Bayes Linear Strategies for the Approximation of Complex Numerical Calculations Arising in Sequential Design and Physical Modelling Problems.

Abstract

In a range of different scientific fields, deterministic calculations for which there is no analytic solution must be approximated numerically. The use of numerical approximations is necessary, but introduces a discrepancy between the true solution and the numerical solution that is generated. Bayesian methods are used to account for uncertainties introduced through numerical approximation in a variety of situations.
To solve problems in Bayesian sequential experimental design, a sequence of complex integration and optimisation steps must be performed; for most problems, these calculations have no closed-form solution. An approximating framework is developed which tracks numerical uncertainty about the result of each calculation through each step of the design procedure. This framework is illustrated through application to a simple linear model, and to a more complex problem in atmospheric dispersion modelling. The approximating framework is also adapted to allow for the situation where beliefs about a model may change at certain points in the future.
Where ordinary or partial differential equation (ODE or PDE) systems are used to represent a real-world system, it is rare that these can be solved directly. A wide variety of different approximation strategies have been developed for such problems; the approximate solution that is generated will differ from the true solution in some unknown way. A Bayesian framework which accounts for the uncertainty induced through numerical approximation is developed, and Bayes linear graphical analysis is used to efficiently update beliefs about model components using observations on the real system. In the ODE case, the framework is illustrated through application to a Lagrangian mechanical model for the interaction between a set of ringing bells and the tower in which they are hung; in the PDE case, the framework is illustrated through application to the heat equation in one spatial dimension.