Speciality:
01.01.07 (Computing mathematics)
Birth date:
12.12.1959
E-mail: ,
Website: https://www.inm.ras.ru/persons/yuri.nechepurenko/ Keywords: spectral analysis; integral performance creterions for dichotomy; separation of spectrum; spectral portraits; generalized Lyapunov equations; singular functions; transient regimes; reactivity; noncoservative systems; aeroelasticity problems; stability of hydrodynamic flows; modal analysis; measures of stability.
Subject:
An interpolation approach to the construction of fast numerically stable algorithms was proposed for the matrix-vector multiplication. Analytical properties of singular functions of polynomial and analytical matrix pencils were been studied as well as the connection between the singular functions and spectral characteristics of the pencils. It was proposed the method of singular functions which reduces the eigenvalue problem for a matrix pencil to calculation of singular vectors corresponding to the smallest singular values of the pencil at fixed values of the parameter. It was proposed a new technology of numerical spectral analysis which is based on the Schur decomposition and required for a detailed spectral analysis of the ODE's system $du/dt=Au$ asymptotically $1/n$ arithmetic operations of the traditional technology, where $n$ means the order of the matrix $A$. New norm bounds for the matrix exponential and the Green matrix that significantly more precise than the known ones were obtained. A number of papers (in collaboration with S. K. Godunov) were concerned with methods based on the integral performance criterions for dichotomy. Particularly, a new approach to proofs of the existence of low-dimensional main parts was proposed for finite-dimensional analoges of the operators whose inverses exist and are finite-order operators according to Keldish's difinition. This approach is significantly simpler than traditional one (based on entire function theory) and makes possible to obtain more precise estimates of the resolvent norm. Based on the integral performance criterions for dichitomy new covergence rate estimates were obtained for a Newton method for computing invariant subspaces of finite-dimentional analoges of a partial differential operator. These estimates allow to connect the rate of convergence with properties of the initial operator which are usually studied in the theory of differential operators.
Main publications:
Nechepurenko Yu. M. On the singular-function approach to eigenproblems // Russian J. Numer. Anal. Math. Modelling, 1998, 13(3), 219–233.
Nechepurenko Yu. M. New spectral analysis technology based on the Schur decomposition // Russian J. Numer. Anal. Math. Modelling, 1999, 14(3), 265–274.
ISTINA
