inverse problems of mathematical physics; dynamical and spectral inverse problems; multidimensional inverse problems of acoustics, electrodynamics, elasticity; boundary control theory and its relations with inverse problems; general linear system theory; questions on operator theory related with inverse problems; theory of operator models in connection with inverse problems.
Subject:
A new approach to the inverse problems based upon their relations with the boundary control theory (the so-called BC-method) was created. The approach is of complex character: besides the control theory, the asymptotic methods (Geometrical Optics, propagation of singularities), the functional analysis (triangular factorization of operators, operator integral), and the general linear system theory (models, canonical realizations) are in use. The BC-method permits to reconstruct the Riemannian manifold of arbitrary dimension and topology trough its spectral or dynamical (wave, heat, electromagnetic) boundary data. In the case of dynamical data the reconstruction by the BC-method is time-optimal: the longer is time of observations at the boundary, the bigger is the depth of reconstruction, that is essential in applications. The algorithms based upon the BC-method demonstrate their efficiency in numerical testing.
Biography
Graduated from Faculty of Physics of the Leningrad State University in 1972 (chair of mathematical physics). Ph.D. thesis was defended in 1978. D.Sci. thesis was defended in 1992. A list of my works contains around 80 titles.
Main publications:
M. I. Belishev. Boundary control in reconstruction of manifolds and metrics (the BC-method) // Inverse Problems, 1997, 13(5), R1–R45.
М. И. Белишев, В. М. Исаков, Л. Н. Пестов, В. А. Шарафутдинов. К реконструкции метрики по внешним электромагнитным измерениям // ДАН РАН, 2000, 372(3), 298–300.
M. I. Belishev and A. K. Glasman. Dynamical inverse problem for the Maxwell system: recovering the velocity in regular zone (the BC-method) // St. Petersburg Math. J., 2001, 12(2), 279–316.
M. I. Belishev. Dynamical systems with boundary control: models and characterization of inverse data // Inverse Problems, 2001, 17, 659–682.
M. I. Belishev. On relations between spectral and dynamical inverse data // J. Inv. Ill-Posed Problems, 2001, 9(6), 1–18.
M. I. Belishev, A. V. Kaplun, “Canonical form of the $C^*$-algebra of eikonals related to a metric graph”, Izv. RAN. Ser. Mat., 86:4 (2022), 3–50; Izv. Math., 86:4 (2022), 621–666
M. I. Belishev, A. V. Kaplun, “Canonical forms of metric graph eikonal algebra and graph geometry”, Zap. Nauchn. Sem. POMI, 519 (2022), 35–66
2021
6.
M. I. Belishev, N. A. Karazeeva, “Toeplitz matrices in the BC-method for the plane domains”, Zap. Nauchn. Sem. POMI, 506 (2021), 21–35
2020
7.
M. I. Belishev, S. A. Simonov, “The wave model of a metric space with measure and an application”, Mat. Sb., 211:4 (2020), 44–62; Sb. Math., 211:4 (2020), 521–538
M. I. Belishev, T. Sh. Khabibullin, “Characterization of data in dynamical inverse problem for the 1d wave equation with matrix potential”, Zap. Nauchn. Sem. POMI, 493 (2020), 48–72
2019
9.
M. I. Belishev, A. F. Vakulenko, “On algebras of harmonic quaternion fields in ${\mathbb R}^3$”, Algebra i Analiz, 31:1 (2019), 1–17; St. Petersburg Math. J., 31:1 (2020), 1–12
M. I. Belishev, S. A. Simonov, “On an evolutionary dynamical system of the first order with boundary control”, Zap. Nauchn. Sem. POMI, 483 (2019), 41–54
M. I. Belishev, A. S. Blagoveshchensky, N. A. Karazeeva, “Simplest test for three-dimensional dynamical inverse problem (the BC-method)”, Zap. Nauchn. Sem. POMI, 483 (2019), 19–40
M. I. Belishev, A. V. Kaplun, “Eikonal algebra on a graph of simple structure”, Eurasian Journal of Mathematical and Computer Applications, 6:3 (2018), 4–33
M. I. Belishev, N. A. Karazeeva, “Simplest test for two-dimensional dynamical inverse problem (the BC-method)”, Zap. Nauchn. Sem. POMI, 471 (2018), 38–58; J. Math. Sci. (N. Y.), 243:5 (2019), 656–670
M. I. Belishev, S. A. Siminov, “Wave model of the Sturm–Liouville operator on the half-line”, Algebra i Analiz, 29:2 (2017), 3–33; St. Petersburg Math. J., 29:2 (2018), 227–248
M. I. Belishev, “Boundary control and tomography of Riemannian manifolds (the BC-method)”, Uspekhi Mat. Nauk, 72:4(436) (2017), 3–66; Russian Math. Surveys, 72:4 (2017), 581–644
M. I. Belishev, “Local boundary controllability in classes of differentiable functions for the wave equation”, Zap. Nauchn. Sem. POMI, 461 (2017), 52–64; J. Math. Sci. (N. Y.), 238:5 (2019), 591–600
M. I. Belishev, “On algebras of three-dimensional quaternionic harmonic fields”, Zap. Nauchn. Sem. POMI, 451 (2016), 14–28; J. Math. Sci. (N. Y.), 226:6 (2017), 701–710
M. I. Belishev, A. F. Vakulenko, A. Ya. Kazakov, “Evolution of wave field jumps near caustics (elementary approach)”, Zap. Nauchn. Sem. POMI, 438 (2015), 46–72; J. Math. Sci. (N. Y.), 224:1 (2017), 27–46
2014
20.
