Computational inference integrates simulations of scientific phenomena with physical measurements in an information feedback control system. Control parameters of the computational model are optimized to fit the physical measurements. These parameters are then used in the simulations to predict and characterize the scientific phenomena. Typical examples include weather forecasting and medical applications such as pacemaker design for cardiac patients. In this talk, I focus on applications where the underlying scientific phenomena is modeled by partial differential equations (PDEs), with unknown critical parameters of the PD Es such as boundary conditions, initial conditions, and material properties. Computational inference poses several challenges such as a) imperfections in models and data, and b) computational cost driven by the need to run several computer simulations of these models making the process expensive. This talk will present efficient computational techniques to mitigate some of these challenges and address other outstanding issues. The proposed methods will be demonstrated on real applications such as numerical weather forecasting.