Computer models for the simulation of physical and environmental phenomena are often regulated by complicated dependences on unknown variables, and these unobservable inputs must be inferred from a comparison of simulator output against physical data. The standard Bayesian statistical approaches to this inference problem require fitting a complicated statistical model to the existing parameter evaluations, usually through use of a Markov chain Monte Carlo sampling scheme. When there already exists a large bank of simulated values, it may be undesirable to develop a sophisticated statistical surrogate or to sample additional output from the computer simulator. In response to this motivation, we discuss a sampling importance resampling algorithm for Bayesian inference in inverse problems that works in conjunction with kernel density estimation to resample, from the original computer output, an approximate posterior sample for the unobservable variables of interest. Given a sufficiently large bank of computer output, our resampling method is able to provide high-quality results at a much lower cost than the standard Bayesian techniques. We present two applications where unobservable inputs are to be inferred from scarce observations and abundant simulated output. One consists of a climate simulator and the other of a groundwater flow model.