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March  2022, 12(1): 81-114. doi: 10.3934/mcrf.2021003

Learning nonlocal regularization operators

Gernot Holler 1,, and  Karl Kunisch 1,2,

1. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2. 

Radon Institute, Austrian Academy of Sciences, RICAM Linz, Altenbergerstrasse 69, 4040 Linz, Austria

* Corresponding author: Gernot Holler

Received  February 2020 Revised  September 2020 Published  March 2022 Early access  January 2021

A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided.

Keywords: Nonlocal operators, optimal control, inverse problems, bilevel optimization, fractional order regularization, learning.
Mathematics Subject Classification: Primary: 49J20, 49N45, 35R11; Secondary 35J05.
Citation: Gernot Holler, Karl Kunisch. Learning nonlocal regularization operators. Mathematical Control & Related Fields, 2022, 12 (1) : 81-114. doi: 10.3934/mcrf.2021003
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

B. Alali, K. Liu and M. Gunzburger, A generalized nonlocal vector calculus, Z. Angew. Math. Phys., 66 (2015), 2807-2828.  doi: 10.1007/s00033-015-0514-1.  Google Scholar

[3]

H. Antil, E. Otárola and A. J. Salgado, Optimization with respect to order in a fractional diffusion model: Analysis, approximation and algorithmic aspects, J. Sci. Comput., 77 (2018), 204-224.  doi: 10.1007/s10915-018-0703-0.  Google Scholar

[4]

H. Antil and C. N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM J. Math. Anal., 51 (2019), 2479-2503. doi: 10.1137/18M1224970.  Google Scholar

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K. Bredies and D. Lorenz, Mathematische Bildverarbeitung, Vieweg+Teubner Verlag, 2011. doi: 10.1007/978-3-8348-9814-2.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

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H. Cartan, Differential Calculus, Hermann, Paris; Houghton Mifflin Co., Boston, Mass., 1971.  Google Scholar

[8]

E. Casas, M. Mateos and A. Rösch, Analysis of control problems of nonmontone semilinear elliptic equations, ESAIM: COCV, 26 (2020). doi: 10.1051/cocv/2020032.  Google Scholar

[9]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math.-Ver., 117 (2015), 3-44.  doi: 10.1365/s13291-014-0109-3.  Google Scholar

[10]

J. Chung, M. Chung and D. P. O'Leary, Designing optimal spectral filters for inverse problems, SIAM J. Sci. Comput., 33 (2011), 3132-3152.  doi: 10.1137/100812938.  Google Scholar

[11]

J. Chung and M. I. Español, Learning regularization parameters for general-form Tikhonov, Inverse Problems, 33 (2017), 21pp. doi: 10.1088/1361-6420/33/7/074004.  Google Scholar

[12]

M. D'Elia, J. C. De Los Reyes and A. Miniguano Trujillo, Bilevel parameter optimization for nonlocal image denoising models, preprint, arXiv: 1912.02347v3. doi: 10.2172/1592945.  Google Scholar

[13]

M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), 1245-1260.  doi: 10.1016/j.camwa.2013.07.022.  Google Scholar

[14]

J. C. De Los Reyes, C.-B. Schönlieb and T. Valkonen, The structure of optimal parameters for image restoration problems, J. Math. Anal. Appl., 434 (2016), 464-500.  doi: 10.1016/j.jmaa.2015.09.023.  Google Scholar

[15]

J. C. De Los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization, Inverse Probl. Imaging, 7 (2013), 1183-1214.  doi: 10.3934/ipi.2013.7.1183.  Google Scholar

[16]

S. Dempe, Foundations of Bilevel Programming, Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/b101970.  Google Scholar

[17]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[18]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696. doi: 10.1137/110833294.  Google Scholar

[19]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[20]

G. B. Folland, Real Analysis. Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1999.  Google Scholar

[21]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[22]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598.  doi: 10.1137/090766607.  Google Scholar

[23]

E. Haber and L. Tenorio, Learning regularization functionals-A supervised training approach, Inverse Problems, 19 (2003), 611-626.  doi: 10.1088/0266-5611/19/3/309.  Google Scholar

