Stable and optimal conductance recovery on networks

• Stable recovery of piecewise constant conductances from boundary data • Influence of regularization parameter on reconstruction stability • Critical and non-critical network behaviors are analyzed • Optimization framework for conductance identification is proposed • Numerical experiments validate the theoretical findings This paper extends our previous work on the discrete inverse conductance problem on spider networks. Our approach leverages polynomial optimization techniques, incorporating a regularization term to stabilize the inverse problem that penalises the deviation from a piecewise constant conductance hypothesis. Here, we show examples of the stability of the conductance recovery even in cases in which the real conductance does not satisfy the piecewise constant property. We provide a comprehensive analysis of the error behavior with respect to the penalty parameter, demonstrating the effectiveness of our method in various network scenarios. Finally, we look for optimality guarantees for the numerical solution of the polynomial optimization problem. This work significantly contributes to the field of applied inverse problems, offering robust strategies for conductance recovery in complex network structures.