A coupled complex boundary expanding compacts method for inverse source problems
-
Ye Zhang
,
Rongfang Gong
,
Mårten Gulliksson
Abstract
In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be determined by additional boundary data. This problem is ill-posed since the dimensionality of the boundary is lower than the dimensionality of the inner domain. To overcome the ill-posed nature, using the a priori information (sourcewise representation), and based on the coupled complex boundary method, we propose a coupled complex boundary expanding compacts method (CCBECM). A finite element method is used for the discretization of CCBECM. The regularization properties of CCBECM for both the continuous and discrete versions are proved. Moreover, an a posteriori error estimate of the obtained finite element approximate solution is given and calculated by a projected gradient algorithm. Finally, numerical results show that the proposed method is stable and effective.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11401304
Award Identifier / Grant number: 11571311
Funding source: Knowledge Foundation
Award Identifier / Grant number: 20170059
Funding statement: The work of the first author is supported by the Alexander von Humboldt foundation through a postdoctoral researcher fellowship. The work of the second author is supported by the Natural Science Foundation of China (grant no. 11401304) and the Fundamental Research Funds for the Central Universities (grant no. NS2018047). The work of the third author is supported by the Swedish Knowledge Foundation (grant no. 20170059). The work of the last author is supported by the Natural Science Foundation of China (grant no. 11571311).
A Appendix
A.1 Proof of Lemma 7
Note that
where the functional ξ is defined in (2.13). Since
Multiply the PDE in (2.20) by
Taking the partial derivative of (2.20) and (A.2) with respect to
and
Further, multiply (A.4) by
Adding (A.3) and (A.5), we obtain
On the other hand, it is not difficult to show that ξ in (2.13) satisfies
by noting that
A.2 Proof of Theorem 2
Similarly to Theorem 5, it is not difficult to show the well-posedness of DCCBECM. However, the regularization property of the approximate solution
Let
holds for all
and
being the unique weak solutions of
and
respectively.
Further, let
Using (3.8) and noticing that
which implies that
by noting that (3.9). Here,
Set
Then
Multiply (A.11) by
Similarly, multiply (A.12) with
Combine (A.13) and (A.14) to get
Therefore, from (A.9), (A.10) and the equality above, we obtain
Similarly, replace
since
by identity (A.15).
On the other hand, similarly to Theorem 1, the following estimator holds for the solution of the adjoint problem (2.20):
Combine the above inequality and (3.1) and (3.3) to derive that for any
Combining (A.17) and (A.18), we can deduce that
Define
which implies
Consequently, combining inequalities (A.16) and (A.19), we obtain
which implies that
and the definition of
we can conclude that
which implies that
Acknowledgements
We express our gratitude to the anonymous reviewers whose valuable comments and suggestions lead to an improvement of the manuscript.
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© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach
- On an inverse spectral problem for one integro-differential operator of fractional order
- Parameter identification for the linear wave equation with Robin boundary condition
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- On the asymptotic study of transmission problem in a thin domain
- A coupled complex boundary expanding compacts method for inverse source problems
- Contrast enhanced tomographic reconstruction of vascular blood flow with first order and second order adjoint methods
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Articles in the same Issue
- Frontmatter
- Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach
- On an inverse spectral problem for one integro-differential operator of fractional order
- Parameter identification for the linear wave equation with Robin boundary condition
- Numerical resolution of optimal control problem for the in-stationary Navier–Stokes equations
- On the asymptotic study of transmission problem in a thin domain
- A coupled complex boundary expanding compacts method for inverse source problems
- Contrast enhanced tomographic reconstruction of vascular blood flow with first order and second order adjoint methods
- On an asymmetric backward heat problem with the space and time-dependent heat source on a disk
- Semi-heuristic parameter choice rules for Tikhonov regularisation with operator perturbations
- The enclosure method for inverse obstacle scattering over a finite time interval: V. Using time-reversal invariance
