A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

Abstract

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problem's important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.

Keywords:

• Energy output least squares
• Inverse problems
• Locking effect
• Mixed finite element method
• Nearly incompressible elasticity
• Output least squares
• Parameter identification
• Regularization
• Saddle point problems
• Total variation
• Tumor identification

Mathematics Subject Classification:

• 35R30
• 65N30

ACKNOWLEDGMENTS

This article is dedicated to Alemdar Hasanov on the occasion of his 60th birthday.

Part of the special issue, “Variational Analysis and Applications.”

Notes

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lnfa.