M. I. Belishev, A. L. Pestov, “Characterization of inverse data for one-dimensional two-velocity dynamical system”, Algebra i Analiz, 26:3 (2014), 89–130; St. Petersburg Math. J., 26:3 (2015), 411–440
M. I. Belishev, M. N. Demchenko, A. N. Popov, “Noncommutative geometry and the tomography of manifolds”, Tr. Mosk. Mat. Obs., 75:2 (2014), 159–180; Trans. Moscow Math. Soc., 75 (2014), 133–149
M. I. Belishev, A. V. Ivanov, “On a calculus of variations problem”, Zap. Nauchn. Sem. POMI, 426 (2014), 12–22; J. Math. Sci. (N. Y.), 214:3 (2016), 252–259
2013
23.
M. I. Belishev, “C*-algebras in reconstruction of manifolds”, Nanosystems: Physics, Chemistry, Mathematics, 4:4 (2013), 484–489
2012
24.
M. I. Belishev, M. N. Demchenko, “Dynamical system with boundary control associated with symmetric semi-bounded operator”, Zap. Nauchn. Sem. POMI, 409 (2012), 17–39; J. Math. Sci. (N. Y.), 194:1 (2013), 8–20
M. I. Belishev, “Determination of distances to virtual source from dynamical boundary data”, Zap. Nauchn. Sem. POMI, 393 (2011), 29–45; J. Math. Sci. (N. Y.), 185:4 (2012), 526–535
M. I. Belishev, “On reconstruction of Riemannian manifold via boundary data: theory and plan of numerical testing”, Zap. Nauchn. Sem. POMI, 380 (2010), 8–30; J. Math. Sci. (N. Y.), 175:6 (2011), 623–636
M. I. Belishev, A. L. Pestov, “Forward dynamical problem for Timoshenko beam”, Zap. Nauchn. Sem. POMI, 369 (2009), 16–47; J. Math. Sci. (N. Y.), 167:5 (2010), 603–621
M. I. Belishev, “Boundary control and inverse problems: one-dimensional variant of the BC-method”, Zap. Nauchn. Sem. POMI, 354 (2008), 19–80; J. Math. Sci. (N. Y.), 155:3 (2008), 343–378
M. I. Belishev, A. F. Vakulenko, “On a control problem for the wave equation in $\mathbf R^3$”, Zap. Nauchn. Sem. POMI, 332 (2006), 19–37; J. Math. Sci. (N. Y.), 142:6 (2007), 2528–2539
M. I. Belishev, S. A. Ivanov, “Determination of the parameters of the system of connected beams from dynamical boundary measurements”, Zap. Nauchn. Sem. POMI, 324 (2005), 20–42; J. Math. Sci. (N. Y.), 138:2 (2006), 5491–5502
M. I. Belishev, “On boundary controllability of dynamical system gouverned by the wave equation on a class of graphs (trees)”, Zap. Nauchn. Sem. POMI, 308 (2004), 23–47; J. Math. Sci. (N. Y.), 132:1 (2006), 11–25
M. I. Belishev, “On relations between data of dynamical and spectral inverse problems”, Zap. Nauchn. Sem. POMI, 297 (2003), 30–48; J. Math. Sci. (N. Y.), 127:6 (2005), 2353–2363
M. I. Belishev, V. M. Isakov, “On uniqueness of recovering the parameters of the Maxwell system via dynamical boundary data”, Zap. Nauchn. Sem. POMI, 285 (2002), 15–32; J. Math. Sci. (N. Y.), 122:5 (2004), 3459–3469
M. I. Belishev, S. A. Ivanov, “A local uniqueness in the dynamical inverse problem for the two–velosity system”, Zap. Nauchn. Sem. POMI, 275 (2001), 41–54; J. Math. Sci. (N. Y.), 117:2 (2003), 3910–3917
