Abstract

This research investigates the estimation of the mean and variance of stochastic coefficients in mixed variational problems, with an application to the elasticity imaging inverse problem for tumor localization. We study an abstract mixed variational problem, where the coefficients and the right-hand side are strongly measurable functions. By establishing pathwise mixed variational problems, we demonstrate unique solvability for each realization and prove the measurability of pathwise solutions. This result facilitates the derivation of an integral formulation, which is crucial for treating the inverse problem as a stochastic optimization problem, as well as developing a discretization framework for both direct and inverse problems. In Sobolev-Bochner spaces, we establish existence results for integral solutions and obtain basic estimates. We analyze the coefficient-to-solution map for the inverse problem, proving its Lipschitz continuity and characterizing its first- and second-order derivatives. Two optimization approaches are introduced: the classical least-squares and energy minimization, with the latter shown to be convex. A pathwise approach addresses the compact embedding constraint, enabling the integral formulation to be discretized and solved using a stochastic Galerkin scheme. Finally, we provide numerical examples that demonstrate the estimation of the unknown coefficient's mean and variance from data using discrete stochastic methods.

Author information

Contact details are reproduced from the original publication and may be historical.

Zi-Jia Gong

School of Mathematics and Statistics, Rochester Institute of Technology, Rochester, New York, U.S.A.

zg3988@g.rit.edu

Akhtar A. Khan

School of Mathematics and Statistics, Rochester Institute of Technology, Rochester, New York, U.S.A.

aaksma@rit.edu

Miguel Sama

Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Madrid, Spain

msama@ind.uned.es

Keywords

  • Stochastic mixed variational problems
  • stochastic inverse problems
  • partial differential equations with random data
  • nearly incompressible elasticity system
  • stochastic Galerkin method
  • regularization
  • finite-dimensional noise

