#### Abstract

Imaging the elastic modulus distributions of soft tissues requires the solution of an elastic inverse problem. A typical approach to formulating this inverse problem is as an optimization problem. We consider in particular formulating this inverse problem as a constrained optimization problem in which we minimize the objective functional (the data mismatch) with the partial differential equations of elasticity as constraints. The Lagrange multiplier method is applied resulting in a two-field variational formulation that is often called a mixed formulation. The constrained optimization formulation differs from the output least squares method that appears frequently in the literature in which the inverse problem is formulated as an unconstrained optimization problem. Formulating this inverse problem as a constrained optimization problem has some advantages over formulating it as an unconstrained one since we can analyze the well posedness of this formulation by using some standards theorems. So we can verify that under some suitable conditions this inverse problem will converge reliably to a unique solution. Under some relatively strong mathematical conditions, this is true for the continuous case. In practice, however, when we solve this problem approximately by using classical discretization techniques, these stability conditions are never satisfied. Thus the discrete solution may converge to the wrong solution or may diverge. A novel solution to this problem is proposed by adding some new residual stabilization terms to the initial functional. Initial progress on proving stability theoretically is presented. Finally we show some reconstruction results using this formulation for the plane stress case that demonstrate that this new formulation can accurately predict the shear modulus distribution.

#### Keywords

Elastic Inverse Problems, Optimization

#### Subject Categories

Diagnostic imaging

#### Disciplines

Engineering

#### Publisher

Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-CenSSIS)

#### Publication Date

2007

#### Rights Holder

Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-CenSSIS)

#### Permanent URL

#### Recommended Citation

Rivas, Carlos E.; Barbone, Paul E.; and Oberai, Assad A., "An Optimization Approach to Elastic Inverse Problems" (2007). *Research Thrust R1 Presentations.* Paper 5. http://hdl.handle.net/2047/d1000872x

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## Notes

Poster presented at the 2007 R1A Nonlinear and Dual Wave Probes Conference