[24]

M. Hintermüller and C. N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory, J. Math. Imaging Vision, 59 (2017), 498-514.  doi: 10.1007/s10851-017-0744-2.  Google Scholar

[25]

G. Holler, K. Kunisch and R. C. Barnard, A bilevel approach for parameter learning in inverse problems, Inverse Problems, 34 (2018), 28pp. doi: 10.1088/1361-6420/aade77.  Google Scholar

[26]

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718614.  Google Scholar

[27]

K. Ito and K. Kunisch, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), 357-376. doi: 10.1137/S0363012902411015.  Google Scholar

[28]

K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM J. Imaging Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.  Google Scholar

[29]

P. Ochs, R. Ranftl, T. Brox and T. Pock, Bilevel optimization with nonsmooth lower level problems, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci., 9087, Springer, Cham, 2015,654-665. doi: 10.1007/978-3-319-18461-6_52.  Google Scholar

[30]

M. Ulbrich and S. Ulbrich, Nichtlineare Optimierung, Mathematik Kompakt, Birkhäuser, Basel, 2012. doi: 10.1007/978-3-0346-0654-7.  Google Scholar

[31]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62.  doi: 10.1007/BF01442543.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[2]

B. Alali, K. Liu and M. Gunzburger, A generalized nonlocal vector calculus, Z. Angew. Math. Phys., 66 (2015), 2807-2828.  doi: 10.1007/s00033-015-0514-1.  Google Scholar

[3]

H. Antil, E. Otárola and A. J. Salgado, Optimization with respect to order in a fractional diffusion model: Analysis, approximation and algorithmic aspects, J. Sci. Comput., 77 (2018), 204-224.  doi: 10.1007/s10915-018-0703-0.  Google Scholar

[4]

H. Antil and C. N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM J. Math. Anal., 51 (2019), 2479-2503. doi: 10.1137/18M1224970.  Google Scholar

[5]

K. Bredies and D. Lorenz, Mathematische Bildverarbeitung, Vieweg+Teubner Verlag, 2011. doi: 10.1007/978-3-8348-9814-2.  Google Scholar

[6]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[7]

H. Cartan, Differential Calculus, Hermann, Paris; Houghton Mifflin Co., Boston, Mass., 1971.  Google Scholar

[8]

E. Casas, M. Mateos and A. Rösch, Analysis of control problems of nonmontone semilinear elliptic equations, ESAIM: COCV, 26 (2020). doi: 10.1051/cocv/2020032.  Google Scholar

[9]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math.-Ver., 117 (2015), 3-44.  doi: 10.1365/s13291-014-0109-3.  Google Scholar

[10]

J. Chung, M. Chung and D. P. O'Leary, Designing optimal spectral filters for inverse problems, SIAM J. Sci. Comput., 33 (2011), 3132-3152.  doi: 10.1137/100812938.  Google Scholar

[11]

J. Chung and M. I. Español, Learning regularization parameters for general-form Tikhonov, Inverse Problems, 33 (2017), 21pp. doi: 10.1088/1361-6420/33/7/074004.  Google Scholar

[12]

M. D'Elia, J. C. De Los Reyes and A. Miniguano Trujillo, Bilevel parameter optimization for nonlocal image denoising models, preprint, arXiv: 1912.02347v3. doi: 10.2172/1592945.  Google Scholar

[13]

M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), 1245-1260.  doi: 10.1016/j.camwa.2013.07.022.  Google Scholar

[14]

J. C. De Los Reyes, C.-B. Schönlieb and T. Valkonen, The structure of optimal parameters for image restoration problems, J. Math. Anal. Appl., 434 (2016), 464-500.  doi: 10.1016/j.jmaa.2015.09.023.  Google Scholar

[15]

J. C. De Los Reyes and C.-B. Schönlieb, Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization, Inverse Probl. Imaging, 7 (2013), 1183-1214.  doi: 10.3934/ipi.2013.7.1183.  Google Scholar

