Variational interpolation of functionals in transport theory inverse problems

V. V. Uchaikin   ; V. A. Litvinov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 3, pp. 363-380

Résumé

It is known that the dual representation of problems (through solutions of the main and the conjugate in the Lagrange sense equations) allows one to formulate the perturbation theory serving as basement for the successive approximation method in the inverse problems theory. If, according to preliminary predictions, the solution of an inverse problem (for example, the structure of the medium of interest) belongs to a certain set $A$, then selecting a suitable (trial, reference) element $a_0$ as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of the solution of the direct problem in this domain and find a subset $A_0$ that best matches the measurement data. However, as the accuracy requirements increase, the domain $A_0$ of the first approximation is rapidly narrowing, expanding it by adding higher terms of the expansion complicates the decision procedure. For this reason, a number of works have been devoted to the search for unperturbed approaches. Among them is the method of variational interpolation (VI-method), in the frame of which not one, but several problems $a_1, a_2,\dots ,a_n$ an are used in order to compose from their solutions the desired one. The functional of interest is represented in the stationary form, and the coefficients of the expansion are determined from the condition of stationarity of the bilinear form. This paper demonstrates the application of VI-method to solving inverse problems in the frame of simple model situations associated with cosmic rays astrophysics.

Bibliographie

[1] Tarantola A., Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, USA, 2005 | MR | Zbl

[2] Banks H. T., Bortz D. M., “Inverse problems for a class of measure dependent dynamical systems”, J. Inverse and Ill-posed Problems, 13 (2005), 103–121 | DOI | MR | Zbl

[3] Linde N., Renard P., Mukerji T., Caers J., “Geological realism in hydrogeological and geophysical inverse modeling: a review”, Advances in Water Resources, 86 (2015), 86–101 | DOI

[4] Adelman H. M., Haftka R. T., “Sensitivity analysis of discrete structural systems”, AIAA Journal, 24 (1986), 823–832 | DOI

[5] Banks H. T., Dediu S., Nguyen H. K., “Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space”, Math. Biosci. and Engineering, 4 (2007), 403–430 | DOI | MR | Zbl

[6] Usachev L. N., “Uravneniya dlya cennosti neitronov, kinetika reaktorov i teoriya vozmuschenii”, Reaktorostroenie i teoriya reaktorov, Izd-vo AN SSSR, M., 1955, 251–268

[7] Marchuk G. I., Orlov V. V., “K teorii sopryazhyonnyh funkcii”, Neitronnaya fizika, Gosatomizdat, M., 1961, 30–45

[8] Lewins J. D., Importance; the Adjoint Function, Pergamon, Oxford, 1965

[9] Stumbur E. F., Primenenie teorii vozmuschenii v fizike yadernyh reaktorov, Atomizdat, M., 1976

[10] Marchuk G. I., Metody raschyota yadernyh reaktorov, Gosatomizdat, M., 1961

[11] Marchuk G. I., Methods of Numerical Mathematics, Springer Verlag, 1982 | MR | Zbl

[12] Marchuk G. I., Adjoint Equations and Analysis of Complex Systems, Springer-Scence+Business Media, B.V., 1995 | MR

[13] Shihov S. B., Shishkov L. K., “Teoriya vozmuschenii vysshih poryadkov dlya resheniya nekotoryh zadach raschyota reaktorov”, Teoriya i fizika reaktorov, ed. L.N. Yurovoi, Atomizdat, M., 1967, 3–29

[14] Uchaikin V. V., “Variational interpolation method in nuclear-engineering calculations”, Soviet Atomic Energy, 67:1 (1989), 558–560 | DOI

[15] Uchaikin V. V., Litvinov V. A., “Variational interpolation method in radiative transfer theory”, Atmosphere and Oceanics Optics, 2:1 (1989), 27–30

[16] Litvinov V. A., “Variacionnoe interpolirovanie v probleme chuvstvitel'nosti harakteristik kaskadnyh processov”, Yadernaya fizika, 56:2 (1993), 244–254

[17] Uchaikin V. V., Ryzhov V. V., Stohasticheskaya teoriya perenosa chastic vysokih energii, Nauka. Sib. otd-nie, Novosibirsk, 1988

[18] Kolchuzhkin A. M., Uchaikin V. V., Vvedenie v teoriyu prohozhdeniya chastic cherez veschestvo, Atomizdat, M., 1978

[19] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, Dover Publications, 1965 | MR

[20] Lagutin A. A., Litvinov V. A., Uchaikin V. V., Teoriya chuvstvitel'nosti v fizike kosmicheskih luchei, Izd-vo AGU, Barnaul, 1995

[21] Uchaikin V. V., Lagutin A. A., Stohasticheskaya cennost', Energoatomizdat, M., 1993

[22] Lagutin A. A., Uchaikin V. V., Metod sopryazhyonnyh uravnenii v teorii perenosa kosmicheskih luchei vysokih energii, Izd-vo AGU, Barnaul, 2013

[23] Uchaikin V. V., Litvinov V. A., Kozhemjakina E. V., “Nonlinear modifications of perturbation theory with applications to complex system analysis”, Discontinuity, Nonlinearity, and Complexity, 7:3 (2018), 327–354 | DOI | MR | Zbl