April  2017, 11(2): 221-246. doi: 10.3934/ipi.2017011

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  March 2017

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.

Citation: Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems and Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011
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. GubbinsD. IversS. 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. HofmannB. KaltenbacherC. 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. KirisitsL.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. KirisitsC. PöschlE. 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. PöschlE. 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.

show all references

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. GubbinsD. IversS. 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. HofmannB. KaltenbacherC. 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. KirisitsL.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. KirisitsC. PöschlE. 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. PöschlE. 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.

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 ${\bf{u}}^{\dagger}=[40x_1^3x_2, -40x_1^4]^T$
Figure 2.  A direct reconstruction without regularization
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
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 ${\bf{u}}^{\dagger}(x)=[10x_2+5x_1 ,5x_2-10x_1]^T=10\tau+5\mathit{n}$
Figure 5.  Approximation of curves by different scales of discretization
Figure 6.  Geometry mapping and vector spaces
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
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
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 $\mathbf{32.7122}$
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 $\mathbf{32.7122}$
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