Let ${\mit\Omega}\subset I\hspace{-3pt}R^2$ be bounded by two heteroclinic orbits, ${\mit\Gamma}_1, {\mit\Gamma}_2$ of the $\nabla u$-flow. Then $\nabla\cdot (c \nabla u)=0$ in ${\mit\Omega}$ implies $c\equiv 0$ in $\bar{\mit\Omega}$ [\textsc{Chicone} and \textsc{Gerlach}, 1987]. Let $u\in {\cal C}^2 ({\mit\Omega})\cap{\cal C}^0 (\bar{\mit\Omega})$ be known. The (unique) conductivity $\hat a$, which complies with $\nabla\cdot(\hat a\nabla u)=f$, can be identified by minimising with respect to $b$ the norm of $\nabla\tilde a [b] \times\nabla u - \nabla b\times\nabla p$ under constraints, where $\nabla\cdot(b\nabla p)=f$ and $\tilde a[b]\partial_j u := b\partial_j p$, $j=$ 1 \textbf{or} 2. This is an attempt at justifying the ``comparison model'' algorithm [\textsc{Scarascia} and \textsc{Ponzini}, 1972], which has seen successful practical applications to inverse hydrogeology ever since.

Crosta, G. (2015). Identification of conductivity by minimising a gradient co-linearity mismatch norm. In Final program and abstracts - SIAM conference on mathematical and computational issues in the geosciences (pp.61-61). Philadelphia : SIAM.

Identification of conductivity by minimising a gradient co-linearity mismatch norm

CROSTA, GIOVANNI FRANCO FILIPPO
2015

Abstract

Let ${\mit\Omega}\subset I\hspace{-3pt}R^2$ be bounded by two heteroclinic orbits, ${\mit\Gamma}_1, {\mit\Gamma}_2$ of the $\nabla u$-flow. Then $\nabla\cdot (c \nabla u)=0$ in ${\mit\Omega}$ implies $c\equiv 0$ in $\bar{\mit\Omega}$ [\textsc{Chicone} and \textsc{Gerlach}, 1987]. Let $u\in {\cal C}^2 ({\mit\Omega})\cap{\cal C}^0 (\bar{\mit\Omega})$ be known. The (unique) conductivity $\hat a$, which complies with $\nabla\cdot(\hat a\nabla u)=f$, can be identified by minimising with respect to $b$ the norm of $\nabla\tilde a [b] \times\nabla u - \nabla b\times\nabla p$ under constraints, where $\nabla\cdot(b\nabla p)=f$ and $\tilde a[b]\partial_j u := b\partial_j p$, $j=$ 1 \textbf{or} 2. This is an attempt at justifying the ``comparison model'' algorithm [\textsc{Scarascia} and \textsc{Ponzini}, 1972], which has seen successful practical applications to inverse hydrogeology ever since.
abstract + slide
inverse problems; seepage flow; hydraulic conductivity; identification of conductivity; variational formulation; minimisation; co-linearity
English
SIAM conference on mathematical and computational issues in the geosciences
2015
Ahkbari, D; Aizinger, V; [...] Zhang, Y; Zoccarato, C
Gerritsen, M; Lie, K-A; Pop, IS
Final program and abstracts - SIAM conference on mathematical and computational issues in the geosciences
2015
61
61
CP3 - 2
https://www.siam.org/meetings/gs15/program.php
open
Crosta, G. (2015). Identification of conductivity by minimising a gradient co-linearity mismatch norm. In Final program and abstracts - SIAM conference on mathematical and computational issues in the geosciences (pp.61-61). Philadelphia : SIAM.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/94589
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