Abstract
Nonlinear ill-posed problems arise in many inverse problems in Hilbert space. We investigate the homotopy method, which can obtain global convergence to solve the problems. The “homotopy with Tikhonov regularization” and “homotopy without derivative” are developed in this paper. The existence of the homotopy curve is proved. Several numerical schemes for tracing the homotopy curve are given, including adaptive tracing skills. Compared to the regularized Newton method, the numerical examples show that our proposed methods are stable and effective.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11626230
Award Identifier / Grant number: 11771180
Funding statement: The first author was supported by the National Natural Science Foundation of China (NSFC) Grant 11626230, and the second author was supported by the National Natural Science Foundation of China (NSFC) Grant 11771180.
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Numerical solution of a source identification problem: Almost coercivity
- Invertibility and stability for a generic class of radon transforms with application to dynamic operators
- A class of homotopy with regularization for nonlinear ill-posed problems in Hilbert space
- Inverse nodal problems for integro-differential operators with a constant delay
- Unique continuation for a reaction-diffusion system with cross diffusion
- On solenoidal-injective and injective ray transforms of tensor fields on surfaces
- The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius
- Identification of point sources in an elliptic equation from interior measurements: Application to a seawater intrusion problem
- Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints
- On the travel time tomography problem in 3D
Articles in the same Issue
- Frontmatter
- Numerical solution of a source identification problem: Almost coercivity
- Invertibility and stability for a generic class of radon transforms with application to dynamic operators
- A class of homotopy with regularization for nonlinear ill-posed problems in Hilbert space
- Inverse nodal problems for integro-differential operators with a constant delay
- Unique continuation for a reaction-diffusion system with cross diffusion
- On solenoidal-injective and injective ray transforms of tensor fields on surfaces
- The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius
- Identification of point sources in an elliptic equation from interior measurements: Application to a seawater intrusion problem
- Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints
- On the travel time tomography problem in 3D