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| Main Authors: | , |
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| Format: | Preprint |
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2015
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| Online Access: | https://arxiv.org/abs/1508.07854 |
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| _version_ | 1866910323187908608 |
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| author | Munch, Arnaud Souza, Diego |
| author_facet | Munch, Arnaud Souza, Diego |
| contents | We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $Ω\times (0,T)$ - $Ω$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the parabolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. The well-posedness of this mixed formulation - in particular the inf-sup property - is a consequence of classical energy estimates. We then reproduce the arguments to a linear first order system, involving the normal flux, equivalent to the linear parabolic equation. The method, valid in any dimension spatial dimension $N$, may also be employed to reconstruct solution for boundary observations.
With respect to the hyperbolic situation considered in \cite{NC-AM-InverseProblems} by the first author, the parabolic situation requires - due to regularization properties - the introduction of appropriate weights function so as to make the problem numerically stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1508_07854 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Inverse problems for linear parabolic equations using mixed formulations - Part 1 : Theoretical analysis Munch, Arnaud Souza, Diego Optimization and Control We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $Ω\times (0,T)$ - $Ω$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the parabolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. The well-posedness of this mixed formulation - in particular the inf-sup property - is a consequence of classical energy estimates. We then reproduce the arguments to a linear first order system, involving the normal flux, equivalent to the linear parabolic equation. The method, valid in any dimension spatial dimension $N$, may also be employed to reconstruct solution for boundary observations. With respect to the hyperbolic situation considered in \cite{NC-AM-InverseProblems} by the first author, the parabolic situation requires - due to regularization properties - the introduction of appropriate weights function so as to make the problem numerically stable. |
| title | Inverse problems for linear parabolic equations using mixed formulations - Part 1 : Theoretical analysis |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/1508.07854 |