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Introduction: The Problem to be Solved Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

11. Wavelets for Inverse Problems on the 3D Ball

verfasst von : Volker Michel

Erschienen in: Lectures on Constructive Approximation

Verlag: Birkhäuser Boston

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Abstract

We present here one particular wavelet method, which was developed by the author. This is certainly not the only wavelet method for tomographic problems on the 3D ball. There exist alternatives, where at least [60, 175, 176] should be mentioned here.

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Vorheriges Kapitel Splines
Nächstes Kapitel The Regularized Functional Matching Pursuit
Fußnoten
1
For this example, we assume that \({\tau }_{m,n}\neq 0\) for all \(m,n \in {\mathbb{N}}_{0}\).
 
2
In the case of tensorial functions, that is, \({b}_{m,n,j} : D \rightarrow {\mathbb{R}}^{3\times 3}\), we can use the isomorphism of \({\mathbb{R}}^{3\times 3}\) and \({\mathbb{R}}^{9}\) and interpret this case as q = 9.
 
3
“SB” refers here to “scaling function” and “ball.”
 
4
This is, for example, caused by the following fact: Typically, g is noisy, where the relative contribution of the noise to the Fourier coefficients \(\langle g,\,{b{}_{m,n,j}\rangle }_{\mathrm{{L}}^{2}(D,\,{\mathbb{R}}^{q})}\) increases with increasing degree m or n. On the one hand, we get that \({\Phi }_{J}^{\wedge }(m,n)\) tends to zero as \(m \rightarrow \infty \) or \(n \rightarrow \infty \) from (SB4). On the other hand, (SB2) and (SB3) require that \(\vert {\Phi }_{J}^{\wedge }(m,n)\vert \) increases with increasing scale J and approaches the (absolute value of the) reciprocal of the singular value. The former causes that Fourier coefficients corresponding to high degrees are equipped with small factors due to the convolution \({\Phi }_{J} {_\ast} g\)—which is good due to the noisy character of these coefficients. The latter, however, causes that this smoothing effect gets weaker and weaker with increasing scale J such that finally the noise is not sufficiently attenuated any more. An optimal scale \({J}^{{_\ast}}\) corresponds to a trade-off between a notable attenuation and a low approximation error.
 
Literatur
1.
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1972)
2.
Akhtar, N.: A multiscale harmonic spline interpolation method for the inverse spheroidal gravimetric problem. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)
3.
Akhtar, N., Michel, V.: Reproducing Kernel based splines for the regularization of the inverse ellipsoidal gravimetric problem. Appl. Anal. (2011). Accepted for publication, pre-published online via doi:10.1080/00036811.2011.590479
4.
Akram, M.: Constructive approximation on the 3-dimensional ball with focus on locally supported kernels and the Helmholtz decomposition. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)
5.
Akram, M., Amina, I., Michel, V.: A study of differential operators for particular complete orthonormal systems on a 3D ball. Int. J. Pure Appl. Math. 73, 489–506 (2011)
6.
Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 5–43 (1996)
7.
Amann, H., Escher, J.: Analysis III, 2nd edn. Birkhäuser, Basel (2008)
8.
9.
Amirbekyan, A., Michel, V.: Splines on the three-dimensional ball and their application to seismic body wave tomography. Inverse Probl. 24, 1–25 (2008)
10.
Antoine, J.P., Demanet, L., Jacques, L., Vandergheynst, P.: Wavelets on the sphere: implementations and approximations. Appl. Comput. Harm. Anal. 13, 177–200 (2002)
11.
Antoine, J.P., Vandergheynst, P.: Wavelets on the 2-sphere: A group-theoretic approach. Appl. Comput. Harm. Anal. 7, 1–30 (1999)
12.
Ballani, L., Engels, J., Grafarend, E.W.: Global base functions for the mass density in the interior of a massive body (Earth). Manuscr. Geodaet. 18, 99–114 (1993)
13.
Bäni, W.: Wavelets: Eine Einführung für Ingenieure. Oldenburg, München (2002)
14.
Barron, A.R., Cohen, A., Dahmen, W., DeVore, R.A.: Approximation and learning by greedy algorithms. Ann. Stat. 36, 64–94 (2008)
15.
