Gibson's primary research interests are computational electromagnetics, finite element and finite difference methods, and inverse problems. Research topics coinciding with primary interests include: wave propagation modeling, direct and indirect (sparse) linear solvers, optimization and regularization techniques, high performance and parallel computing, parameter identification and sensitivity analysis, and modeling uncertainty in coefficients of PDE's. His current work with NASA Langley Research Center involves inverse problem techniques related to non-destructive evaluation (NDE) of Space Shuttle foam.