Convergence rates under a range invariance condition with application to electrical impedance tomography (2403.18704v1)
Abstract: This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schr\"odinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT.
- A. Auslender. Méthodes numériques pour la décomposition et la minimisation de fonctions non différentiables. Numer. Math., 18:213–223, 1971/72.
- Iterative Methods for Approximate Solution of Inverse Problems, volume 577 of Mathematics and Its Applications. Springer, Dordrecht, 2004.
- Parallel and distributed computation: numerical methods. Athena Scientific, Belmont, MA, 2014. Originally published by Prentice-Hall, Inc. in 1989. Includes corrections (1997).
- Barbara Blaschke-Kaltenbacher. Some Newton Type Methods for the Regularization of Nonlinear Ill–posed Problems. PhD thesis, University of Linz, 1996.
- Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems. SIAM J.Numer.Anal., 44:153–182, 2006.
- Alberto P Calderón. On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, 25:65–73, 1980.
- G. Chavent and K. Kunisch. On weakly nonlinear inverse problems. SIAM J. Appl. Math., 56(2):542–572, 1996.
- A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Problems, 14:1081–1106, 1998.
- Regularization of Inverse Problems. Kluwer, Dordrecht, 1996.
- Globally convergent block-coordinate techniques for unconstrained optimization. Optim. Methods Softw., 10(4):587–637, 1999.
- A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math., 72(1):21–37, 1995.
- Jon Johnsen. Pointwise multiplication of besov and triebel-lizorkin spaces. Mathematische Nachrichten, 175(1):85–133, 1995.
- Iterative Regularization Methods for Nonlinear Problems. de Gruyter, Berlin, New York, 2008. Radon Series on Computational and Applied Mathematics.
- Barbara Kaltenbacher. Some Newton type methods for the regularization of nonlinear ill–posed problems. Inverse Problems, 13:729–753, 1997.
- Barbara Kaltenbacher. On Broyden’s method for nonlinear ill–posed problems. Numerical Functional Analysis and Optimization, 19:807–833, 1998.
- Barbara Kaltenbacher. Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton-type methods via range invariance. IMA Journal of Numerical Analysis, page drad044, 07 2023. see also arXiv:2209.12596 [math.NA].
- Barbara Kaltenbacher and Mario Previatti de Souza. Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space. Numerische Mathematik, 140:449–478, 2018. arxiv:1707.07589 [math.NA].
- Simultaneous reconstruction of sound speed and nonlinearity parameter in a paraxial model of vibro-acoustography in frequency domain. Computational Methods in Applied Mathematics, 2023. to appear; see also arXiv:2303.17236 [math.NA].
- On the simultanenous reconstruction of two space dependent coefficients in acoustic nonlinearity parameter tomography. Inverse Problems, 39:105001, 2023. see also arXiv:2210.08063 [math.NA].
- Andreas Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences. Springer International Publishing, 3rd ed. edition, 2021.
- Z. Q. Luo and P. Tseng. On the convergence of the coordinate descent method for convex differentiable minimization. J. Optim. Theory Appl., 72(1):7–35, 1992.
- Linear and Nonlinear Inverse Problems with Practical Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012.
- Finite-dimensional approximation of Tikhonov regularized solutions of nonlinear ill-posed problems. Numer. Funct. Anal. Optim., 11(1-2):85–99, 1990.
- Iterative solution of nonlinear equations in several variables. Academic Press, New York-London, 1970.
- Gauss–newton method for solving linear inverse problems with neural network coders. Sampl. Theory Signal Process. Data Anal., 21(2), 2023.
- Regularization Methods in Banach Spaces. Walter de Gruyter, Berlin, 2012.
- Frederic Weidling. Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs. Universitätsverlag Göttingen, Göttingen, 2019.