Beckmann, J., Mhaskar, H.N., Prestin, J.: Quadrature formulas for integration of multivariate trigonometric polynomials on spherical triangles. Int. J. Geomath. 3, 119–138 (2012)
16.
Berg, A.P., Mikhael, W.B.: A survey of mixed transform techniques for speech and image coding. In: Proceedings of the 1999 IEEE International Symposium on Circuits and Systems, vol. 4, pp. 106–109 (1999)
17.
Berkel, P.: Multiscale methods for the combined inversion of normal mode and gravity variations. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2009)
18.
Berkel, P., Fischer, D., Michel, V.: Spline multiresolution and numerical results for joint gravitation and normal mode inversion with an outlook on sparse regularisation. Int. J. Geomath. 1, 167–204 (2011)
19.
Berkel, P., Michel, V.: On mathematical aspects of a combined inversion of gravity and normal mode variations by a spline method. Math. Geosci. 42, 795–816 (2010)
20.
Blatter, C.: Wavelets: Eine Einführung. Vieweg, Braunschweig (1998)
21.
Blick, C., Freeden, W.: Spherical spline application to radio occultation data. J. Geodetic Sci 1, 379–395 (2011)
22.
Bogdanova, I., Vandergheynst, P., Antoine, J.P., Jacques, L., Morvidone, M.: Stereographic wavelet frames on the sphere. Appl. Comput. Harm. Anal. 19, 223–252 (2005)
23.
Böhme, M., Potts, D.: A fast algorithm for filtering and wavelet decomposition on the sphere. Electron. Trans. Numer. Anal. 16, 70–93 (2003)
24.
Chambodut, A., Panet, I., Mandea, M., Diament, M., Holschneider, M., Jamet, O.: Wavelet frames: An alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163, 875–899 (2005)
25.
Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Rev. 43, 129–159 (2001)
26.
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)
27.
Chui, C.K.: An Introduction to Wavelets. Academic, San Diego (1992)
28.
Cohen, A.: Numerical Analysis of Wavelet Methods. Elsevier, Amsterdam (2003)
29.
Coifman, R., Meyer, Y., Wickerhauser, V.: Adapted wave form analysis; waveletpackets and applications. In: ICIAM 91, Proceedings of the Second International Conference on Industrial and Applied Mathematics, pp. 41–50 (1992)
30.
Coifman, R., Wickerhauser, V.: Entropy-based algorithms for best basis selection. IEEE Trans. Inform. Theory 38, 713–718 (1992)
31.
Conrad, M., Prestin, J.: Multiresolution on the sphere. In: Iske, A., Quak, E., Floater, M.S. (eds.) Summer School Lecture Notes on Principles of Multiresolution in Geometric Modelling, pp. 165–202, Munich (2001)
32.
Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19, 297–301 (1965)
33.
Cui, J., Freeden, W.: Equidistribution on the sphere. SIAM J. Sci. Comput. 18, 595–609 (1997)
34.
Dahlen, F.A., Simons, F.J.: Spectral estimation on a sphere in geophysics and cosmology. Geophys. J. Int. 174, 774–807 (2008)
35.
Dahlen, F.A., Tromp, J.: Theoretical Global Seismology. Princeton University Press, Princeton (1998)
36.
Dahlke, S., Dahmen, W., Schmitt, E., Weinreich, I.: Multiresolution analysis and wavelets on S 2 and S 3. Numer. Func. Anal. Opt. 16, 19–41 (1995)
37.
Dahlke, S., Fornasier, M., Raasch, T.: Multilevel preconditioning and adaptive sparse solution of inverse problems. Math. Comput. 81, 419–446 (2009)
38.
Dahlke, S., Steidl, G., Teschke, G.: Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere. Adv. Comput. Math. 21, 147–180 (2004)
39.
Dahlke, S., Steidl, G., Teschke, G.: Frames and coorbit theory on homogeneous spaces with a special guidance on the sphere. J. Fourier Anal. Appl. 13, 387–403 (2007)
40.
Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
41.
