The unknown is the position-dependent storage coefficient s[.] which appears in a linear, partial differential equation of parabolic type with respect to hydraulic potential p[..,.] and multiplies the time derivative of p[.,.]. The spatial derivatives of p.,.] appear in the form div ( a grad p ), where the position-dependent conductivity a[.] is known. The identification of s[.] from knowledge of the measured hydraulic potential z[.,.] relies on the minimisation of the output error functional. The minimisation algorithm is derived from the variational formulation of the inverse problem [J. L. Lions, 1968; G. Chavent, 1971]. A discrete gradient algorithm and the corresponding computer code are described. A stopping ctiterion based on data noise ``energy'' is defined. Numerical and graphical results obtained from (computationally) exact and noisy data are presented. The methods and results apply to the modeling of aquifers and of petroleum reservoirs.

Crosta, G. (1985). On the identification of a spatially varying coefficient appearing in a parabolic partial differential equation. In F. Kappel, K. Kunisch, W. Schappacher (a cura di), Distributed parameter systems (pp. 92-108). Berlin : Springer Verlag [10.1007/BFb0005646].

On the identification of a spatially varying coefficient appearing in a parabolic partial differential equation

Crosta, GFF
1985

Abstract

The unknown is the position-dependent storage coefficient s[.] which appears in a linear, partial differential equation of parabolic type with respect to hydraulic potential p[..,.] and multiplies the time derivative of p[.,.]. The spatial derivatives of p.,.] appear in the form div ( a grad p ), where the position-dependent conductivity a[.] is known. The identification of s[.] from knowledge of the measured hydraulic potential z[.,.] relies on the minimisation of the output error functional. The minimisation algorithm is derived from the variational formulation of the inverse problem [J. L. Lions, 1968; G. Chavent, 1971]. A discrete gradient algorithm and the corresponding computer code are described. A stopping ctiterion based on data noise ``energy'' is defined. Numerical and graphical results obtained from (computationally) exact and noisy data are presented. The methods and results apply to the modeling of aquifers and of petroleum reservoirs.
No
Scientifica
Capitolo o saggio
inverse problems; storage coefficient; parabolic partial differential equation; output error; optimal control; variational formulation; functional gradient; discrete gradient; minimisation; stopping criteria; numerical results; noise figure;
English
Distributed parameter systems
978-3-540-15872-1
ZentralBlatt fuer Mathematik: review Zbl 0602.93021 signed by Martin Brokate (Muenchen, DE) ``The author considers the parabolic equation in two space dimensions sp_t=div(a grad p)+f with standard initial and Dirichlet boundary conditions. Here a and f are given, s is an unknown function. He is interested in the output least squares solution s* of min J_e(s)=∫ΩdtdD(p(s)−z)^2 where z is given. For numerical solution, he uses a projected gradient method. He discusses his algorithm in detail and presents results for some test problems chosen in order to have simple closed form solutions. He also compares with the case where this data are subject to noise.'' --- Mathematical Reviews: citation MR0897553. --- Research supported by the following projects: 1) ``ICES'' (Identificazione e Controllo dei Sistemi, Elaborazione dei Segnali) funded by the Italian Ministry of Education and coordinated by Prof. Giovanni Marro (Universita` di Bologna, IT) and Prof. Edoardo Mosca (Universita` di Firenze, IT); 2) ``Strumenti informatici e modelli matematici per la gestione e il controllo dell'acqua della falda Milanese'' funded by the Department of Ecology and Environmental Hygiene - City of Milan. Some of the computational results were obtained by Ms. Chiara Bertamoni [Ref. 1] while working at her Master thesis in Physics, Universita` di Milano, IT. Online ISBN 978-3-540-39661-1
Crosta, G. (1985). On the identification of a spatially varying coefficient appearing in a parabolic partial differential equation. In F. Kappel, K. Kunisch, W. Schappacher (a cura di), Distributed parameter systems (pp. 92-108). Berlin : Springer Verlag [10.1007/BFb0005646].
Crosta, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10281/93808
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