Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces

Guozhi Dong; Bert Jüttler; Otmar Scherzer; Thomas Takacs

doi:10.3934/ipi.2017011

Article Contents

2017, Volume 11, Issue 2: 221-246. Doi: 10.3934/ipi.2017011

This issue Previous Article Next Article A two-step mixed inpainting method with curvature-based anisotropy and spatial adaptivity

Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces

1.
Computational Science Center, University of Vienna, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria
2.
Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, A-4040 Linz, Austria
3.
Computational Science Center, University of Vienna and, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria Academy of Science, Altenberger Str. 69, A-4040 Linz, Austria

Received: October 2015

Revised: December 2016

Published: May 2017

Abstract / Introduction Full Text(HTML) Figure(6) / Table(2) Related Papers Cited by

Abstract

We study Tikhonov regularization for solving ill-posed operator equations where the solutions are functions defined on surfaces. One contribution of this paper is an error analysis of Tikhonov regularization which takes into account perturbations of the surfaces, in particular when the surfaces are approximated by spline surfaces. Another contribution is that we highlight the analysis of regularization for functions with range in vector bundles over surfaces. We also present some practical applications, such as an inverse problem of gravimetry and an imaging problem for denoising vector fields on surfaces, and show the numerical verification.

Keywords:

Mathematics Subject Classification: Primary: 47A52, 65J20; Secondary: 53B20, 46C99, 65D07.

Citation:

Full Text(HTML)

Figure 1. The left image illustrates the problem setting. The right image shows some noisy magnetic potential data (with NSR=0.5) corresponding to the magnetization ${\bf{u}}^{\dagger}=[40x_1^3x_2, -40x_1^4]^T$

Download: Full-size image PowerPoint slide

Figure 2. A direct reconstruction without regularization

Download: Full-size image PowerPoint slide

Figure 3. The results obtained by minimizing the Tikhonov functional to approximate ${\bf{u}}^{\dagger}=[40x_1^3x_2, -40x_1^4]^T,\;[x_1,x_2]^T \in S_1^+$ with a decreasing level of noise, for decreasing regularization parameters and discretization sizes

Download: Full-size image PowerPoint slide

Figure 4. The above two images plot the reconstructed vector field with the squared ${{\mathcal{H}}^{\text{1}}}{(S_1)}$-seminorm in the ambient space coordinates. The below ones are results with ordinary squared ${{\mathit{H}}^{\text{1}}}{(S_1)}$-seminorm (46). Here ${\bf{u}}^{\dagger}(x)=[10x_2+5x_1 ,5x_2-10x_1]^T=10\tau+5\mathit{n}$

Download: Full-size image PowerPoint slide

Figure 5. Approximation of curves by different scales of discretization

Download: Full-size image PowerPoint slide

Figure 6. Geometry mapping and vector spaces

Download: Full-size image PowerPoint slide

Table 2. Notation corresponding to the geometry

Notation	Remark	Notation	Remark
${\cal M}$	a parametrizable surface	$g$	metric tensor
$\mathit{m}$	mapping $\mathit{m}:\Omega\rightarrow {\cal M}$	$\partial \mathit{m}$	Jacobian of $\mathit{m}$
$\mathcal{TM}$	tangent vector bundle	$\mathcal{NM}$	normal vector bundle
${{\mathcal{P}}_{\mathit{\tau }}}$	tangent projection	${{\mathcal{P}}_{\mathit{n}}}$	normal projection
$\mathit{n}$	unit normal vector	$g(\cdot,\cdot)$	inner product on $\mathcal{TM}$
$\partial_i \mathit{m}$	tangent basis vector

| Show Table

DownLoad: CSV

Table 1. Convergence rates of vector field denoising on curves

NSR($\frac{\left\\| {\delta} \right\\|_{L^2}}{\left\\| {{\bf{u}}^\dagger} \right\\|_{L^2}}$)	1	0.5	0.25	0.125	0.625
$\alpha$	0.04	0.02	0.01	0.005	0.0025
	$h_{s,1}=0.5\pi $	$\gamma_1=1.8371$	$h_{{\bf{u}},1}=0.02\pi $
$\left\| {\check{{\bf{u}}}_1-{\bf{u}}^\dagger} \right\|^2$	$\mathbf{366.3082}$	228.1245	133.0704	77.8783	48.6980
	$h_{s,2}=0.25\pi $	$\gamma_2=0.8211$	$h_{{\bf{u}},2}=0.01\pi $
$\left\| {\check{{\bf{u}}}_2-{\bf{u}}^\dagger} \right\|^2$	347.8737	$\mathbf{200.5511}$	110.7511	62.4273	37.5464
	$h_{s,3}=0.125\pi $	$\gamma_3=0.3866 $	$h_{{\bf{u}},3}=0.005\pi $
$\left\| {\check{{\bf{u}}}_3-{\bf{u}}^\dagger} \right\|^2$	276.7971	166.7043	$\mathbf{94.3003}$	54.0387	33.1077
	$h_{s,4}=0.0625\pi $	$\gamma_4=0.1922 $	$h_{{\bf{u}},4}=0.0025\pi $
$\left\| {\check{{\bf{u}}}_4-{\bf{u}}^\dagger} \right\|^2$	242.3850	150.7489	90.4440	$\mathbf{55.0508}$	34.8112
	$h_{s,5}=0.03125\pi $	$\gamma_5=0.0971 $	$h_{{\bf{u}},5}=0.00125\pi $
$\left\| {\check{{\bf{u}}}_5-{\bf{u}}^\dagger} \right\|^2$	268.2314	158.8666	90.0663	52.4830	$\mathbf{32.7122}$

