Quadratic Basis Pursuit in Model Updating of Underconstrained Problems

Abstract

Abstract This paper considers the extraction of sparse solutions using the L1 minimization surrogate for two approximations of the nonlinear relation between parameters and features; in the first, nonline-arity is discarded altogether and in the second a quadratic relation is assumed. The associated algorithms are the well-known Basis Pursuit (BP) and the more recently introduced Quadratic Basis Pursuit (QBP). It is contended that measuring success by comparing the extracted solutions with the underlying truth is unnecessary, as all that’s needed from the extractor is to identify a sufficiently small parameter set to render the problem determined. Specifically, with p as the number of features, one is interest in the likelihood that the algorithm leads to Ω p ∩ Θ = Θ, where Θ = true set and Ω p = set of the largest p non-zero entries in the solution. When operating in this manner the disruption that nonlinearity brings into the BP solution derives only from the nonlinearity induced rotation of the right-hand side away from the span of the column partition of the Jacobian for the non-zero entries. From this perspective one expects performance to deteriorate only weakly with the extent of damage and this result is numerically con-firmed. It is found that although QBP has the quadratic premise as an advantage over BP, the perfor-mance in simulations, due to the difficulty in obtaining optimal values for the free parameters (and the fact that damage extent prediction was not relevant) proved somewhat poorer than BP.