On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911544765317120 |
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| author | Kaltenbacher, Barbara |
| author_facet | Kaltenbacher, Barbara |
| contents | In this paper we provide some uniqueness results for the (multi-)coefficient identification problem of reconstructing the spatially varying spin density as well as the spin-lattice and spin-spin relaxation times and the local field inhomogeneity in the Bloch-Torrey equation, as relevant in magnetic resonance imaging MRI. To this end, we follow two approaches: (a) Relying on sampling of the k-space and (approximately) explicit reconstruction formulas in the simplified (Bloch) ODE setting, along with perturbation estimates; (b) Relying on infinite speed of propagation due to diffusion. The results on well-posendess and Lipschitz continuous differentiability of the coefficient-to-state map derived for this purpose, are expected to be useful also in the convergence analysis of reconstruction schemes as well in mathematical optimization of the experimental design in MRI. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging Kaltenbacher, Barbara Analysis of PDEs 35R30, 35K20, 95C55 In this paper we provide some uniqueness results for the (multi-)coefficient identification problem of reconstructing the spatially varying spin density as well as the spin-lattice and spin-spin relaxation times and the local field inhomogeneity in the Bloch-Torrey equation, as relevant in magnetic resonance imaging MRI. To this end, we follow two approaches: (a) Relying on sampling of the k-space and (approximately) explicit reconstruction formulas in the simplified (Bloch) ODE setting, along with perturbation estimates; (b) Relying on infinite speed of propagation due to diffusion. The results on well-posendess and Lipschitz continuous differentiability of the coefficient-to-state map derived for this purpose, are expected to be useful also in the convergence analysis of reconstruction schemes as well in mathematical optimization of the experimental design in MRI. |
| title | On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging |
| topic | Analysis of PDEs 35R30, 35K20, 95C55 |
| url | https://arxiv.org/abs/2506.13708 |