[16]

S. Dempe, Foundations of Bilevel Programming, Nonconvex Optimization and its Applications, 61, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/b101970.  Google Scholar

[17]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[18]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696. doi: 10.1137/110833294.  Google Scholar

[19]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[20]

G. B. Folland, Real Analysis. Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1999.  Google Scholar

[21]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[22]

M. Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598.  doi: 10.1137/090766607.  Google Scholar

[23]

E. Haber and L. Tenorio, Learning regularization functionals-A supervised training approach, Inverse Problems, 19 (2003), 611-626.  doi: 10.1088/0266-5611/19/3/309.  Google Scholar

[24]

M. Hintermüller and C. N. Rautenberg, Optimal selection of the regularization function in a weighted total variation model. Part I: Modelling and theory, J. Math. Imaging Vision, 59 (2017), 498-514.  doi: 10.1007/s10851-017-0744-2.  Google Scholar

[25]

G. Holler, K. Kunisch and R. C. Barnard, A bilevel approach for parameter learning in inverse problems, Inverse Problems, 34 (2018), 28pp. doi: 10.1088/1361-6420/aade77.  Google Scholar

[26]

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718614.  Google Scholar

[27]

K. Ito and K. Kunisch, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), 357-376. doi: 10.1137/S0363012902411015.  Google Scholar

[28]

K. Kunisch and T. Pock, A bilevel optimization approach for parameter learning in variational models, SIAM J. Imaging Sci., 6 (2013), 938-983.  doi: 10.1137/120882706.  Google Scholar

[29]

P. Ochs, R. Ranftl, T. Brox and T. Pock, Bilevel optimization with nonsmooth lower level problems, in Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci., 9087, Springer, Cham, 2015,654-665. doi: 10.1007/978-3-319-18461-6_52.  Google Scholar

[30]

M. Ulbrich and S. Ulbrich, Nichtlineare Optimierung, Mathematik Kompakt, Birkhäuser, Basel, 2012. doi: 10.1007/978-3-0346-0654-7.  Google Scholar

[31]

J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), 49-62.  doi: 10.1007/BF01442543.  Google Scholar