Mathematics Subject Classification

35R30, 49N45, 65J20, 65J22, 65M30

References

  1. M. A. Aguiló, L. P. Swiler, A. Urbina: First-order formulations for large-scale stochastic parameter estimation within the frameworks of steady state dynamics: the elastic and viscoelastic case, Inverse Problems 28/7 (2012), art.no. 075003, 26p.https://doi.org/10.1088/0266-5611/28/7/075003Google Scholar
  2. M. A. Aguiló, L. P. Swiler, A. Urbina: An overview of inverse material identification within the frameworks of deterministic and stochastic parameter estimation, Int. J. Uncertain. Quantif. 3/4 (2013) 289–319.https://doi.org/10.1615/Int.J.UncertaintyQuantification.2012003668Google Scholar
  3. H. Ammari, E. Bretin, J. Garnier, W. Jing, H. Kang, A. Wahab: Localization, stability, and resolution of topological derivative based imaging functionals in elasticity, SIAM J. Imaging Sci. 6/4 (2013) 2174–2212.https://doi.org/10.1137/120899303Google Scholar
  4. H. Ammari, E. Bretin, J. Garnier, H. Kang, H. Lee, A. Wahab: Mathematical Methods in Elasticity Imaging, Princeton Series in Applied Mathematics, Princeton University Press, Princeton (2015).https://doi.org/10.23943/princeton/9780691165318.001.0001Google Scholar
  5. H. Ammari, P. Garapon, F. Jouve: Separation of scales in elasticity imaging: a numerical study, J. Comput. Math. 28/3 (2010) 354–370.https://doi.org/10.4208/jcm.2009.12-m1001Google Scholar
  6. W. Arendt, M. Kreuter: Mapping theorems for Sobolev spaces of vector-valued functions, Studia Math. 240/3 (2018) 275–299.https://doi.org/10.4064/sm8757-4-2017Google Scholar
  7. A. Arnold, S. Reichling, O. Bruhns, J. Mosler: Efficient computation of the elastography inverse problem by combining variational mesh adaption and clustering technique, Phys. Med. Biol. 55 (2010) 2035–2056.https://doi.org/10.1088/0031-9155/55/7/016Google Scholar
  8. C. Awasthi, A. Lamperski, T. M. Kowalewski: Multi-material inverse design of soft deformable bodies via functional optimization, Inverse Problems 39/3 (2023), art.no. 035006, 34p.https://doi.org/10.1088/1361-6420/acaa31Google Scholar
  9. V. A. Badri Narayanan, N. Zabaras: Stochastic inverse heat conduction using a spectral approach, Internat. J. Numer. Methods Engrg. 60/9 (2004) 1569–1593.https://doi.org/10.1002/nme.1015Google Scholar
  10. P. E. Barbone, N. H. Bamber: Quantitative elasticity imaging: what can and cannot be inferred from strain images, Phys. Med. Biol. 47 (2002) 2147–2164.https://doi.org/10.1088/0031-9155/47/12/310Google Scholar
  11. P. E. Barbone, N. H. Gokhale: Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions, Inverse Problems 20/1 (2004) 283–296.https://doi.org/10.1088/0266-5611/20/1/017Google Scholar
  12. P. E. Barbone, A. A. Oberai: A review of the mathematical and computational foundations of biomechanical imaging, in: Computational Modeling in Biomechanics, S. De et al. (eds.), Springer, Dordrecht (2010) 375–408.https://doi.org/10.1007/978-90-481-3575-2_13Google Scholar
  13. A. T. Bharucha-Reid: Random Integral Equations, Mathematics in Science and Engineering Vol. 96, Academic Press, New York (1972).Google Scholar
  14. N. Bochud, G. Rus: Probabilistic inverse problem to characterize tissue-equivalent material mechanical properties, IEEE Trans. Ultrasonics Ferro. Frequency Control 59/7 (2012) 1443–1456.https://doi.org/10.1109/TUFFC.2012.2345Google Scholar
  15. J. Borggaard, H.-W. van Wyk: Gradient-based estimation of uncertain parameters for elliptic partial differential equations, Inverse Problems 31/6 (2015), art.no. 065008, 33p.https://doi.org/10.1088/0266-5611/31/6/065008Google Scholar
  16. M. A. Branch, T. F. Coleman, Y. Li: A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems, SIAM J. Sci. Comput. 21/1 (1999) 1–23.https://doi.org/10.1137/S1064827595289108Google Scholar
  17. E. Bretin, P. Millien, L. Seppecher: Stability for finite element discretization of some inverse parameter problems from internal data: application to elastography, SIAM J. Imaging Sci. 16/1 (2023) 340–367.https://doi.org/10.1137/21M1428522Google Scholar
  18. P. Chen, A. Quarteroni, G. Rozza: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, Numer. Math. 133/1 (2016) 67–102.https://doi.org/10.1007/s00211-015-0743-4Google Scholar