Daubechies, I., Defrise, M., DeMol, C.: An iterative thresholding algorithm for linear inverse problems with sparsity constraint. Commun. Pur. Appl. Math. 57, 1413–1457 (2004)
42.
Daubechies, I., Fornasier, M., Loris, I.: Accelerated projected gradient method for linear inverse problems with sparsity constraints. J. Fourier Anal. Appl. 14, 764–792 (2008)
43.
Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)
44.
Deuflhard, P.: On algorithms for the summation of certain special functions. Computing 17, 37–48 (1975)
45.
DeVore, R.A.: Nonlinear approximation. Acta Numerica 7, 51–150 (1998)
46.
Driscoll, J.R., Healy, R.M.: Computing Fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math. 15, 202–250 (1994)
47.
Dufour, H.M.: Fonctions orthogonales dans la sphère. résolution théorique du problème du potentiel terrestre. B. Geod. 51, 227–237 (1977)
48.
Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. In: Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)
49.
Engl, H.W., Grever, W.: Using the L-curve for determining optimal regularization parameters. Numer. Math. 69, 25–31 (1994)
50.
Fasshauer, G.E., Schumaker, L.L.: Scattered data fitting on the sphere. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 117–166. Vanderbilt University Press, Nashville, TN (1998)
51.
Feinerman, R.P., Newman, D.J.: Polynomial Approximation. The Williams and Wilkins Company, Baltimore (1974)
52.
Fengler, M.J., Freeden, W., Kohlhaas, A., Michel, V., Peters, T.: Wavelet modelling of regional and temporal variations of the Earth’s gravitational potential observed by GRACE. J. Geodesy 81, 5–15 (2007)
53.
Fengler, M.J., Michel, D., Michel, V.: Harmonic spline-wavelets on the 3-dimensional ball and their application to the reconstruction of the Earth’s density distribution from gravitational data at arbitrarily shaped satellite orbits. Z. Angew. Math. Mech. 86, 856–873 (2006)
54.
Fischer, D.: Sparse regularization of a joint inversion of gravitational data and normal mode anomalies. Ph.D. thesis, University of Siegen, Department of Mathematics, Geomathematics Group (2011). Verlag Dr. Hut, München
55.
Fischer, D., Michel, V.: How to combine spherical harmonics and localized bases for regional gravity modelling and inversion. In: Siegen Preprints on Geomathematics, vol. 8. University of Siegen, Germany (2012, Preprint)
56.
Fischer, D., Michel, V.: Inverting GRACE gravity data for local climate effects. In: Siegen Preprints on Geomathematics, vol. 9. University of Siegen, Germany (2012, Preprint)
57.
Fischer, D., Michel, V.: Sparse regularization of inverse gravimetry — case study: spatial and temporal mass variations in South America. Inverse Probl. 28 (2012). 065012
58.
Fletcher, N.H., Rossing, T.D.: The Physics of Musical Instruments, 2nd edn. Springer, New York (1998)
59.
Fokas, A.S., Hauk, O., Michel, V.: Electro-magneto-encephalography for the three-shell model: numerical implementation via splines for distributed current in spherical geometry. Inverse Probl. 28 (2012). 035009 (28 pp.)
60.
Fornasier, M., Pitolli, F.: Adaptive iterative thresholding algorithms for magnetoencephalography (MEG). J. Comput. Appl. Math. 221, 386–395 (2008)
61.
Freeden, W.: On approximation by harmonic splines. Manuscr. Geodaet. 6, 193–244 (1981)
62.
Freeden, W.: On spherical spline interpolation and approximation. Math. Methods Appl. Sci. 3, 551–575 (1981)
63.
Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B G Teubner. Stuttgart, Leipzig (1999)
64.
Freeden, W., Gerhards, C.: Poloidal and toroidal field modeling in terms of locally supported vector wavelets. Math. Geosci. 42, 817–838 (2010)
65.
Freeden, W., Gervens, T., Schreiner, M.: Tensor spherical harmonics and tensor spherical splines. Manuscr. Geodaet. 19, 70–100 (1994)
66.
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)
67.
Freeden, W., Mayer, C.: Wavelets generated by layer potentials. Appl. Comput. Harm. Anal. 14, 195–237 (2003)
68.
