Dose and range verification have become important tools to bring carbon ion therapy to a higher level of confidence in clinical applications. Positron emission tomography is among the most commonly used approaches for this purpose and relies on the creation of positron emitting nuclei in nuclear interactions of the primary ions with tissue. Predictions of these positron emitter distributions are usually obtained from time-consuming Monte Carlo simulations or measurements from previous treatment fractions, and their comparison to the current, measured image allows for treatment verification. Still, a direct comparison of planned and delivered dose would be highly desirable, since the dose is the quantity of interest in radiation therapy and its confirmation improves quality assurance in carbon ion therapy. In this work, we present a deconvolution approach to predict dose distributions from PET images in carbon ion therapy. Under the assumption that the one-dimensional PET distribution is described by a convolution of the depth dose distribution and a filter kernel, an evolutionary algorithm is introduced to perform the reverse step and predict the depth dose distribution from a measured PET distribution. Filter kernels are obtained from either a library or are created for any given situation on-the-fly, using predictions of the β + -decay and depth dose distributions, and the very same evolutionary algorithm. The applicability of this approach is demonstrated for monoenergetic and polyenergetic carbon ion irradiation of homogeneous and heterogeneous solid phantoms as well as a patient computed tomography image, using Monte Carlo simulated distributions and measured in-beam PET data. Carbon ion ranges are predicted within less than 0.5 mm and 1 mm deviation for simulated and measured distributions, respectively.