Figure 1.  Optimal weights for linear state equation in case (A) (first row) and case (B) (second row) using $ \beta = 0 $. The training set consisted of 512 data vectors given in a single batch
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Figure 2.  Ground truth and reconstructed controls for one data vector from the validation set for the linear state equation in case (B) using $ \beta = 0 $. The training set consisted of 512 data vectors given in a single batch
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Figure 3.  Optimal weights for $ s = 0.1 $ and various $ L^1 $ regularization parameters $ \beta $ in case (B). The training and validation set both consisted of 512 data vectors given in a single batch. The average validation errors were $ 1.42 \times {10^{ - 2}} $ ($ \beta = 10^{-2} $), $ 1.18\times {10^{ - 2}} $ ($ \beta = 10^{-3} $), and $1.03\times {10^{ - 2}} $ ($ \beta = 10^{-4} $)
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Table 1.  Average training and validation error for optimal regularization parameter $ \nu^* $ (second and third column) and optimal weight $ \sigma^* $ (fourth and fifth column) in case (A) using $ \beta = 0 $. The training and validation set both consisted of 512 data vectors
(a) s = 0:1
batchsize train error reg val error reg train error weight val error weight
8 1.81 × 10−2 2.19 × 10−2 1.42 × 10−2 2.04 × 10−2
64 1.88 × 10−2 2.11 × 10−2 1.61 × 10−2 1.86 × 10−2
512 1.90 × 10−2 2.09 × 10−2 1.65 × 10−2 1.82 × 10−2
(b) s = 0:9
8 1.53 × 10−2 1.98 × 10−2 1.51 × 10−2 1.97 × 10−2
641.63 × 10−2 1.85 × 10−2 1.61 × 10−2 1.84 × 10−2
5121.65 × 10−2 1.82 × 10−2 1.64 × 10−2 1.80 × 10−2
(c) L2 and H1 regularization with optimal ν*
batchsize train error L2 val error L2 train error H1 val error H1
8 1.93 × 10−2 2.30 × 10−2 1.51 × 10−2 1.97 × 10−2
641.99 × 10−2 2.22 × 10−2 1.61 × 10−2 1.84 × 10−2
5122.01 × 10−2 2.20 × 10−2 1.64 × 10−2 1.80 × 10−2
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(a) s = 0:1
batchsize train error reg val error reg train error weight val error weight
8 1.81 × 10−2 2.19 × 10−2 1.42 × 10−2 2.04 × 10−2
64 1.88 × 10−2 2.11 × 10−2 1.61 × 10−2 1.86 × 10−2
512 1.90 × 10−2 2.09 × 10−2 1.65 × 10−2 1.82 × 10−2
(b) s = 0:9
8 1.53 × 10−2 1.98 × 10−2 1.51 × 10−2 1.97 × 10−2
641.63 × 10−2 1.85 × 10−2 1.61 × 10−2 1.84 × 10−2
5121.65 × 10−2 1.82 × 10−2 1.64 × 10−2 1.80 × 10−2
(c) L2 and H1 regularization with optimal ν*
batchsize train error L2 val error L2 train error H1 val error H1
8 1.93 × 10−2 2.30 × 10−2 1.51 × 10−2 1.97 × 10−2
641.99 × 10−2 2.22 × 10−2 1.61 × 10−2 1.84 × 10−2
5122.01 × 10−2 2.20 × 10−2 1.64 × 10−2 1.80 × 10−2
Table 2.  Average training and validation error for optimal regularization parameter $ \nu^* $ (second and third column) and optimal weight $ \sigma^* $ (fourth and fifth column) in case (B) using $ \beta = 0 $. The training and validation set both consisted of 512 data vectors
(a) s = 0.1
batchsize train error reg val error reg train error weight val error weight
8 2.03 × 10−2 2.12 × 10−2 1.02 × 10−2 1.09 × 10−2
64 2.05 × 10−2 2.09 × 10−2 1.02 × 10−2 1.04 × 10−2
512 2.06 × 10−2 2.08 × 10−2 1.14 × 10−2 1.15 × 10−2
(b) s = 0.9
8 1.37 × 10−2 1.44 × 10−2 1.33 × 10−2 1.41 × 10−2
64 1.38 × 10−2 1.42 × 10−2 1.35 × 10−2 1.38 × 10−2
512 1.39 × 10−2 1.41 × 10−2 1.35 × 10−2 1.38 × 10−2
(c) L2 and H1 regularization with optimal ν*
8 2.48 × 10−2 2.60 × 10−2 1.33 × 10−2 1.40 × 10−2
64 2.51 × 10−2 2.56 × 10−2 1.35 × 10−2 1.38 × 10−2
512 2.52 × 10−2 2.55 × 10−2 1.35 × 10−2 1.38 × 10−2
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(a) s = 0.1
batchsize train error reg val error reg train error weight val error weight
8 2.03 × 10−2 2.12 × 10−2 1.02 × 10−2 1.09 × 10−2
64 2.05 × 10−2 2.09 × 10−2 1.02 × 10−2 1.04 × 10−2
512 2.06 × 10−2 2.08 × 10−2 1.14 × 10−2 1.15 × 10−2
(b) s = 0.9
8 1.37 × 10−2 1.44 × 10−2 1.33 × 10−2 1.41 × 10−2
64 1.38 × 10−2 1.42 × 10−2 1.35 × 10−2 1.38 × 10−2
512 1.39 × 10−2 1.41 × 10−2 1.35 × 10−2 1.38 × 10−2
(c) L2 and H1 regularization with optimal ν*
8 2.48 × 10−2 2.60 × 10−2 1.33 × 10−2 1.40 × 10−2
64 2.51 × 10−2 2.56 × 10−2 1.35 × 10−2 1.38 × 10−2
512 2.52 × 10−2 2.55 × 10−2 1.35 × 10−2 1.38 × 10−2
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