Freeden, W., Michel, V.: Constructive approximation and numerical methods in geodetic research today—an attempt at a categorization based on an uncertainty principle. J. Geodesy 73, 452–465 (1999)
69.
Freeden, W., Michel, V.: Multiscale Potential Theory (with Applications to Geoscience). Birkhäuser, Boston (2004)
70.
Freeden, W., Michel, V.: Orthogonal zonal, tesseral and sectorial wavelets on the sphere for the analysis of satellite data. Adv. Comput. Math. 21, 181–217 (2004)
71.
Freeden, W., Michel, V., Nutz, H.: Satellite-to-satellite tracking and satellite gravity gradiometry (advanced techniques for high-resolution geopotential field determination). J. Eng. Math. 43, 19–56 (2002)
72.
Freeden, W., Nutz, H.: Satellite gravity gradiometry as tensorial inverse problem. Int. J. Geomath. 2, 177–218 (2011)
73.
Freeden, W., Schneider, F.: Regularization wavelets and multiresolution. Inverse Probl. 14, 225–243 (1998)
74.
Freeden, W., Schreiner, M.: Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constr. Appr. 14, 493–515 (1998)
75.
Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences, a Scalar, Vectorial, and Tensorial Setup. Springer, Berlin (2009)
76.
Freeden, W., Windheuser, U.: Earth’s gravitational potential and its MRA approximation by harmonic singular integrals. Z. Angew. Math. Mech. 75, 633–634 (1995)
77.
Freeden, W., Windheuser, U.: Spherical wavelet transform and its discretization. Adv. Comput. Math. 5, 51–94 (1996)
78.
Freeden, W., Windheuser, U.: Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl. Comput. Harm. Anal. 4, 1–37 (1997)
79.
Gerhards, C.: Spherical decompositions in a global and local framework: theory and application to geomagnetic modeling. Int. J. Geomath. 1, 205–256 (2011)
80.
Gerhards, C.: Spherical multiscale methods in terms of locally supported wavelets: theory and application to geomagnetic modeling. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group (2011). Verlag Dr. Hut, München
81.
Gledhill, J.A.: Aeronomic effects of the South Atlantic anomaly. Rev. Geophys. 14, 173–187 (1976)
82.
Göttelmann, J.: Locally supported wavelets on the sphere. Z. Angew. Math. Mech. 78, 919–920 (1998)
83.
Goupillaud, P., Grossmann, A., Morlet, J.: Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23, 85–102 (1984/85)
84.
Greville, T.N.E.: Introduction to spline functions. In: Greville, T.N.E. (ed.) Theory and Applications of Spline Functions, pp. 1–35. Academic, New York (1969)
85.
Gronwall, T.: On the degree of convergence of Laplace series. Trans. Am. Math. Soc. 15, 1–30 (1914)
86.
Haar, A.: Zur Theorie der orthogonalen Funktionen-Systeme. Math. Ann. 69, 331–371 (1910)
87.
Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34, 561–580 (1992)
88.
Hansen, P.C.: The L-curve and its use in the numerical treatment of inverse problems. In: Johnston, P. (ed.) Computational Inverse Problems in Electrocardiology, pp. 119–142. WIT Press, Southampton (2000)
89.
Hebinger, G., Michel, V., Richter, M., Simon, A.: Speech Recognition Support of Assisted Living. Schriften zur Funktionalanalysis und Geomathematik 40 (2008)
90.
Heirtzler, J.R.: The future of the South Atlantic anomaly and implications for radiation damage in space. J. Atmos. Sol.-Terr. Phy. 64, 1701–1708 (2002)
91.
Heiskanen, W.A., Moritz, H.: Physical Geodesy, Reprint. Institute of Physical Geodesy, Technical University Graz/Austria (1981)
92.
Hesse, K., Sloan, I.H., Womersly, R.S.: Numerical integration on the sphere. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 1187–1219. Springer, Heidelberg (2010)
93.
Heuser, H.: Funktionalanalysis, 3rd edn. B G Teubner, Stuttgart (1992)
94.
Hobson, E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Chelsea, New York (1965)
95.
Holschneider, M.: Continuous wavelet transforms on the sphere. J. Math. Phys. 37, 4156–4165 (1996)
96.
