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A new inverse problem formulation based on the compliance error functional is presented. The proposed formulation permits the derivation of an explicit expression for the Lagrange multipliers. Thus, the computation of the Lagrange multipliers does not require the solution of the computationally intensive adjoint system of equations. This enables the computation of the first order derivative information at the expense of just one model evaluation. Leading to significant speedups for large-scale, gradient-based inverse problems. Second, if second order optimization algorithms are available, Newton’s method can be applied to the first order necessary optimality conditions. Newton’s method relies on second order derivative information to compute descent directions during optimization. This paper presents two Hessian formulations based on the compliance error functional. The first formulation relies on the mathematical properties of the compliance error functional to compute the application of the trial step to the analytical Hessian at the expense of one model evaluation per Newton iteration. The second formulation is based on a Gauss-Newton approximation to the Hessian. The Gauss-Newton approximation further speedups analysis since the second order derivative information is computed without performing additional model evaluations during the Newton iterations. Third, examples in heat transfer are presented to demonstrate the effectiveness of the compliance error functional. The compliance error minimization formulation is compared against the data misfit minimization formulation. Results will show that the compliance error minimization formulation outperforms the data misfit formulation.

AN INVERSE PROBLEM STRATEGY BASED ON FORWARD MODEL EVALUATIONS: GRADIENT-BASED OPTIMIZATION WITHOUT ADJOINT SOLVES