Nonlinear Modifications of Perturbation Theory with Applications to Complex System Analysis - Uchaikin, V. V.; Litvinov, V. A.; Kozhemjakina, E.V. (Discontinuity, Nonlinearity, and Complexity, 2026)
The article reviews some nonlinear generalizations of the perturbation theory performed by various authors (including some co-authors of this article) in various years, and can be useful in analyzing complex systems and inverse problems. There is considered the roblem of widening the range of perturbation theory applicability with simultaneous reducing the complexity of computational algorithm. The consideration is based on the duality principle, including adjoint (in the Lagrange sense) operators and importance unctions. The article contains an exposition of perturbation theory of higher orders, variational methods of higher orders, symmetry transformation and some other question. Special attention is paid to such a promising approach as multi-reference methods. It is shown that this approach extends the range of applicability of perturbation theory and saves calculations time. To demonstrate details of the method, some examples of transport problems from reactor engineering and astrophysics are considered.