zbMATH — the first resource for mathematics

Identification of piecewise constant filtration parameters and boundaries of their constancy domains. (English. Russian original) Zbl 1377.93059
Autom. Remote Control 78, No. 8, 1404-1416 (2017); translation from Avtom. Telemekh. 2017, No. 8, 60-75 (2017).
Summary: Consideration is given to the numerical solution of the problem of parametric identification of the processes obeying the parabolic equations using an example of the processes of underground oil filtration. The identified parameters belong to the given functional classes such as the piecewise constant and piecewise linear functions. In the problem, not only to determine the values of the coefficients is needed, but also to identify the constancy boundaries of the coefficients. For numerical solution of the problem, an approach is suggested based on reduction of the initial problem to that of finite-dimensional optimization with a special structure of constraints. Formulas are obtained for the gradient of the objective functional in the discretized problem allowing one to apply the efficient methods of first-order optimization. The results of numerical experiments on the model problems are presented.
93B30 System identification
35K20 Initial-boundary value problems for second-order parabolic equations
49K45 Optimality conditions for problems involving randomness
49M25 Discrete approximations in optimal control
Full Text: DOI
[1] Charnii, I.A., Podzemnaya gidpogazodinamika (Underground Hydro-Gas Dynamics), Moscow: Gostoptekhizdat, 1963.
[2] Aziz, Kh. and Settari, E., Matematicheskoe modelirovanie plastovykh sistem (Mathematical Modeling of Reservoir Systems), Moscow: Nedra, 1982.
[3] Virnovskii, G.A.; Levitan, E.I., Identification of a two-dimensional model of the flow of a homogeneous liquid in a porous medium, USSR Comput. Math. Math. Phys., 30, 64-70, (1990) · Zbl 0850.76725
[4] Khairullin, M.Kh., Solution of the inverse problems of Subterranean hydromechanics by Tikhonov regularizing algorithms, USSR Comput. Math. Math. Phys., 26, 90-92, (1986) · Zbl 0637.76102
[5] Aida-zade, K.R.; Pashaev, E.T.; Babaeva, S.R., Numerical method of parametric identification in the problem of subsurface hydromechanics, Izv. Nats. Akad. Nauk Azerb., Ser. FTMN, 1-2, 30-35, (1995)
[6] Aida-zade, K.R.; Pashaev, E.T., Problem of identification of the parameter constancy domains, Izv. Nats. Akad. Nauk Azerb., Ser. FTMN, 6, 8-13, (1997)
[7] Rahimov, A.B., On an approach to inverse oil filtration problems, Izv. Nats. Akad. Nauk Azerb., Ser. FTMN, 24, 274-279, (2004)
[8] Kolehmainen, V.; Arridge, S.R.; Lionheart, W.R.B.; etal., Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data, Inverse Probl. J., 15, 1375-1391, (1999) · Zbl 0936.35195
[9] Ayda-zade, K.R.; Bagirov, A.G., On the problem of spacing of oil wells and control of their production rates, Autom. Remote Control, 67, 44-53, (2006) · Zbl 1126.49326
[10] Akhmetzyanov, A.V.; Kulibanov, V.N., Optimal choice of coordinates for oil well drilling, Autom. Remote Control, 63, 1699-1706, (2002) · Zbl 1107.49305
[11] Akhmetzyanov, A.V.; Kulibanov, V.N., Optimal placement of sources for stationary scalar fields, Autom. Remote Control, 60, 797-804, (1999) · Zbl 1273.49028
[12] Litman, A.; Lesselier, D.; Santosa, F., Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set, Inverse Probl. J., 14, 685-706, (1998) · Zbl 0912.35158
[13] Litman, A., Reconstruction by level sets of n-ary sattering obstacles, Inverse Probl. J., 21, s131-s152, (2005) · Zbl 1112.78015
[14] Vasil’ev, F.P., Metody optimizatsii (Methods of Optimization), Moscow: Faktorial Press, 2002.
[15] Samarskii, A.A. and Vabishchevich, P.N., Chislennye metody resheniya obratnykh zadach matematicheskoi fiziki (Numerical Methods to Solve the Inverse Problems of Mathematical Physics), Moscow: LKI, 2009.
[16] Aida-zade, K.R.; Rahimov, A.B., An approach to numerical solution of some inverse problems for parabolic equations, Inverse Probl. Sci. Eng., 22, 96-111, (2014) · Zbl 1312.65150
[17] Aida-zade, K.R.; Abdullayev, V.M., Solution to a class of inverse problems for system of loaded ordinary differential equations with integral conditions, J. Inverse Ill-Posed Probl., 24, 543-558, (2016) · Zbl 1352.65192
[18] Aida-zade, K.R., Study and numerical solution of finite difference approximations of distributed- system control problems, USSR Comput. Math. Math. Phys., 29, 15-21, (1989) · Zbl 0706.49021
[19] Aida-zade, K.R., Investigation of nonlinear optimization problems of networks structure, Autom. Remote Control, 51, 135-145, (1990) · Zbl 0717.49028
[20] Aida-zade, K.R.; Evtushenko, Yu.G., Fast computer-aided automatic differentialtion, Mat. Model., 1, 120-131, (1989) · Zbl 0972.65504
[21] Aida-zade, K.R.; Rahimov, A.B., On an approach to problems of parametric identification in the distributed systems, Izv. Nats. Akad. Nauk Azerb., Ser. FTMN, 25, 145-150, (2005)
[22] Samarskii, A.A., Teoriya raznostnykh skhem, Moscow: Nauka,1977. Translated under the title The Theory of Difference Schemes, New York: Marcel Dekker,2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.