  19. Z. Chen, W. Zhang, J. Zou: Stochastic convergence of regularized solutions and their finite element approximations to inverse source problems, SIAM J. Numer. Analysis 60/2 (2022) 751–780.https://doi.org/10.1137/21M1409779Google Scholar
  20. Y. Choi, H.-C. Lee: Error analysis of finite element approximations of the optimal control problem for stochastic Stokes equations with additive white noise, Appl. Numer. Math. 133 (2018) 144–160.https://doi.org/10.1016/j.apnum.2018.03.002Google Scholar
  21. M. Dambrine, A. A. Khan, M. Sama, H.-J. Starkloff: Stochastic elliptic inverse problems. Solvability, convergence rates, discretization, and applications, J. Convex Analysis 30/3 (2023) 851–885.https://doi.org/10.68381/jca30040Google Scholar
  22. P. J. Davies, E. Barnhill, I. Sack: The MRE inverse problem for the elastic shear modulus, SIAM J. Appl. Math. 79/4 (2019) 1367–1388.https://doi.org/10.1137/18M1201160Google Scholar
  23. M. M. Doyley: Model-based elastography: a survey of approaches to the inverse elasticity problem, Physics Med. Biology 57/3 (2012) R35.https://doi.org/10.1088/0031-9155/57/3/R35Google Scholar
  24. B. Faverjon, B. Puig, T. N. Baranger: Identification of boundary conditions by solving Cauchy problem in linear elasticity with material uncertainties, Comput. Math. Appl. 73/3 (2017) 494–504.https://doi.org/10.1016/j.camwa.2016.12.011Google Scholar
  25. J. Fehrenbach, M. Masmoudi, R. Souchon, P. Trompette: Detection of small inclusions by elastography, Inverse Problems 22/3 (2006) 1055–1069.https://doi.org/10.1088/0266-5611/22/3/020Google Scholar
  26. I. M. Franck, P. S. Koutsourelakis: Sparse variational Bayesian approximations for nonlinear inverse problems: applications in nonlinear elastography, Comput. Methods Appl. Mech. Engrg. 299 (2016) 215–244.https://doi.org/10.1016/j.cma.2015.10.015Google Scholar
  27. N. H. Gokhale, P. E. Barbone, A. Oberai: Solution of the nonlinear elasticity imaging inverse problem: the compressible case, Inverse Problems 24/4 (2008), art.no. 045010, 26p.https://doi.org/10.1088/0266-5611/24/4/045010Google Scholar
  28. M. D. Gunzburger, H.-C. Lee, J. Lee: Error estimates of stochastic optimal Neumann boundary control problems, SIAM J. Numer. Analysis 49/4 (2011) 1532–1552.https://doi.org/10.1137/100801731Google Scholar
  29. J. Gwinner, B. Jadamba, A. A. Khan, F. Raciti: Uncertainty Quantification in Variational Inequalities – Theory, Numerics, and Applications, CRC Press, Boca Raton (2022).https://doi.org/10.1201/9781315228969Google Scholar
  30. S. Hubmer, E. Sherina, A. Neubauer, O. Scherzer: Lamé parameter estimation from static displacement field measurements in the framework of nonlinear inverse problems, SIAM J. Imaging Sci. 11/2 (2018) 1268–1293.https://doi.org/10.1137/17M1154461Google Scholar
  31. B. Jadamba, A. A. Khan, M. Richards, M. Sama: A convex inversion framework for identifying parameters in saddle point problems with applications to inverse incompressible elasticity, Inverse Problems 36/7 (2020), art.no. 074003, 25p.https://doi.org/10.1088/1361-6420/ab8482Google Scholar
  32. B. Jadamba, A. A. Khan, M. Richards, M. Sama, C. Tammer: Analyzing the role of the Inf-Sup condition for parameter identification in saddle point problems with application in elasticity imaging, Optimization 69/12 (2020) 2577–2610.https://doi.org/10.1080/02331934.2020.1789128Google Scholar
  33. B. Jadamba, A. A. Khan, G. Rus, M. Sama, B. Winkler: A new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location, SIAM J. Appl. Math. 74/5 (2014) 1486–1510.https://doi.org/10.1137/130928261Google Scholar
  34. B. Jadamba, A. A. Khan, M. Sama, H.-J. Starkloff, C. Tammer: A convex optimization framework for the inverse problem of identifying a random parameter in a stochastic partial differential equation, SIAM/ASA J. Uncertainty Quant. 9/2 (2021) 922–952.https://doi.org/10.1137/20M1323953Google Scholar
  35. L. Ji, J. McLaughlin: Recovery of the Lamé parameter µ in biological tissues, Inverse Problems 20/1 (2004) 1–24.https://doi.org/10.1088/0266-5611/20/1/001Google Scholar
  36. B. Jin, J. Zou: Inversion of Robin coefficient by a spectral stochastic finite element approach, J. Comput. Phys. 227/6 (2008) 3282–3306.https://doi.org/10.1016/j.jcp.2007.11.042Google Scholar
  37. F. Kallel, M. Bertrand: Tissue elasticity reconstruction using linear perturbation method, Medical Imaging, IEEE Trans. Med. Imaging 15 (1996) 299–313.https://doi.org/10.1109/42.500139Google Scholar