Holschneider, M., Chambodut, A., Mandea, M.: From global to regional analysis of the magnetic field on the sphere using wavelet frames, Phys. Earth Planet. In. 135, 107–124 (2003)
97.
Holschneider, M., Iglewska-Nowak, I.: Poisson wavelets on the sphere. J. Fourier Anal. Appl. 13, 405–419 (2007)
98.
Johnston, I.: Measured Tones. The Interplay of Physics and Music. Institute of Physics Publishing, Bristol (1989)
99.
Jones, F.: Lebesgue Integration on Euclidean Spaces. Jones and Bartlett Publishers, Boston (1993)
100.
Keiner, J., Prestin, J.: A Fast Algorithm for Spherical Basis Approximation. In: Govil, N.K., Mhaskar, H.N., Mohapatra, R.N., Nashed, Z., Szabados, J. (eds.) Frontiers in Interpolation and Approximation, pp. 259–286. Chapman & Hall/CRC, Boca Raton (2006)
101.
Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin (1967)
102.
Kress, R.: Numerical Analysis. Springer, New York (1998)
103.
Kufner, A., John, O., Fučík, S.: Function Spaces. Noordhoff International Publishing, Leyden (1977)
104.
Kunis, S., Potts, D.: Fast spherical Fourier algorithms. J. Comput. Appl. Math. 161, 75–98 (2003)
105.
Lai, M.J., Shum, C.K., Baramidze, V., Wenston, P.: Triangulated spherical splines for geopotential reconstruction. J. Geodesy 83, 695–708 (2009)
106.
Laín Fernández, N.: Optimally space-localized band-limited wavelets on \({\mathbb{S}}^{q-1}\). J. Comput. Appl. Math. 199, 68–79 (2007)
107.
Landau, H.J., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—II. Bell Syst. Tech. J. 40, 65–84 (1961)
108.
Lang, S.: Undergraduate Analysis, 2nd edn. Springer, New York (2001)
109.
Le Gia, Q.T., Mhaskar, H.N.: Localized linear polynomial operators and quadrature formulas on the sphere. SIAM J. Numer. Anal. 47, 440–466 (2008)
110.
Lemoine, F.G., Smith, D.E., Kunz, L., Smith, R., Pavlis, E.C., Pavlis, N.K., Klosko, S.M., Chinn, D.S., Torrence, M.H., Williamson, R.G., Cox, C.M., Rachlin, K.E., Wang, Y.M., Kenyon, S.C., Salman, R., Trimmer, R., Rapp, R.H., Nerem, R.S.: The development of the NASA GSFC and NIMA joint geopotential model. In: Proceedings of the International Symposium on Gravity, Geoid, and Marine Geodesy (GRAGEOMAR 1996), The University of Tokyo. Springer (1996)
111.
Li, T.H.: Multiscale representation and analysis of spherical data by spherical wavelets. SIAM J. Sci. Comput. 21, 924–953 (1999)
112.
Louis, A.K., Maaß, P., Rieder, A.: Wavelets: Theory and Applications. Wiley, Chichester (1997)
113.
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966)
114.
Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic, Burlington (2009)
115.
Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)
116.
Masters, G., Richards-Dinger, K.: On the efficient calculation of ordinary and generalized spherical harmonics. Geophys. J. Int. 135, 307–309 (1998)
117.
Maus, S., Rother, M., Hemant, K., Stolle, C., Lühr, H., Kuvshinov, A., Olsen, N.: Earth’s lithospheric magnetic field determined to spherical harmonic degree 90 from CHAMP satellite measurements. Geophys. J. Int. 164, 319–330 (2006)
118.
Maus, S., Rother, M., Holme, R., Lühr, H., Olsen, N., Haak, V.: First scalar magnetic anomaly map from CHAMP satellite data indicates weak lithospheric field. Geophys. Res. Lett. 29, 47–1 to 47–4 (2002)
119.
McShane, E.J.: Integration. Princeton University Press, Princeton (1974)
120.
Mhaskar, H.N.: Local quadrature formulas on the sphere. J. Complex. 20, 753–772 (2004)
121.
