Towards optimal allocation of computer resources: Trade-offs between uncertainty quantification, discretization and model reduction

• We jointly consider physical, temporal and stochastical model errors, and their associated CPU budgets.
• For every given computational budget there exists an optimum for the allocation of the CPU resources.
• We find that this optimum depends on the specific physical problem and prediction task.
• Our results indicate a smaller number of Monte Carlo runs when compared to those found in the literature.
• Allocating CPU budgets wisely leads, for our specific examples, to a speedup in the order of 5–100.

The computational complexity of numerical models can be broken down into contributions ranging from spatial, temporal and stochastic resolution, e.g., spatial grid resolution, time step size and number of repeated simulations dedicated to quantify uncertainty. Controlling these resolutions allows keeping the computational cost at a tractable level whilst still aiming at accurate and robust predictions. The objective of this work is to introduce a framework that optimally allocates the available computational resources in order to achieve highest accuracy associated with a given prediction goal. Our analysis is based on the idea to jointly consider the discretization errors and computational costs of all individual model dimensions (physical space, time, parameter space). This yields a cost-to-error surface which serves to aid modelers in finding an optimal allocation of the computational resources (ORA). As a pragmatic way to proceed, we propose running small cost-efficient pre-investigations in order to estimate the joint cost-to-error surface, then fit underlying complexity and error models, decide upon a computational design for the full simulation, and finally to perform the designed simulation at near-optimal costs-to-accuracy ratio. We illustrate our approach with three examples from subsurface hydrogeology and show that the computational costs can be substantially reduced when allocating computational resources wisely and in a situation-specific and task-specific manner. We conclude that the ORA depends on a multitude of parameters, assumptions and problem-specific features and, hence, ORA needs to be determined carefully prior to each investigation.