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	J. H. Ahlberg and E. N. Nilson, Convergence properties of the spline fit, J. Soc. Indust. Appl. Math., 11 (1963), 95-104. doi: 10.1137/0111007.
[2]	A. B. Bakushinskii and A. V. Goncharskii, Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994. doi: 10.1007/978-94-011-1026-6.
[3]	R. Blakely, Potential Theory in Gravity and Magnetic Applications, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511549816.
[4]	M. Burger and S. Osher, Convergence rates of convex variational regularization, Inverse Probl., 20 (2004), 1411-1421. doi: 10.1088/0266-5611/20/5/005.
[5]	M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992. doi: 10.1007/978-1-4757-2201-7.
[6]	B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry -Methods and Applications: Part Ⅰ: The Geometry of Surfaces, Transformation Groups, and Fields, Graduate Texts in Mathematics. Spinger, 2 edition, 1991. Tanslated by Burns, R. G. doi: 10.1007/978-1-4684-9946-9.
[7]	H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, volume 375 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, 1996. doi: 10.1007/978-94-009-1740-8.
[8]	C. Gerhards, On the unique reconstruction of induced spherical magnetizations, Inverse Probl., 32 (2016), 1-24. doi: 10.1088/0266-5611/32/1/015002.
[9]	G. H. Golub and Ch. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996.
[10]	C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
[11]	D. Gubbins, D. Ivers, S. M. Masterton and D. E. Winch, Analysis of lithospheric magnetization in vector spherical harmonics, Geophys. J. Int., 187 (2011), 99-117. doi: 10.1111/j.1365-246X.2011.05153.x.
[12]	P. C. Hansen, Discrete Inverse Problems, volume 7 of Fundamentals of Algorithms, SIAM, Philadelphia, PA, 2010. doi: 10.1137/1.9780898718836.
[13]	E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, volume 5 of CIMS Lecture Notes, New York University, 1999.
[14]	B. Hofmann, B. Kaltenbacher, C. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Probl., 23 (2007), 987-1010. doi: 10.1088/0266-5611/23/3/009.
[15]	J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, volume 160 of Applied Mathematical Sciences, Springer Verlag, New York, 2005. doi: 10.1007/b138659.
[16]	C. Kirisits, L. F. Lang and O. Scherzer, Optical flow on evolving surfaces with an application to the analysis of 4D microscopy data, In A. Kuijper, K. Bredies, T. Pock, and H. Bischof, editors, SSVM'13: Proceedings of the 4th International Conference on Scale Space and Variational Methods in Computer Vision, volume 7893 of Lecture Notes in Computer Science, pages 246-257, Berlin, Heidelberg, 2013. Springer-Verlag. doi: 10.1007/978-3-642-38267-3_21.
[17]	C. Kirisits, L.F. Lang and O. Scherzer, Optical flow on evolving surfaces with space and time regularisation, J. Math. Imaging Vision, 52 (2015), 55-70. doi: 10.1007/s10851-014-0513-4.
[18]	C. Kirisits, C. Pöschl, E. Resmerita and O. Scherzer, Finite-dimensional approximation of convex regularization via hexagonal pixel grids, Appl. Anal., 94 (2015), 612-636. doi: 10.1080/00036811.2014.958998.
[19]	J. M. Lee, Riemannian Surfaces, volume 176 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1997. doi: 10.1007/b98852.
[20]	J. Lefévre and S. Baillet, Optical flow and advection on 2-Riemannian surfaces: A common framework, IEEE Trans. Pattern Anal. Mach. Intell., 30 (2008), 1081-1092. doi: 10.1109/TPAMI.2008.51.
[21]	V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York, Berlin, Heidelberg, 1984. doi: 10.1007/978-1-4612-5280-1.
[22]	J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, volume 10 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972344.
[23]	A. Neubauer and O. Scherzer, Finite-dimensional approximation of Tikhonov regularized solutions of nonlinear ill-posed problems, Numer. Funct. Anal. Optim., 11 (1990), 85-99. doi: 10.1080/01630569008816362.
[24]	C. Pöschl, E. Resmerita and O. Scherzer, Discretization of variational regularization in Banach spaces, Inverse Probl, 26 (2010), 105017, 18pp. doi: 10.1088/0266-5611/26/10/105017.
[25]	E. Resmerita and O. Scherzer, Error estimates for non-quadratic regularization and the relation to enhancement, Inverse Probl., 22 (2006), 801-814. doi: 10.1088/0266-5611/22/3/004.
[26]	O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, volume 167 of Applied Mathematical Sciences, Springer, New York, 2009. doi: 10.1007/978-0-387-69277-7.
[27]	T. Schuster, B. Kaltenbacher, B. Hofmann and K. S. Kazimierski, Regularization Methods in Banach Spaces, volume 10 of Radon Series on Computational and Applied Mathematics, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. doi: 10.1515/9783110255720.
[28]	V. P. Tanana, Methods for Solution of Nonlinear Operator Equations, Inverse and Ill-posed Problems Series. VSP, Utrecht, 1997. doi: 10.1515/9783110900156.
[29]	A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, John Wiley & Sons, Washington, D. C., 1977.
[30]	A. N. Tikhonov, A. Goncharsky, V. Stepanov and A. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7.
[31]	A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Ill-Posed Problems, volume 14 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1998. Translated from the Russian. doi: 10.1007/978-94-017-5167-4.