TY - JOUR

T1 - Backward Iterations for Solving Integral Equations with Polynomial Nonlinearity

AU - Ermakov, S. M.

AU - Surovikina, T. O.

PY - 2022/3/1

Y1 - 2022/3/1

N2 - Abstract: The theory of adjoint operators is widely used in solving applied multidimensional problems with the Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.I. Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods are mostly used for that purpose. The results for Lyapunov–Schmidt nonlinear polynomial equations are obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area remain open. New results about dual processes used for solving polynomial equations with the Monte Carlo method are presented. In particular, the adjoint Markov process for the branching process and corresponding unbiased estimate of the functional of the solution to the equation are constructed in the general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.

AB - Abstract: The theory of adjoint operators is widely used in solving applied multidimensional problems with the Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.I. Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods are mostly used for that purpose. The results for Lyapunov–Schmidt nonlinear polynomial equations are obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area remain open. New results about dual processes used for solving polynomial equations with the Monte Carlo method are presented. In particular, the adjoint Markov process for the branching process and corresponding unbiased estimate of the functional of the solution to the equation are constructed in the general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.

KW - adjoint equations

KW - balance equation

KW - dual estimate

KW - Lyapunov–Schmidt nonlinear equations

KW - Monte Carlo method

UR - http://www.scopus.com/inward/record.url?scp=85131365521&partnerID=8YFLogxK

U2 - 10.1134/S1063454122010046

DO - 10.1134/S1063454122010046

M3 - Article

AN - SCOPUS:85131365521

VL - 55

SP - 16

EP - 26

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -