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June  2021, 15(3): 415-443. doi: 10.3934/ipi.2020074

Some application examples of minimization based formulations of inverse problems and their regularization

Department of Mathematics, Alpen-Adria-Universität Klagenfurt, 9020 Klagenfurt, Austria

* Corresponding author: Barbara Kaltenbacher

Received  April 2020 Revised  August 2020 Published  June 2021 Early access  November 2020

Fund Project: supported by the Austrian Science Fund FWF under grant P30054

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

Citation: Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems and Imaging, 2021, 15 (3) : 415-443. doi: 10.3934/ipi.2020074
References:
[1]

S. AndrieuxT. N. Baranger and A. BenAbda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22 (2006), 115-133.  doi: 10.1088/0266-5611/22/1/007.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.

[3]

K. Bredies and H. K. Pikkarainen, Inverse problems in spaces of measures, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 190-218.  doi: 10.1051/cocv/2011205.

[4]

T. F. Brooks and W. M. Humphreys, A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays, AIAA, 2012. doi: 10.2514/6.2004-2954.

[5]

B. M. Brown and M. Jais, A variational approach to an electromagnetic inverse problem, Inverse Problems, 27 (2011), 045011, 29pp. doi: 10.1088/0266-5611/27/4/045011.

[6]

B. M. BrownM. Jais and I. W. Knowles, A variational approach to an elastic inverse problem, Inverse Problems, 21 (2005), 1953-1973.  doi: 10.1088/0266-5611/21/6/010.

[7]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.

[8]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications. Springer Netherlands, 1996.

[10]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.1090/S0025-5718-1977-0436612-4.

[11]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[12]

T. Hohage and F. Werner, Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data, Numerische Mathematik, 123 (2013), 745-779.  doi: 10.1007/s00211-012-0499-z.

[13]

P. HungerländerB. Kaltenbacher and F. Rendl, Regularization of inverse problems via box constrained minimization, Inverse Problems and Imaging, 14 (2020), 437-461.  doi: 10.3934/ipi.2020021.

[14]

V. K. Ivanov, V. V. Vasin and V. P. Tanana, Theory of Linear Ill-posed Problems and Its Applications, Inverse and ill-posed problems series. Utrecht; Boston, 2002. doi: 10.1515/9783110944822.

[15]

B. JinY. Xu and J. Zou, A convergent adaptive finite element method for electrical impedance tomography, IMA Journal of Numerical Analysis, 37 (2016), 1520-1550.  doi: 10.1093/imanum/drw045.

[16]

B. KaltenbacherM. Kaltenbacher and S. Gombots, Inverse scheme for acoustic source localization using microphone measurements and finite element simulations, Acta Acustica united with Acustica, 104 (2018), 647-656.  doi: 10.3813/AAA.919204.

[17]

B. Kaltenbacher, Regularization based on all-at-once formulations for inverse problems, SIAM Journal on Numerical Analysis, 54 (2016), 2594-2618.  doi: 10.1137/16M1060984.

[18]

B. Kaltenbacher, Minimization based formulations of inverse problems and their regularization, SIAM Journal on Optimization, 28 (2018), 620-645.  doi: 10.1137/17M1124036.

[19]

B. KaltenbacherM. Kaltenbacher and S. Reitzinger, Identification of nonlinear B–H curves based on magnetic field computations and multigrid methods for ill-posed problems, European Journal of Applied Mathematics, 14 (2003), 15-38.  doi: 10.1017/S0956792502005089.

[20]

B. Kaltenbacher, A. Kirchner and B. Vexler, Goal oriented adaptivity in the irgnm for parameter identification in PDEs: II. all-at-once formulations, Inverse Problems, 30 (2014), 045002, 33pp. doi: 10.1088/0266-5611/30/4/045002.

[21]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 055008, 24pp. doi: 10.1088/1361-6420/aab739.

[22]

B. KaltenbacherF. Rendl and E. Resmerita, Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems, Journal of Inverse and Ill-posed Problems, 24 (2016), 435-447.  doi: 10.1515/jiip-2015-0087.

[23]

A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity, Inverse Problems, 32 (2016), 085001, 24pp. doi: 10.1088/0266-5611/32/8/085001.