Mhaskar, H.N.: Local quadrature formulas on the sphere, II. In: Neamtu M., Saff, E.B. (eds.) Advances in Constructive Approximation, pp. 333–344. Nashboro Press, Brentwood (2004)
122.
Mhaskar, H.N., Narcowich, F.J., Prestin, J., Ward, J.D.: Polynomial frames on the sphere. Adv. Comput. Math. 13, 387–403 (2000)
123.
Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comput. 70, 1113–1130 (2000)
124.
Mhaskar, H.N., Prestin, J.: Polynomial frames: a fast tour. In: Chui, C.K., Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XI: Gatlinburg 2004, pp. 101–132. Nashboro Press, Brentwood (2004)
125.
Michel, D.: Framelet based multiscale operator decomposition. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (2006)
126.
Michel, V.: A wavelet based method for the gravimetry problem. In: Freeden, W. (ed.) Progress in Geodetic Science, Proceedings of the Geodetic Week, pp. 283–298. Shaker, Aachen (1998)
127.
Michel, V.: A multiscale method for the gravimetry problem: theoretical and numerical aspects of harmonic and anharmonic modelling. Ph.D. thesis, University of Kaiserslautern, Department of Mathematics, Geomathematics Group. Shaker, Aachen (1999)
128.
Michel, V.: A multiscale approximation for operator equations in separable Hilbert spaces—case study: reconstruction and description of the Earth’s interior, Habilitation thesis. Shaker, Aachen (2002)
129.
Michel, V.: Scale continuous, scale discretized and scale discrete harmonic wavelets for the outer and the inner space of a sphere and their application to an inverse problem in geomathematics. Appl. Comput. Harm. Anal. 12, 77–99 (2002)
130.
Michel, V.: Regularized wavelet-based multiresolution recovery of the harmonic mass density distribution from data of the Earth’s gravitational field at satellite height. Inverse Probl. 21, 997–1025 (2005)
131.
Michel, V.: Wavelets on the 3-dimensional ball. Proc. Appl. Math. Mech. 5, 775–776 (2005)
132.
Michel, V.: Tomography—problems and multiscale solutions. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 949–972. Springer, Heidelberg (2010)
133.
Michel, V.: Optimally localized approximate identities on the 2-sphere. Numer. Func. Anal. Opt. 32, 877–903 (2011)
134.
Michel, V., Fokas, A.S.: A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods. Inverse Probl. 24 (2008). 045019 (25 pp.)
135.
Michel, V., Wolf, K.: Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophys. J. Int. 173, 1–16 (2008)
136.
Mikhlin, S.G.: Mathematical Physics, an Advanced Course. North-Holland Publishing Company, Amsterdam (1970)
137.
Mohlenkamp, M.J.: A fast transform for spherical harmonics. J. Fourier Anal. Appl. 5, 159–184 (1999)
138.
Müller, C.: Über die ganzen Lösungen der Wellengleichung. Math. Ann. 124, 235–264 (1952)
139.
Müller, C.: Spherical Harmonics. Springer, Berlin (1966)
140.
Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin (1969)
141.
Narcowich, F.J., Petrushev, P., Ward, J.D.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)
142.
Narcowich, F.J., Ward, J.D.: Nonstationary wavelets on the m-sphere for scattered data. Appl. Comput. Harm. Anal. 3, 324–336 (1996)
143.
Nievergelt, Y.: Wavelets Made Easy. Birkhäuser, Boston (1999)
144.
Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics—A Unified Introduction with Applications. Birkhäuser, Basel (1988). Translated from the Russian by R. P. Boss
145.
Olson, H.F.: Music, Physics and Engineering, 2nd edn. Dover, New York (1967)
146.
147.
Plato, R.: Numerische Mathematik kompakt, 4th edn. Vieweg + Teubner, Wiesbaden (2010)
148.
Potts, D., Steidl, G., Tasche, M.: Kernels of spherical harmonics and spherical frames. In: Fontanella, F., Jetter, K., Laurent, P.J. (eds.) Advanced Topics in Multivariate Approximation, pp. 287–301. World Scientific, Singapore (1996)
149.
