An Application of Random Projection to Parameter Estimation in Partial Differential Equations

In this article, we use a dimension reduction technique called random projection to reduce the computational cost of estimating unknown parameters in models based on partial differential equations (PDEs). In this setting, the repeated numerical solution of the discrete PDE model dominates the cost of parameter estimation. In turn, the size of the discretized PDE corresponds directly to the number of physical experiments. As the number of experiments grows, parameter estimation becomes prohibitively expensive. In order to reduce this cost, we develop an algorithmic technique based on random projection that solves the parameter estimation problem using a much smaller number of so-called encoded experiments. Encoded experiments amount to nothing more than random sums of physical experiments. Using this construction, we provide a lower bound for the required number of encoded experiments. This bound holds in a probabilistic sense and is independent of the number of physical experiments. Finally, we demonstrat...