[24]

I. Knowles, A variational algorithm for electrical impedance tomography, Inverse Problems, 14 (1998), 1513-1525.  doi: 10.1088/0266-5611/14/6/010.

[25]

I. Knowles and R. Wallace, A variational solution for the aquifer transmissivity problem, Inverse Problems, 12 (1996), 953-963.  doi: 10.1088/0266-5611/12/6/010.

[26]

R. V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems, 6 (1990), 389-414.  doi: 10.1088/0266-5611/6/3/009.

[27]

R. V. Kohn and M. Vogelius, Relaxation of a variational method for impedance computed tomography, Communications on Pure and Applied Mathematics, 40 (1987), 745-777.  doi: 10.1002/cpa.3160400605.

[28]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularization schemes, Inverse Problems, 29 (2013), 075016, 19pp. doi: 10.1088/0266-5611/29/7/075016.

[29]

P. Monk, Finite Element Methods For Maxwell's Equations, Clarendon Press - Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.

[30]

T. J. Mueller, Aeroacoustic Measurements, Springer-Verlag, 2002.

[31]

A. Neubauer and R. Ramlau, On convergence rates for quasi-solutions of ill-posed problems, ETNA. Electronic Transactions on Numerical Analysis, 41 (2014), 81-92. 

[32]

K. Pieper, B. Q. Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, Computational Optimization and Applications, 77 2020,213–249, arXiv: 1805.03310. doi: 10.1007/s10589-020-00205-y.

[33]

L. Rondi, A variational approach to the reconstruction of cracks by boundary measurements, Journal de Mathématiques Pures et Appliquées, 87 (2007), 324-342.  doi: 10.1016/j.matpur.2007.01.007.

[34]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences. Springer-Verlag New York, 2009.

[35]

A. SchuhmacherK. Rasmussen and C. Hansen, Sound source reconstruction using inverse boundary element calculations, J. Acoust. Soc. Am., 113 (2003), 114-127.  doi: 10.1121/1.1529668.

[36]

P. Sijtsma, CLEAN based on spatial source coherence, Int. J. Aeroacoustics, 6 (2009), 357-374.  doi: 10.2514/6.2007-3436.

[37]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.

show all references

References:
[1]

S. AndrieuxT. N. Baranger and A. BenAbda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems, 22 (2006), 115-133.  doi: 10.1088/0266-5611/22/1/007.

[2]

L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99–R136. doi: 10.1088/0266-5611/18/6/201.

[3]

K. Bredies and H. K. Pikkarainen, Inverse problems in spaces of measures, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), 190-218.  doi: 10.1051/cocv/2011205.

[4]

T. F. Brooks and W. M. Humphreys, A deconvolution approach for the mapping of acoustic sources (DAMAS) determined from phased microphone arrays, AIAA, 2012. doi: 10.2514/6.2004-2954.

[5]

B. M. Brown and M. Jais, A variational approach to an electromagnetic inverse problem, Inverse Problems, 27 (2011), 045011, 29pp. doi: 10.1088/0266-5611/27/4/045011.

[6]

B. M. BrownM. Jais and I. W. Knowles, A variational approach to an elastic inverse problem, Inverse Problems, 21 (2005), 1953-1973.  doi: 10.1088/0266-5611/21/6/010.

[7]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM Journal on Control and Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.

[8]

I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990. doi: 10.1007/978-94-009-2121-4.

[9]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications. Springer Netherlands, 1996.

[10]

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.  doi: 10.1090/S0025-5718-1977-0436612-4.

[11]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[12]

T. Hohage and F. Werner, Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data, Numerische Mathematik, 123 (2013), 745-779.  doi: 10.1007/s00211-012-0499-z.

[13]

P. HungerländerB. Kaltenbacher and F. Rendl, Regularization of inverse problems via box constrained minimization, Inverse Problems and Imaging, 14 (2020), 437-461.  doi: 10.3934/ipi.2020021.

[14]

V. K. Ivanov, V. V. Vasin and V. P. Tanana, Theory of Linear Ill-posed Problems and Its Applications, Inverse and ill-posed problems series. Utrecht; Boston, 2002. doi: 10.1515/9783110944822.

[15]

B. JinY. Xu and J. Zou, A convergent adaptive finite element method for electrical impedance tomography, IMA Journal of Numerical Analysis, 37 (2016), 1520-1550.  doi: 10.1093/imanum/drw045.