M. I. Belishev, “On a unitary transform in the space $L_2(\Omega,\mathbb R^3)$ connected with the Weyl decomposition”, Zap. Nauchn. Sem. POMI, 275 (2001), 25–40; J. Math. Sci. (N. Y.), 117:2 (2003), 3900–3909
M. I. Belishev, A. K. Glasman, “A dynamic inverse problem for the Maxwell system: reconstruction of the velocity in the regular zone (the BC-method).”, Algebra i Analiz, 12:2 (2000), 131–187; St. Petersburg Math. J., 12:2 (2001), 279–316
M. I. Belishev, A. V. Zurov, “The effects connected with coincidence of velocities in the two-velocities dynamical system”, Zap. Nauchn. Sem. POMI, 264 (2000), 44–65; J. Math. Sci. (New York), 111:4 (2002), 3645–3656
M. I. Belishev, “On the triangular factorization of isomorphisms”, Zap. Nauchn. Sem. POMI, 264 (2000), 33–43; J. Math. Sci. (New York), 111:4 (2002), 3639–3644
M. I. Belishev, S. A. Ivanov, “Characterization of data in the dynamical inverse problem for two-velocity system”, Zap. Nauchn. Sem. POMI, 259 (1999), 19–45; J. Math. Sci. (New York), 109:5 (2002), 1814–1834
M. I. Belishev, A. K. Glasman, “On the projecting in the space of solenoidal vector fields”, Zap. Nauchn. Sem. POMI, 257 (1999), 16–43; J. Math. Sci. (New York), 108:5 (2002), 642–664
M. I. Belishev, A. K. Glasman, “Visualization of waves in the Maxwell dynamical system (the BC-method)”, Zap. Nauchn. Sem. POMI, 250 (1998), 49–61; J. Math. Sci. (New York), 102:4 (2000), 4166–4174
S. A. Avdonin, M. I. Belishev, Yu. S. Rozhkov, “Dynamical inverse problem for nonselfadjoint Sturm–Liouville operator”, Zap. Nauchn. Sem. POMI, 250 (1998), 7–21; J. Math. Sci. (New York), 102:4 (2000), 4139–4148
M. I. Belishev, “On a uniqueness of the recovering low-order terms in the wave equation via dynamical boundary measurements”, Zap. Nauchn. Sem. POMI, 249 (1997), 55–76; J. Math. Sci. (New York), 101:5 (2000), 3408–3421
M. I. Belishev, A. B. Pushnitskii, “On a triangular factorization of positive operators”, Zap. Nauchn. Sem. POMI, 239 (1997), 45–60; J. Math. Sci. (New York), 96:4 (1999), 3312–3320
M. I. Belishev, “A canonical model of a dynamical system with boundary control in the inverse heat conduction problem”, Algebra i Analiz, 7:6 (1995), 3–32; St. Petersburg Math. J., 7:6 (1996), 869–890
M. I. Belishev, “The conservative model of a dissipative dynamical system”, Zap. Nauchn. Sem. POMI, 230 (1995), 21–35; J. Math. Sci. (New York), 91:2 (1998), 2711–2721
M. I. Belishev, S. A. Ivanov, “Boundary control and canonical realizations of a two-velosity dynamical system”, Zap. Nauchn. Sem. POMI, 222 (1995), 18–44; J. Math. Sci. (New York), 87:5 (1997), 3788–3805
M. I. Belishev, V. A. Ryzhov, V. B. Philippov, “A spectral variant of the VS-method: theory and numerical
experiment”, Dokl. Akad. Nauk, 337:2 (1994), 172–176; Dokl. Math., 39:7 (1994), 466–470
49.