Prestin, J., Rosca, D.: On some cubature formulas on the sphere. J. Approx. Theory 142, 1–19 (2006)
150.
Protter, M.H., Morrey, C.B.: A First Course in Real Analysis, 2nd edn. Springer, New York (1977)
151.
Purucker, M.E., Dyment, J.: Satellite magnetic anomalies related to seafloor spreading in the South Atlantic ocean. Geophys. Res. Lett. 27, 2765–2768 (2000)
152.
Qian, S., Chen, D.: Signal representation using adaptive normalized Gaussian functions. Signal Process. 36, 1–11 (1994)
153.
Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd edn. Springer, Berlin (2007)
154.
Reigber, C., Balmino, G., Schwintzer, P., Biancale, R., Bode, A., Lemoine, J-M., König, R., Loyer, S., Neumayer, H., Marty, J-C., Barthelmes, F., Perosanz, F., Zhu, S.Y.: A high-quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophys. Res. Lett. 29, 37–1 to 37–4 (2002)
155.
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Springer, New York (1996)
156.
Renka, R.J.: Interpolation of data on the surface of a sphere. ACM T. Math. Software 10, 417–436 (1984)
157.
Reuter, R.: Über Integralformeln der Einheitssphäre und harmonische Splinefunktionen. Ph.D. thesis, Veröff. Geod. Inst. RWTH Aachen, RWTH Aachen, vol. 33 (1982)
158.
Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods for Physics and Engineering, 4th edn. Cambridge University Press, Cambridge (2008)
159.
Rivlin, T.J.: An Introduction to the Approximation of Functions. Blaisdell Publishing Company, Waltham (1969)
160.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 1. Gauthier-Villars, Paris (1957)
161.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 2. Gauthier-Villars, Paris (1958)
162.
Robin, L.: Fonctions Sphérique de Legendre et Fonctions Sphéroidale, vol. 3. Gauthier-Villars, Paris (1959)
163.
Sard, A.: Linear Approximation. American Mathematical Society, Providence (1963)
164.
Schaeben, H., Bernstein, S., Hielscher, R., Beckmann, J., Keiner, J., Prestin, J.: High resolution texture analysis with spherical wavelets. Mater. Sci. Forum 495–497, 245–254 (2005)
165.
Schmidt, M., Fengler, M., Mayer-Gürr, T., Eicker, A., Kusche, J., Sánchez, L., Han, S-C.: Regional gravity modeling in terms of spherical base functions. J. Geodesy 81, 17–38 (2007)
166.
Schneider, F.: Inverse problems in satellite geodesy and their approximate solution by splines and wavelets. Ph.D. thesis, University of Kaiserslautern, Geomathematics Group. Shaker, Aachen (1997)
167.
Schoenberg, I.J.: On best approximations of linear operators. Nederl. Akad. Wetensch. Proc. Ser. A 67, 155–163 (1964)
168.
Schreiner, M.: On a new condition for strictly positive definite functions on spheres. Proc. Am. Math. Soc. 125, 531–539 (1997)
169.
Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: SIGGRAPH’95 Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques pp. 161–172. ACM, New York (1995)
170.
Schwarz, H.R.: Numerical Analysis: A Comprehensive Introduction. Wiley, Chichester (1989)
171.
Sethares, W.A.: Tuning, Timbre, Spectrum, Scale. Springer, London (2005)
172.
Simons, F.J.: Slepian functions and their use in signal estimation and spectral analysis. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, pp. 891–923. Springer, Heidelberg (2010)
173.
Simons, F.J., Dahlen, F.A.: Spherical Slepian functions and the polar gap in geodesy. Geophys. J. Int. 166, 1039–1061 (2006)
174.
Simons, F.J., Dahlen, F.A., Wieczorek, M.A.: Spatiospectral concentration on a sphere. SIAM Rev. 48, 504–536 (2006)
175.
Simons, F.J., Loris, I., Brevdo, E., Daubechies, I.C.: Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion. Proc. SPIE 8138 (2011). 81380X
176.
Simons, F.J., Loris, I., Nolet, G., Daubechies, I.C., Voronin, S., Judd, J.S., Vetter, P.A., Charléty, J., Vonesch, C.: Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity. Geophys. J. Int. 187, 969–988 (2011)
177.
