Abstract
We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures.
R. Jaroudi—Support by EU funding under the program ALYSSA (ERASMUS MUNDUS Action 2, Lot 6) is gratefully acknowledged.
F. Åström—Support by the German Science Foundation and the Research Training Group (GRK 1653) is gratefully acknowledged.
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Notes
- 1.
Note that the PDE models considered in this work have their origin in ordinary differential equation (ODE) models and the terminology is adopted from ODE models with corresponding source terms as their solution. For example, \(u_t = \rho u\) has the exponential function \(u = C e^{\rho t}\) as a solution. Furthermore, the logistic equation \(u_t = u (1-u)\) has the logistic function \(u = 1/(1+e^{-t})\) as a solution.
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Jaroudi, R., Baravdish, G., Åström, F., Johansson, B.T. (2016). Source Localization of Reaction-Diffusion Models for Brain Tumors. In: Rosenhahn, B., Andres, B. (eds) Pattern Recognition. GCPR 2016. Lecture Notes in Computer Science(), vol 9796. Springer, Cham. https://doi.org/10.1007/978-3-319-45886-1_34
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