[16]

B. KaltenbacherM. Kaltenbacher and S. Gombots, Inverse scheme for acoustic source localization using microphone measurements and finite element simulations, Acta Acustica united with Acustica, 104 (2018), 647-656.  doi: 10.3813/AAA.919204.

[17]

B. Kaltenbacher, Regularization based on all-at-once formulations for inverse problems, SIAM Journal on Numerical Analysis, 54 (2016), 2594-2618.  doi: 10.1137/16M1060984.

[18]

B. Kaltenbacher, Minimization based formulations of inverse problems and their regularization, SIAM Journal on Optimization, 28 (2018), 620-645.  doi: 10.1137/17M1124036.

[19]

B. KaltenbacherM. Kaltenbacher and S. Reitzinger, Identification of nonlinear B–H curves based on magnetic field computations and multigrid methods for ill-posed problems, European Journal of Applied Mathematics, 14 (2003), 15-38.  doi: 10.1017/S0956792502005089.

[20]

B. Kaltenbacher, A. Kirchner and B. Vexler, Goal oriented adaptivity in the irgnm for parameter identification in PDEs: II. all-at-once formulations, Inverse Problems, 30 (2014), 045002, 33pp. doi: 10.1088/0266-5611/30/4/045002.

[21]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 055008, 24pp. doi: 10.1088/1361-6420/aab739.

[22]

B. KaltenbacherF. Rendl and E. Resmerita, Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems, Journal of Inverse and Ill-posed Problems, 24 (2016), 435-447.  doi: 10.1515/jiip-2015-0087.

[23]

A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity, Inverse Problems, 32 (2016), 085001, 24pp. doi: 10.1088/0266-5611/32/8/085001.

[24]

I. Knowles, A variational algorithm for electrical impedance tomography, Inverse Problems, 14 (1998), 1513-1525.  doi: 10.1088/0266-5611/14/6/010.

[25]

I. Knowles and R. Wallace, A variational solution for the aquifer transmissivity problem, Inverse Problems, 12 (1996), 953-963.  doi: 10.1088/0266-5611/12/6/010.

[26]

R. V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography, Inverse Problems, 6 (1990), 389-414.  doi: 10.1088/0266-5611/6/3/009.

[27]

R. V. Kohn and M. Vogelius, Relaxation of a variational method for impedance computed tomography, Communications on Pure and Applied Mathematics, 40 (1987), 745-777.  doi: 10.1002/cpa.3160400605.

[28]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularization schemes, Inverse Problems, 29 (2013), 075016, 19pp. doi: 10.1088/0266-5611/29/7/075016.

[29]

P. Monk, Finite Element Methods For Maxwell's Equations, Clarendon Press - Oxford, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.

[30]

T. J. Mueller, Aeroacoustic Measurements, Springer-Verlag, 2002.

[31]

A. Neubauer and R. Ramlau, On convergence rates for quasi-solutions of ill-posed problems, ETNA. Electronic Transactions on Numerical Analysis, 41 (2014), 81-92. 

[32]

K. Pieper, B. Q. Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, Computational Optimization and Applications, 77 2020,213–249, arXiv: 1805.03310. doi: 10.1007/s10589-020-00205-y.

[33]

L. Rondi, A variational approach to the reconstruction of cracks by boundary measurements, Journal de Mathématiques Pures et Appliquées, 87 (2007), 324-342.  doi: 10.1016/j.matpur.2007.01.007.

[34]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Applied Mathematical Sciences. Springer-Verlag New York, 2009.

[35]

A. SchuhmacherK. Rasmussen and C. Hansen, Sound source reconstruction using inverse boundary element calculations, J. Acoust. Soc. Am., 113 (2003), 114-127.  doi: 10.1121/1.1529668.

[36]

P. Sijtsma, CLEAN based on spatial source coherence, Int. J. Aeroacoustics, 6 (2009), 357-374.  doi: 10.2514/6.2007-3436.

[37]

E. SomersaloM. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52 (1992), 1023-1040.  doi: 10.1137/0152060.

Figure 1.  Electrodes (in red) on the boundary with $ L = 4 $
Figure 2.  Typical $ B $$ H $-curve shape (left) and measurement setup (right)
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