American Institute of Mathematical Sciences

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 & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011
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The left image illustrates the problem setting. The right image shows some noisy magnetic potential data (with NSR=0.5) corresponding to the magnetization ${\bf{u}}^{\dagger}=[40x_1^3x_2, -40x_1^4]^T$
A direct reconstruction without regularization
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
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 ${\bf{u}}^{\dagger}(x)=[10x_2+5x_1 ,5x_2-10x_1]^T=10\tau+5\mathit{n}$
Approximation of curves by different scales of discretization
Geometry mapping and vector spaces
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
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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