Abstract
We study the global convergence of the gradient descent method of the minimization of strictly convex functionals on an open and bounded set of a Hilbert space. Such results are unknown for this type of sets, unlike the case of the entire Hilbert space. Then, we use our result to establish a general framework to numerically solve boundary value problems for quasi-linear partial differential equations with noisy Cauchy data. The procedure involves the use of Carleman weight functions to convexify a cost functional arising from the given boundary value problem and thus to ensure the convergence of the gradient descent method above. We prove the global convergence of the method as the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this method to solve a highly nonlinear and severely ill-posed coefficient inverse problem, which is the so-called back scattering inverse problem. This problem has many real-world applications. Numerical examples are presented.
Similar content being viewed by others
Data Availability
The data is generated computationally by solving a number of Partial Differential Equations. It is available from the corresponding author on reasonable request.
References
Alifanov, O.M.: Inverse Heat Conduction Problems. Springer, New York (1994)
Alifanov, O.M., Artukhin, A.E., Rumyantcev, S.V.: Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems. Begell House, New York (1995)
Bakushinskii, A.B., Klibanov, M.V., Koshev, N.A.: Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs. Nonlinear Anal. Real World Appl. 34, 201–224 (2017)
Beilina, L., Klibanov, M.V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York (2012)
Bukhgeim, A.L., Klibanov, M.V.: Uniqueness in the large of a class of multidimensional inverse problems. Sov. Math. Dokl. 17, 244–247 (1981)
Colton, David, Kress, Rainer: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, 3rd edn. Springer, New York (2013)
Isakov, V.: Inverse Problems for Partial Differential Equations, 3rd edn. Springer, New York (2017)
Khoa, V.A., Bidney, G.W., Klibanov, M.V., Nguyen, L.H., Nguyen, L., Sullivan, A., Astratov, V.N.: Convexification and experimental data for a 3D inverse scattering problem with the moving point source. Inverse Probl. 36, 085007 (2020)
Khoa, V.A., Bidney, G.W., Klibanov, M.V., Nguyen, L.H., Nguyen, L., Sullivan, A., Astratov, V.N.: An inverse problem of a simultaneous reconstruction of the dielectric constant and conductivity from experimental backscattering data. Inverse Probl. Sci. Eng. 29(5), 712–735 (2021)
Khoa, V.A., Klibanov, M.V., Nguyen, L.H.: Convexification for a 3D inverse scattering problem with the moving point source. SIAM J. Imaging Sci. 13(2), 871–904 (2020)
Klibanov, M.V.: Inverse problems and Carleman estimates. Inverse Probl. 8, 575–596 (1992)
Klibanov, M.V.: Global convexity in a three-dimensional inverse acoustic problem. SIAM J. Math. Anal. 28, 1371–1388 (1997)
Klibanov, M.V.: Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill Posed Probl. 21, 477–560 (2013)
Klibanov, M.V.: Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs. Inverse Probl. 31, 125007 (2015)
Klibanov, M.V.: Convexification of restricted Dirichlet to Neumann map. J. Inverse Ill Posed Probl. 25(5), 669–685 (2017)
Klibanov, M.V., Ioussoupova, O.V.: Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem. SIAM J. Math. Anal. 26, 147–179 (1995)
Klibanov, M.V., Khoa, V.A., Smirnov, A.V., Nguyen, L.H., Bidney, G.W., Nguyen, L., Sullivan, A., Astratov, V.N.: Convexification inversion method for nonlinear SAR imaging with experimentally collected data. J. Appl. Ind. Math. 15, 413–436 (2021)
Klibanov, M.V., Li, J.: Inverse Problems and Carleman Estimates: Global Uniqueness, Global Convergence and Experimental Data. De Gruyter, Berlin (2021)
Klibanov, M.V., Li, J., Zhang, W.: Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data. Inverse Probl. 35, 035005 (2019)
Klibanov, M.V., Li, Z., Zhang, W.: Convexification for the inversion of a time dependent wave front in a heterogeneous medium. SIAM J. Appl. Math. 79, 1722–1747 (2019)
Klibanov, M.V., Romanov, V.G.: Reconstruction procedures for two inverse scattering problems without the phase information. SIAM J. Appl. Math. 76, 178–196 (2016)
Klibanov, M.V., Timonov, A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. Inverse and Ill-Posed Problems Series. VSP, Utrecht (2004)
Kuzhuget, A., Klibanov, M.V.: Global convergence for a 1-D inverse problem with application to imaging of land mines. Appl. Anal. 89(1), 125–157 (2010)
Le, T.T., Nguyen, L.H.: A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data. J. Inverse Ill Posed Probl. (2020). https://doi.org/10.1515/jiip-2020-0028
Lechleiter, A., Nguyen, D.-L.: A trigonometric Galerkin method for volume integral equations arising in TM grating scattering. Adv. Comput. Math. 40, 1–25 (2014)
Nguyen, D.L.: A volume integral equation method for periodic scattering problems for anisotropic Maxwell’s equations. Appl. Numer. Math. 98, 59–78 (2015)
Nguyen, L.H.: An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method. Inverse Probl. 35, 035007 (2019)
Nguyen, L.H.: A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements. Comput. Math. Appl. 80, 2135–2149 (2020)
Schubert, H., Kuznetsov, A.: Detection and Disposal of Improvised Explosives. Springer, Dordrecht (2006)
Triggiani, R., Yao, P.F.: Carleman estimates with no lower order terms for general Riemannian wave equations. Global uniqueness and observability in one shot. Appl. Math. Optim. 46, 331–375 (2002)
Truong, T., Nguyen, D.-L., Klibanov, M.V.: Convexification numerical algorithm for a 2D inverse scattering problem with backscatter data. Inverse Probl. Sci. Eng. 29, 2656–2675 (2021)
Weatherall, J.C., Barber, J., Smith, B.T.: Identifying explosives by dielectric properties obtained through wide-band millimeter-wave illumination. In: Passive and Active Millimeter-Wave Imaging XVIII. Proc. SPIE 9462 (2015)
Yamamoto, M.: Carleman estimates for parabolic equations. Top. Rev. Inverse Probl. 25, 123013 (2009)
Acknowledgements
The authors sincerely thank Dr. Michael V. Klibanov for many fruitful discussions.
Funding
The work is supported in part by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044 and by funds provided by the Faculty Research Grant program at UNC Charlotte, Fund No. 111272.
Author information
Authors and Affiliations
Contributions
All authors contributed to the manuscript equally.
Corresponding author
Ethics declarations
Conflict of interests
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Le, T.T., Nguyen, L.H. The Gradient Descent Method for the Convexification to Solve Boundary Value Problems of Quasi-Linear PDEs and a Coefficient Inverse Problem. J Sci Comput 91, 74 (2022). https://doi.org/10.1007/s10915-022-01846-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-022-01846-3