  The termination in the white dwarf luminosity function is a standard
diagnostic tool for measuring the total age of nearby stellar populations. In
this paper, an algorithm is presented for inverting the full white dwarf
luminosity function to obtain a maximum likelihood estimate of the time varying
star formation rate of the host stellar population. Tests with synthetic data
demonstrate that the algorithm converges over a wide class of underlying star
formation rate forms. The algorithm successfully estimates the moving average
star formation rate as a function of lookback time in the presence of realistic
measurement noise, though suffers from degeneracies around discontinuities in
the underlying star formation rate. The inversion results are most sensitive to
the choice of white dwarf cooling models, with the models produced by different
groups giving quite different results. The results are relatively insensitive
to the progenitor metallicity, initial mass function, initial-final mass
relation and ratio of H/He atmosphere white dwarfs. Application to two
independent determinations of the Solar neighbourhood white dwarf luminosity
function gives similar results. The star formation rate has a bimodal form,
with broad peaks at 2-3 Gyr and 7-9 Gyr in the past, separated by a significant
lull of magnitude 30-90% depending on choice of cooling models. The onset of
star formation occurs around 8-10 Gyr ago. The total integrated star formation
rate is ~0.014 stars/pc3 in the Solar neighbourhood, for stars more massive
than 0.6M_{solar}.
