Basis pursuit set selection for nonlinear underconstrained problems: An application to damage characterization

The Basis Pursuit (BP) is an optimization statement used to search for the sparsest solution of underconstrained problems of the form Φx=b using ℓ 1 minimization. This paper shows that approximating nonlinear problems as linear can result in considerable compromise in the capacity of the BP to attain the minimum cardinality solution and that this impairment can be mitigated by treating the strategy as a selector of a set of unknowns that renders the problem fully constrained, subsequently solving with due consideration for the nonlinearity. It is shown that gains realized by the selector approach derive primarily from the fact that the part of the nonlinearity that projects onto the right-hand side,b, has no effect on the relative values of the solution,x, and that the orthogonal residual seldom modifies the largest entries. The paper considers damage characterization as the application domain and shows that normalization of the Jacobian to equal column norm, a step that is standard when BP is applied in most fields, but not implemented so far in structural engineering, leads to large improvements when the probability of damage across the parameter space is essentially uniform.