A first-order adjoint and a second-order hybrid method for an energy output least-squares elastography inverse problem of identifying tumor location

In this paper we investigate the elastography inverse problem of identifying cancerous tumors within the human body. From a mathematical standpoint, the elastography inverse problem consists of identifying the variable Lamé parameter μ in a system of linear elasticity where the underlying object exhibits nearly incompressible behavior. This problem is subsequently posed as an optimization problem using an energy output least-squares (EOLS) functional, but the nonlinearity that arises makes the computation of the EOLS functional’s derivatives challenging. We employ an adjoint method for the computation of the gradient, something shown to be an efficient method in recent studies, and also give a parallelizable hybrid method for the computation of the EOLS functional’s second derivative. Detailed discrete formulas and nontrivial computational examples are provided to show the feasibility of both the adjoint and hybrid approaches. Furthermore, all results are given in the framework of a general saddle point problem allowing easy adaptation to numerous other inverse problems. MSC: 35R30, 65N30.