I will first review the "data assimilation" problem of updating the predictions of computational models based on observations (data) and give some examples of applications in Earth science and engineering. The numerical solution of a data assimilation problem requires that one be able to draw samples from a Bayesian posterior distribution. This is difficult because the posterior distribution is (i) high-dimensional; and (ii) the posterior distribution is usually not a "standard" distribution (e.g., a Gaussian). In particular a high-dimension is known to cause numerical difficulties and slow convergence of sampling algorithms (importance sampling and Markov chain Monte Carlo). I will explain how ideas from numerical weather prediction can be leveraged to design Markov chain Monte Carlo samplers whose convergence rates are independent of the problem dimension for a well-defined class of problems. This is joint work with Xin Tong, National University of Singapore and Youssef Marzouk, MIT.