Computer Methods for Partial Differential Equations, Volume L. Elliptic Equations and the Finite-Element Method (Robert Vichnevetsky)

SIAM Review ◽  
1983 ◽  
Vol 25 (3) ◽  
pp. 434-435
Author(s):  
R. B. Kellogg
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Equations ◽  
The Finite Element Method ◽  
Partial Differential ◽  
Computer Methods ◽  
Element Method

Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander S. Leonov ◽  
Alexander N. Sharov ◽  
Anatoly G. Yagola
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Inverse Problems ◽  
Differential Equations ◽  
Differential Operators ◽  
The Finite Element Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Element Method

Abstract A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young’s modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.

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