  38. M. Keyanpour, A. M. Nehrani: Optimal thickness of a cylindrical shell subject to stochastic forces, J. Optim. Theory Appl. 167/3 (2015) 1032–1050.https://doi.org/10.1007/s10957-014-0663-yGoogle Scholar
  39. D. P. Kouri, T. M. Surowiec: Existence and optimality conditions for risk-averse PDE-constrained optimization, SIAM/ASA J. Uncert. Quant. 6/2 (2018) 787–815.https://doi.org/10.1137/16M1086613Google Scholar
  40. G. J. Lord, C. E. Powell, T. Shardlow: An Introduction to Computational Stochastic PDEs, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York (2014).https://doi.org/10.1017/CBO9781139017329Google Scholar
  41. A. Nouy, C. Soize: Random field representations for stochastic elliptic boundary value problems and statistical inverse problems, Eur. J. Appl. Math. 25/3 (2014) 339–373.https://doi.org/10.1017/S0956792514000072Google Scholar
  42. D. V. Patel, D. Ray, A. A. Oberai: Solution of physics-based Bayesian inverse problems with deep generative priors, Comput. Methods Appl. Mech. Engrg. 400 (2022), art.no. 115428, 25p.https://doi.org/10.1016/j.cma.2022.115428Google Scholar
  43. K. R. Raghavan, A. E. Yagle: Forward and inverse problems in elasticity imaging of soft tissuess, IEEE. Trans. Nuclear Sci. 41 (1994) 1639-–1648.https://doi.org/10.1109/23.322961Google Scholar
  44. D. T. Seidl: A computational framework for elliptic inverse problems with uncertain boundary conditions, Ph. D. Thesis, Boston University, ProQuest LLC, Ann Arbor (2015).Google Scholar
  45. D. T. Seidl, A. A. Oberai, P. E. Barbone: The coupled adjoint-state equation in forward and inverse linear elasticity: incompressible plane stress, Comput. Methods Appl. Mech. Engrg. 357 (2019), art.no. 112588, 22p.https://doi.org/10.1016/j.cma.2019.112588Google Scholar
  46. D. T. Seidl, B. G. Van Bloemen Waanders, T. M. Wildey: Simultaneous inversion of shear modulus and traction boundary conditions in biomechanical imaging, Inverse Probl. Sci. Engrg. 28/2 (2020) 256–276.https://doi.org/10.1080/17415977.2019.1603222Google Scholar
  47. W. Shen, L. Ge, W. Liu: Stochastic Galerkin method for optimal control problem governed by random elliptic PDE with state constraints, J. Sci. Comput. 78/3 (2019) 1571–1600.https://doi.org/10.1007/s10915-018-0823-6Google Scholar
  48. E. Sherina, L. Krainz, S. Hubmer, W. Drexler, O. Scherzer: Displacement field estimation from OCT images utilizing speckle information with applications in quantitative elastography, Inverse Problems 36/12 (2020), art.no. 124003, 27p.https://doi.org/10.1088/1361-6420/abaf65Google Scholar
  49. C. Touzeau, B. Magnain, G. Lubineau, E. Florentin: Robust method for identifying material parameters based on virtual fields in elastodynamics, Comput. Math. Appl. 77/11 (2019) 3021–3042.https://doi.org/10.1016/j.camwa.2018.08.005Google Scholar
  50. W. I. T. Uy, M. Grigoriu: Specification of additional information for solving stochastic inverse problems, SIAM J. Sci. Comput. 41/5 (2019) A2880–A2910.https://doi.org/10.1137/18M120155XGoogle Scholar
  51. J. Van Neerven: Stochastic Evolution Equations, ISEM Lecture Notes, Delft (2007).Google Scholar
  52. J. Van Neerven: Compactness in the Lebesgue-Bochner spaces Lp (µ; X), Indag. Math. (N.S.) 25/2 (2014) 389–394.https://doi.org/10.1016/j.indag.2013.06.002Google Scholar
  53. H.-W. Van Wyk: A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems, Doctoral Dissertation, Virginia Tech, Blacksburg (2012).Google Scholar
  54. Y. Wang, X. Zhu, P. Kloeden: Compactness in Lebesgue-Bochner spaces of random variables and the existence of mean-square random attractors, Stoch. Dynamics 19/4 (2019), art.no. 1950032, 16p.https://doi.org/10.1142/S0219493719500321Google Scholar
  55. J. E. Warner, W. Aquino, M. D. Grigoriu: Stochastic reduced order models for inverse problems under uncertainty, Comput. Methods Appl. Mech. Engrg. 285 (2015) 488–514.https://doi.org/10.1016/j.cma.2014.11.021Google Scholar

Suggested citation

Z.-J. Gong, A. A. Khan, M. Sama, A. Zălinescu. “Estimating Random Coefficients in Mixed Variational Problems with an Application to the Stochastic Elasticity Imaging Inverse Problem for Tumor Identification.” Journal of Convex Analysis 32 (2025), No. 2, 321–357. https://doi.org/10.68381/jca32017

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press.