Slepian, D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43, 3009–3057 (1964)
178.
Slepian, D., Pollak, H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—I. Bell Syst. Tech. J. 40, 43–63 (1961)
179.
Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)
180.
Szegö, G.: Orthogonal Polynomials, vol. XXIII, 14th edn. AMS Colloquium Publications, Providence (1975)
181.
Temlyakov, V.N.: Greedy algorithms and m-term approximation. J. Approx. Theor. 98, 117–145 (1999)
182.
Temlyakov, V.N.: Greedy algorithms with regard to multivariate systems with special structure. Constr. Approx. 16, 399–425 (1999)
183.
Temlyakov, V.N.: Nonlinear methods of approximation. Found. Comput. Math. 3, 33–107 (2003)
184.
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Johann Ambrosius Barth Verlag, Heidelberg (1995)
185.
Trim, D.: Calculus. Prentice Hall, Scarborough (1993)
186.
Tscherning, C.C.: Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions. Math. Geol. 28, 161–168 (1996)
187.
Tygert, M.: Fast algorithms for spherical harmonic expansions II. J. Comput. Phys. 227, 4260–4279 (2008)
188.
Voigt, A., Wloka, J.: Hilberträume und elliptische Differentialoperatoren. Bibliographisches Institut, Mannheim (1975)
189.
Walnut, D.F.: An Introduction to Wavelet Analysis. Birkhäuser, Boston (2002)
190.
Walter, W.: Einführung in die Potentialtheorie. Bibliographisches Institut, Mannheim (1971)
191.
Walter, W.: Analysis 2, 3rd edn. Springer, Berlin (1992)
192.
Wang, Z., Dahlen, F.A.: Spherical-spline parameterization of three-dimensional Earth models. Geophys. Res. Lett. 22, 3099–3102 (1995)
193.
Wang, Z.X., Guo, D.R.: Special Functions. World Scientific, Singapore (1989)
194.
Weinreich, I.: A construction of C1-wavelets on the two-dimensional sphere. Appl. Comput. Harm. Anal. 10, 1–26 (2001)
195.
Werner, J.: Numerische Mathematik I: Lineare und nichtlineare Gleichungssysteme, Interpolation, numerische Integration. Vieweg, Braunschweig, Wiesbaden (1992)
196.
Wesfried, E., Wickerhauser, M.V.: Adapted local trigonometric transforms and speech processing. IEEE Trans. Signal Process. 41, 3596–3600 (1993)
197.
Wickerhauser, M.V.: INRIA lectures on wavelet packet algorithms. In: Minicourse lecture notes. INRIA, Rocquencourt (1991)
198.
Wieczorek, M.A., Simons, F.J.: Localized spectral analysis on the sphere. Geophys. J. Int. 162, 655–675 (2005)
199.
Wieczorek, M.A., Simons, F.J.: Minimum-variance spectral analysis on the sphere. J. Fourier Anal. Appl. 13, 665–692 (2007)
200.
Windheuser, U.: Sphärische Wavelets: Theorie und Anwendung in der Physikalischen Geodäsie. Ph.D. thesis, University of Kaiserslautern, Geomathematics Group (1995)
201.
Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)
202.
Wood, A.: The Physics of Music. University Paperbacks, London (1962)
203.
204.
205.
206.
Xu, Y., Cheney, E.W.: Strictly positive definite functions on spheres. Proc. Am. Math. Soc. 116, 977–981 (1992)
207.
Yosida, K.: Functional Analysis, 6th edn. Classics in Mathematics. Springer, Berlin (1995)
208.
Zeidler, E. (ed.): Teubner-Taschenbuch der Mathematik, originally from I.N. Bronstein and K.A. Semendjajew. Teubner, Leipzig (1996)
Metadaten
Titel
Wavelets for Inverse Problems on the 3D Ball
verfasst von
Volker Michel
Copyright-Jahr
2013
Verlag
Birkhäuser Boston
Buch
Lectures on Constructive Approximation

Print ISBN: 978-0-8176-8402-0

Electronic ISBN: 978-0-8176-8403-7

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-0-8176-8403-7_11