to the surface under consideration thereby represents the scale level in the scaling function. Tensorial scaling functions and wavelets show space localizing properties. Thus, they can be used to represent vector fields locally on a regular surface. Furthermore, these functions can be used as ansatz functions for the discretization of Fredholm integral equations of the second kind which result from the Stokes boundary-value problems with respect to a regular surface. By this, scaling functions and wavelets enable us to give a multiscale representation of the solution of the Stokes problem. This representation will be demonstrated in a concrete example.

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“…Our constructions [18], [19], [8], [9], [12], [13] are intrinsically based on the specific properties concerning the geometry of the sphere and the theory of "spherical polynomials", i.e. in the jargon of mathematical (geo-)physics spherical harmonics.…”

Section: Introductionmentioning

confidence: 99%

Multiresolution Analysis by Spherical Up Functions

Freeden

Schreiner

2005

Constr Approx

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A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of "isotropic finite elements". The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square-integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere. AMS-Classification: 41A30, 41A58, 42C15, 42C40, 44A35, 45E10, 65T60.

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Tree Algorithms in Wavelet Approximations by Helmholtz Potential Operators

Cited by 20 publications

References 10 publications

Mathematische Methoden in der Geothermie

Mathematische Methoden in der Geothermie

Wavelets on Regular Surfaces Generated by Stokes Potentials

Multiresolution Analysis by Spherical Up Functions

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