﻿ 基于变分伴随的扩散方程系数反问题的数值反演算法 An Inversion Algorithm for Inverse Coefficient Problems in the Diffusion Equation Based on Variational Adjoint Method

Vol.07 No.04(2018), Article ID:24735,10 pages
10.12677/AAM.2018.74057

An Inversion Algorithm for Inverse Coefficient Problems in the Diffusion Equation Based on Variational Adjoint Method

Zhendong Wang, Gongsheng Li, Yingmei Wang

School of Mathematics and Statistics, Shandong University of Technology, Zibo Shandong

Received: Apr. 12th, 2018; accepted: Apr. 23rd, 2018; published: Apr. 30th, 2018

ABSTRACT

A numerical inversion algorithm is proposed based on the variational adjoint method for solving the backward problem and inverse source problem in the diffusion equation. With the help of the adjoint problem, a variational identity connecting the known data with the unknown is derived with which a linear system for the unknown is set forth by suitably choosing the solution of the adjoint equation. Numerical solutions to the inverse problems are obtained by solving the linear system with Tikhonov regularization, and numerical examples are presented to demonstrate the effectiveness of the inversion algorithm.

Keywords:Diffusion Equation, Inverse Problem, Adjoint Method, Variational Identity, Numerical Inversion

1. 引言

2. 正问题与反问题

2.1. 正问题

(2.1)

(2.2)

(2.3)

2.2. 逆时问题与源项反问题

(2.4)

(2.5)

3. 变分恒等式与反演算法

3.1. 变分恒等式

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

3.2. 反演算法

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

4. 数值反演

4.1. 逆时问题的数值反演

(4.1)

(4.2)

。取，利用上一节所述的反演方法进行直接计算。图1绘制了

Figure 1. Inversion result for the backward problem with accurate data in Ex.1.

Figure 2. Inversion result for the backward problem with noisy data in Ex.1. (a); (b)

4.2. 源项的数值反演

(4.3)

Figure 3. Inversion results for the backward problem with noisy data in Ex.2. (a); (b)

Figure 4. Exact source and inversion source solutions with accurate data in Ex.3. (a) Exact source; (b) Inversion source with accurate data

(4.4)

5. 结论

Figure 5. The inversion source solutions with noisy data in Ex.3. (a); (b)

Figure 6. Exact source and the inversion source solutions with accurate data in Ex.4. (a) Exact source; (b) Inversion source solution with accurate data

Figure 7. Inversion source solutions with noisy data in Ex.4. (a); (b)

An Inversion Algorithm for Inverse Coefficient Problems in the Diffusion Equation Based on Variational Adjoint Method[J]. 应用数学进展, 2018, 07(04): 466-475. https://doi.org/10.12677/AAM.2018.74057

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