Overview
Alexander Bukhgeym, Professor
Inverse Problems; PhD, Novosibirsk, 1974
Information
- PDEs
- Inverse and Ill-Posed Problems
- Tomography
Dr. Bukhgeym research interests lie in the area of inverse problems and integral geometry. In inverse problems one tries to reconstruct the causes that yield some observable quantities. Example: the cause for the orbits of the planets to be elliptical is Newton's universal gravitation law. If a mathematical model is given by a partial differential equation with some initial and/or boundary conditions and has only one solution, then the complementary traces of this solution on the boundary bring us the information about coefficients of this equation, or its right hand side, or initial conditions, etc. These coefficients or the right hand side or the initial conditions can be considered as the reason that causes corresponding observable traces. Integral geometry (tomography) problems consist in determining some function or more generally vector or tensor field on a manifold given its integrals over a prescribed family of submanifolds. Here the cause is a tensor field that yields corresponding integrals over submanifolds. Such kind of inverse problems arise in geophysical exploration, material testing, medical imaging, etc.
- Partial Differential Equations
- Linear Algebra
- Complex Analysis
- Inverse and Ill-Posed Problems
- Applied Functional Analysis
Selected Publications
- A.L. Bukhgeim, Inverse gravimetry approach to attenuated tomography, Contemporary Mathematics, V. 559, 2011, pp.49-63.
- A.L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, Journal of Inverse & Ill-Posed Problems,V. 16, 2008, pp. 19-33.
- A.L. Bukhgeim, Dyatlov G.V., Uhlmann G., Unique continuation & controllability for hyperbolic equations with memory, Jounral of Inverse & Ill-Posed Problems, V. 15, 2007, pp. 587-598.
- A.L. Bukhgeim, G.V. Dyatlov, G. Uhlmann, Reconstruction of the Memory from Partial Boundary Measurements, Contemporary Mathematics, V.307, 2002, 39-46.
- A.L. Bukhgeim, G. Uhlmann, Recovering a Potential from Partial Cauchy Data, Communications in Partial Differential Equations, V.27, No.3&4, 2002, 653-668.
- A.L. Bukhgeim, J. Chen, M. Yamamoto, Stability for an inverse boundary problem of determining a part of boundary, Inverse Problems, V.15, No.4, 1999, 1021-1032.
- A.L. Bukhgeim, V.E. Andreev, A.S. Terekhov, Recovery of electron velocity distribution in vacuum photodetectors, Journal Inverse Ill-Posed Problems, V.7, No.5, 1999, 427-434.
- A.L. Bukhgeim, B.V. Kardakov, Stability for the inverse problem of finding the boundary condition, Journal Inverse Ill-Posed Problems, V.6, No.4, 1998, 309-318.
- A.L. Bukhgeim, E.V. Arbuzov, S.G. Kazantsev, Two-dimensional tomography problems and the theory of a-analytic functions, Siberian Adv. Math., V.8(4), 1998, 1-20.