A posteriori error estimates for numerical solutions to inverse problems of elastography

Abstract

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be found is smooth or discontinuous. Tikhonov regularization with appropriate regularizing functionals is applied to solve these problems. In particular, discontinuous Young’s modulus distribution can be found on the class of 2D functions with bounded variation of Hardy–Krause type. It is shown in the paper that a variant of Tikhonov regularization provides for such discontinuous distributions the so-called piecewise uniform convergence of approximate solutions as the error levels of the data vanish. The problem of practical a posteriori estimation of the accuracy for obtained approximate solutions is under consideration as well. A method of such estimation is presented. As illustrations, model inverse problems with smooth and discontinuous solutions are solved along with a posteriori estimations of the accuracy.