From sparse measurements to probabilistic solution manifolds: an FCWV inverse framework for parameter identification and full-field reconstruction

Strongly nonlinear and underdetermined inverse problems commonly arise in structural health monitoring, model updating, and deformation reconstruction, where sparse measurements may correspond to multiple admissible parameter combinations. This work presents a probabilistic finite cell weight variation (FCWV)-based inverse framework capable of reconstructing full admissible solution manifolds together with their associated probability distributions. The formulation is derived from the conservation law of probability and the transformation law of probability density, and recursively decomposes a multi-dimensional inverse problem into coupled one-dimensional searches. In this way, admissible solution manifolds can be identified efficiently without reliance on large-scale global optimization or extensive Bayesian sampling. The framework is developed for multi-input, multi-output systems and extended to higher-dimensional inverse problems through a recursive sampling strategy. Two benchmark examples are considered. The first involves identification of unknown parameters in a nonlinear analytical leak-rate model for a cracked pipe using prescribed leak-rate measurements, where the inverse formulation is validated against direct uncertainty propagation. The second considers a beam-element piping system under seismic excitation, in which six unknown support stiffnesses are identified from prescribed target displacement responses at a node, using a surrogate response model constructed from finite element simulations. In both examples, the proposed method successfully reconstructs admissible solution manifolds and quantifies the relative likelihood of competing solutions. The results demonstrate that the FCWV-based inverse framework provides an efficient and physically transparent alternative for solving strongly nonlinear and underdetermined inverse problems with sparse measurements.