M. I. Belishev, “On a justification of the Huygens's Rule”, Zap. Nauchn. Sem. POMI, 218 (1994), 17–24; J. Math. Sci. (New York), 86:3 (1997), 2667–2672
M. I. Belishev, A. P. Kachalov, “Operator integral in multidimensional spectral Inverse Problem”, Zap. Nauchn. Sem. POMI, 215 (1994), 9–37; J. Math. Sci. (New York), 85:1 (1997), 1559–1577
S. A. Avdonin, M. I. Belishev, S. A. Ivanov, “The controllability in a filled domain for a multidimensional wave equation with a singular boundary control”, Zap. Nauchn. Sem. POMI, 210 (1994), 7–21; J. Math. Sci., 83:2 (1997), 165–174
M. I. Belishev, A. P. Katchalov, “Boundary controls and quasiphotons in a Riemannian manifold reconstruction problem via dynamical data”, Zap. Nauchn. Sem. POMI, 203 (1992), 21–50; J. Math. Sci., 79:4 (1996), 1172–1190
S. A. Avdonin, M. I. Belishev, S. A. Ivanov, “Boundary control and a matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$”, Mat. Sb., 182:3 (1991), 307–331; Math. USSR-Sb., 72:2 (1992), 287–310
M. I. Belishev, T. L. Sheronova, “The methods of the boundary control theory in inverse problem for unhomogeneous string”, Zap. Nauchn. Sem. LOMI, 186 (1990), 37–49; J. Math. Sci., 73:3 (1995), 320–329
M. I. Belishev, M. V. Putov, “Finite-dimensional spectral inverse problem for the bundle of Hermite quadratic forms”, Zap. Nauchn. Sem. LOMI, 186 (1990), 33–36; J. Math. Sci., 73:3 (1995), 317–319
M. I. Belishev, A. P. Katchalov, “Application of boundary control theory methods to spectral inverse problem for inhomogeneous string”, Zap. Nauchn. Sem. LOMI, 179 (1989), 14–22; J. Soviet Math., 57:3 (1991), 3072–3077
M. I. Belishev, Ya. V. Kurylev, “The inverse spectral problem of the scattering of plane waves in a half-space with local inhomogeneity”, Zh. Vychisl. Mat. Mat. Fiz., 29:7 (1989), 1045–1056; U.S.S.R. Comput. Math. Math. Phys., 29:4 (1989), 56–64
M. I. Belishev, “To the Kac problem of a reconstruction of the area shape by the spectrum of the Dirichlet problem”, Zap. Nauchn. Sem. LOMI, 173 (1988), 30–41; J. Soviet Math., 55:3 (1991), 1663–1672
M. I. Belishev, “An approach to multidimensional inverse problems for the wave
equation”, Dokl. Akad. Nauk SSSR, 297:3 (1987), 524–527; Dokl. Math., 36:3 (1988), 481–484
M. I. Belishev, “Inverse spectral indefinite problem for the equation $y''+\lambda p(x)y=0$ on an interval”, Funktsional. Anal. i Prilozhen., 21:2 (1987), 68–69; Funct. Anal. Appl., 21:2 (1987), 146–148
M. I. Belishev, Ya. V. Kurylev, “Nonstationary inverse problem for the multidimensional wave equation “in the large””, Zap. Nauchn. Sem. LOMI, 165 (1987), 21–30
M. I. Belishev, “Equations of Gel'fand–Levitan type in multidimensional inverse problem for the wave equation”, Zap. Nauchn. Sem. LOMI, 165 (1987), 15–20
M. I. Belishev, Ya. V. Kurylev, “Inverse problem for acoustical scattering in space with local inhomogeneity”, Zap. Nauchn. Sem. LOMI, 156 (1986), 24–34
M. I. Belishev, “Inverse spectral problem for a trinomial recurrence formula (the case of variable sign)”, Izv. Vyssh. Uchebn. Zaved. Mat., 1982, no. 5, 70–72; Soviet Math. (Iz. VUZ), 26:5 (1982), 89–93
M. I. Belishev, “Inverse problem of the scattering of plane waves for a class of layered media”, Zap. Nauchn. Sem. LOMI, 78 (1978), 30–53; J. Soviet Math., 22:1 (1983), 1014–1031
M. I. Belishev, “Infringement of the condition for solvability of the converse problem for an inhomogeneous string”, Funktsional. Anal. i Prilozhen., 9:4 (1975), 57–58; Funct. Anal. Appl., 9:4 (1975), 318–319
M. I. Belishev, S. Yu. Dobrokhotov, I. A. Ibragimov, A. P. Kiselev, S. V. Kislyakov, M. A. Lyalinov, Yu. V. Matiyasevich, V. G. Romanov, V. P. Smyshlyaev, T. A. Suslina, N. N. Ural'tseva, “Vasilii Mikhailovich Babich (on his ninetieth birthday)”, Uspekhi Mat. Nauk, 76:1(457) (2021), 201–202; Russian Math. Surveys, 76:1 (2021), 193–194
69.
V. M. Babich, M. I. Belishev, M. N. Demchenko, G. L. Zavorokhin, V. V. Zalipaev, N. Ya. Kirpichnikova, A. P. Kiselev, D. V. Korikov, A. S. Mikhailov, S. A. Simonov, Z. A. Yanson, “Mikhail Mikhailovich Popov”, Zap. Nauchn. Sem. POMI, 506 (2021), 7–8
Спектры $C^*$-алгебр в задачах реконструкции многообразий M. I. Belishev International conference "Contemporary Problems of Mathematics, Mechanics, and Mathematical Physics" dedicated to the 150th anniversary of V. A. Steklov May 16, 2013 16:00
Boundary control and inverse problems M. I. Belishev General Mathematics Seminar of the St. Petersburg Division of Steklov Institute of Mathematics, Russian Academy of Sciences May 18, 1998
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