Highlights
- New MLMC+Parareal algorithm combining the computation time reduction of multilevel Monte Carlo with the wall-clock time reduction of parallel-in-time methods
- Open-source software implementation of this algorithm that can easily be used with alternative simulation codes
- Benchmark cases to highlight computational efficiency
| Keywords | Manufacturing & Engineering, Mechanical Engineering |
Challenge
Uncertainty Quantification with Monte Carlo methods requires drawing many random samples of input parameters, and solving a problem (typically a system of differential equations) numerically for each sample, leading to a substantial increase in energy and time required to obtain a solution. Since the numerical solutions of individual samples are independent of each other, the time-to-solution can be reduced via parallel sample evaluation in a parallel computing environment.
The Multi-Level Monte Carlo method uses different evaluation levels, with increasing accuracy and cost. By evaluating most samples with a cheaper method and retaining only relatively few sample evaluations of the desired accuracy, the method manages to retain the low bias error associated with the accurate but costly sample evaluations on the fine level and the low variance error associated with a high number of evaluated samples. At the same time, the method drastically reduces the energy required to complete the calculation. The same effect can be observed for the time-to- solution if samples are evaluated sequentially, but the effect diminishes if a highly parallel computing environment is used. In this case, the time-to-solution is dominated by the remaining fine samples, and adding more parallel computing capacity does not speed up the procedure.
Research Topic
While Monte Carlo type methods for Uncertainty Quantification are generally considered computationally expensive (both in terms of energy use and time-to-solution), they still remain desirable for problems with a high dimension of uncertainty, i.e., more than a few uncertain input parameters. The goal of this research project is to develop a Monte Carlo type method for Uncertainty Quantification which can substantially reduce the energy use and time-to-solution compared to naive Monte Carlo sampling by combining it with Parallel-in-Time time integration methods.
Solution
Since the time-to-solution of the Multi-Level Monte Carlo method is dominated by the evaluation time of the few remaining high-accuracy samples, reducing the time-to-solution of these samples has a noticeable impact on the overall time-to-solution of the method.
In the Parareal algorithm, a Parallel-in-Time integration scheme, the considered time interval is split into subintervals with initial values determined via a sequential but coarse and therefore numerically cheap method. Afterwards, time integration is performed in parallel on different processor cores for each subinterval and a sequential correction term is applied to reduce the discontinuities at subinterval boundaries. This procedure is repeated iteratively until a certain tolerance is reached. By using Parareal to speed up the solution of remaining high-accuracy samples, additional processor cores are leveraged to reduce the overall time-to-solution of the Multi-Level Monte Carlo method.