edoc: No conditions. Results ordered -Date Deposited. 2024-10-08T16:06:08ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2020-06-24T15:20:58Z2020-07-06T14:56:45Zhttps://edoc.unibas.ch/id/eprint/75283This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/752832020-06-24T15:20:58ZAdaptive Eigenspace for Multi-Parameter Inverse Scattering Problems A nonlinear optimization method is proposed for inverse scattering problems in the frequency domain, when the unknown medium is characterized by one or several spatially varying parameters. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type method combined with frequency stepping. Instead of a grid-based discrete representation, each parameter is projected to a separate finite-dimensional subspace, which is iteratively adapted during the optimization. Each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori. The (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi-parameter problems, including the well-known Marmousi model from geosciences. Marcus J. GroteUri Nahum2019-03-28T09:51:36Z2019-04-22T14:40:36Zhttps://edoc.unibas.ch/id/eprint/69952This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699522019-03-28T09:51:36ZAdaptive eigenspace method for inverse scattering problems in the frequency domainA nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method. Marcus GroteMarie KrayUri Nahum2019-03-28T09:51:28Z2019-04-20T15:17:24Zhttps://edoc.unibas.ch/id/eprint/69934This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699342019-03-28T09:51:28ZAdaptive eigenspace regularization for inverse scattering problemsA nonlinear optimization method is proposed for inverse scattering problems in the frequency domain, when the unknown medium is characterized by one or several spatially varying parameters. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type method combined with frequency stepping. Instead of a grid-based discrete representation, each parameter is projected to a separate finite-dimensional subspace, which is iteratively adapted during the optimization. Each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori. The (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi-parameter problems, including the well-known Marmousi problem from geophysics. Marcus GroteUri Nahum2018-04-16T14:52:21Z2018-04-16T14:52:21Zhttps://edoc.unibas.ch/id/eprint/63437This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/634372018-04-16T14:52:21ZAdaptive Eigenspace Inversion for the Helmholtz EquationMarcus GroteUri Nahum2017-10-16T06:45:18Z2017-10-16T06:45:18Zhttps://edoc.unibas.ch/id/eprint/53653This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/536532017-10-16T06:45:18ZAdaptive eigenspace method for inverse scattering problems in the frequency domainA nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method. Marcus J. GroteMarie KrayUri Nahum2016-12-22T10:00:19Z2018-01-22T15:52:42Zhttps://edoc.unibas.ch/id/eprint/44674This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/446742016-12-22T10:00:19ZAdaptive eigenspace for inverse problems in the frequency domainInverse scattering problems are used in a vast number of applications, such as geophysical exploration and medical imaging. The goal is to recover unknown media using wave prop- agation. The inverse problem is designed to minimize simulated data with observation data, using partial differential equations (PDE) as constrains. The resulting minimiza- tion problem is often severely ill-posed and contains a large number of local minima. To tackle ill-posedness, several optimization and regularization techniques have been explored. However, the applications are still asking for improvement and stability. In this thesis, a nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton- type method. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. We actually show how to build an AE from the gradients of Tikhonov-regularization functionals. Both analytical and numerical evidence underpin the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion (AEI) method. We also consider missing frequency data and apply the AEI to the multi-parameter inverse scattering problem. Uri Nahum