title: ÐÑÐ±Ð¾Ñ ÐºÐ²Ð°Ð·Ð¸Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÑ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ð¹ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ Ð¾Ð±ÑÐ°ÑÐ½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸
title: Choice of quasi-optimal values of the regularization parameter in solving the inverse problem
Creators: name: ÐÑ ÐÐ°Ð½ Ð¥ÑÐµÐ½
name: ÐÐ³ÑÐµÐ½ Ð¢Ñ Ð¸ Ð¢Ñ Ñ ÐÑÐ½Ð³
name: Ð§ÐµÑÐ½ÐµÐ½ÑÐºÐ°Ñ , ÐÑÐ´Ð¼Ð¸Ð»Ð°
name: "Ð¡Ð¸ÑÑÐµÐ¼Ð½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð· Ð² Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ð¸ Ð¸ ÑÐ¿ÑÐ°Ð²Ð»ÐµÐ½Ð¸Ð¸", Ð¼ÐµÐ¶Ð´ÑÐ½Ð°ÑÐ¾Ð´Ð½Ð°Ñ Ð½Ð°ÑÑÐ½Ð¾-Ð¿ÑÐ°ÐºÑÐ¸ÑÐµÑÐºÐ°Ñ ÐºÐ¾Ð½ÑÐµÑÐµÐ½ÑÐ¸Ñ (26; 2022; Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³)
Publisher: name: ÐÐÐÐÐ¢ÐÐ¥-ÐÐ ÐÐ¡Ð¡
Year: 2023
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
Ð¿Ð°ÑÐ°Ð¼ÐµÑÑ ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸
ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ñ Ð¢Ð¸Ñ Ð¾Ð½Ð¾Ð²Ð°
ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð¾Ð²Ð°Ð½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ
Ð¾Ð±ÑÐ°ÑÐ½Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ°
regularization parameter
Tikhonov regularization
regularized solution
inverse problem

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ ÐºÐ²Ð°Ð·Ð¸Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ Ð¾Ð±ÑÐ°ÑÐ½ÑÑ Ð·Ð°Ð´Ð°Ñ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð¢Ð¸ÑÐ¾Ð½Ð¾Ð²Ð°. Ð¦ÐµÐ»ÑÑ Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ â Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ðµ Ð¿ÑÐ¾ÑÐµÐ´ÑÑÑ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ñ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ ÑÐ°ÐºÐ¸Ð¼ Ð¾Ð±ÑÐ°Ð·Ð¾Ð¼, ÑÑÐ¾Ð±Ñ Ð¾ÑÐ¸Ð±ÐºÐ° Ð¼ÐµÐ¶Ð´Ñ ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð¾Ð²Ð°Ð½Ð½ÑÐ¼ ÑÐµÑÐµÐ½Ð¸ÐµÐ¼ Ð¸ ÑÐ¾ÑÐ½ÑÐ¼ ÑÐµÑÐµÐ½Ð¸ÐµÐ¼ Ð¾Ð±ÑÐ°ÑÐ½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð±ÑÐ»Ð° Ð¿ÑÐ¸Ð±Ð»Ð¸Ð·Ð¸ÑÐµÐ»ÑÐ½Ð¾ ÑÐ°Ð²Ð½Ð° Ð½ÑÐ»Ñ. ÐÐ»Ñ ÑÑÐ¾Ð³Ð¾, ÑÐ½Ð°ÑÐ°Ð»Ð° Ð±ÑÐ´ÐµÑ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð° Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ°Ñ Ð¼Ð¾Ð´ÐµÐ»Ñ, Ð¾Ð¿Ð¸ÑÑÐ²Ð°ÑÑÐ°Ñ Ð² Ð²Ð¸Ð´Ðµ ÑÐ¸ÑÑÐµÐ¼Ñ Ð¾Ð±ÑÐºÐ½Ð¾Ð²ÐµÐ½Ð½ÑÑ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÑ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹. ÐÐ°Ð»ÐµÐµ, Ð±ÑÐ´ÐµÑ ÑÑÐ¾ÑÐ¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð° Ð¾Ð±ÑÐ°ÑÐ½Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ° Ð² ÑÐ°Ð¼ÐºÐ°Ñ ÑÑÐ¾Ð¹ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð¼Ð¾Ð´ÐµÐ»Ð¸. ÐÐ°ÑÐµÐ¼, Ð¼ÐµÑÐ¾Ð´Ñ ÐºÐ¾Ð½ÐµÑÐ½Ð¾Ð¹ ÑÐ°Ð·Ð½Ð¾ÑÑÐ¸, Ð¼ÐµÑÐ¾Ð´ ÐºÑÐ±Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ ÑÐ¿Ð»Ð°Ð¹Ð½Ð° Ð¸ Ð¼ÐµÑÐ¾Ð´ ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð¢Ð¸ÑÐ¾Ð½Ð¾Ð²Ð° Ð±ÑÐ´ÑÑ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ñ Ð´Ð»Ñ Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¾Ð±ÑÐ°ÑÐ½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð² Ð·Ð°Ð´Ð°ÑÑ Ð¼Ð¸Ð½Ð¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ ÑÑÐ½ÐºÑÐ¸Ð¾Ð½Ð°Ð»Ð°. ÐÐ°ÐºÐ¾Ð½ÐµÑ, Ð±ÑÐ´ÐµÑ Ð¾Ð¿Ð¸ÑÐ°Ð½Ñ Ð¼ÐµÑÐ¾Ð´ Ð²ÑÐ±Ð¾ÑÐ° ÐºÐ²Ð°Ð·Ð¸Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð¸ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½ Ð¿ÑÐ¾ÑÐµÑÑ ÐµÐ³Ð¾ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð±ÑÐ´ÐµÑ Ð½Ð°Ð¹Ð´ÐµÐ½ ÐºÐ²Ð°Ð·Ð¸Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ðµ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ðµ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ° ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð°ÑÐ¸Ð¸, Ð° ÑÐ°ÐºÐ¶Ðµ ÑÐµÐ³ÑÐ»ÑÑÐ¸Ð·Ð¾Ð²Ð°Ð½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð¾Ð±ÑÐ°ÑÐ½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸. Ð Ð±ÑÐ´ÑÑÐµÐµ Ð½Ð°ÑÐ° ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð¾Ð¶ÐµÑ Ð¿Ð¾Ð¼Ð¾ÑÑ ÑÐµÑÐ°ÑÑ Ð¾Ð±ÑÐ°ÑÐ½ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð½Ð° Ð¿ÑÐ°ÐºÑÐ¸ÐºÐµ, Ð½Ðµ Ð·Ð½Ð°Ñ Ð·Ð°ÑÐ°Ð½ÐµÐµ Ð¿Ð¾Ð³ÑÐµÑÐ½Ð¾ÑÑÐ¸ Ð¸ÑÑÐ¾Ð´Ð½ÑÑ Ð´Ð°Ð½Ð½ÑÑ.
description: This work is devoted to determining the quasi-optimal regularization parameter in solving inverse problems by the Tikhonov regularization method. The purpose of this work is to construct a procedure for determining the value of the regularization parameter in such a way that the error between the regularized solution and the exact solution of the inverse problem is approximately equal to zero. To do this, we will first consider a mathematical model that describes in the form of a system of ordinary differential equations. Further, the inverse problem will be formed within the framework of this mathematical model. Then, the finite difference methods, the cubic spline method and the Tikhonov regularization method will be used to transform the inverse problem into a functional minimization problem. Finally, a method for choosing a quasi-optimal value of the regularization parameter will be described and a process for finding it will be constructed. As a result of this work, a quasi-optimal value of the regularization parameter will be found, as well as a regularized solution of the inverse problem. In the future, our work may help to solve inverse problems in practice, without knowing in advance the errors of the initial data.