A general framework for quantifying uncertainty at scale

Quantifying uncertainty and performing sensitivity analysis in real-world, large-scale numerical simulations is virtually impossible with standard methods. We show that our recently developed sensitivity-driven sparse grid interpolation methods enables these two important tasks at scale.

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The tremendous advancements in hardware, software, and algorithms in recent decades established computing as the third pillar of science, alongside theory and experiment. Nowadays, simulations that seemed far-fetched in the past, such as turbulent transport in fusion devices or combustion in rocket engines, can nowadays be performed at scale on parallel supercomputers.  However, it has become well established that an obstacle on the path towards predictive and quantitative numerical simulations of real-world problems is uncertainty. Whether stemming from incomplete knowledge, measurement errors, inherent variability, or any other source, uncertainty is intrinsic to most problems, and this aspect needs to be included in respective modeling efforts. Accounting for, understanding, and reducing uncertainties in numerical simulations is performed within the framework of Uncertainty Quantification (UQ). Moreover, another important task related to UQ is sensitivity analysis (SA) which is used to quantify the importance of input uncertainty in the output(s) of interest of the respective simulation. Performing these two tasks in real-world simulations is, however, far from trivial. This is because UQ and SA generally require ensembles of high-fidelity simulations, and when the ensemble size is large, the total computational cost becomes prohibitive. For example, if a single high-fidelity simulation takes 10,000 core-hours or more on compute clusters or supercomputers, performing even 100 such simulations can be computationally too expensive.

Our goal in this work was to show that our recently developed sensitivity-driven dimension-adaptive sparse grid interpolation algorithm can significantly reduce the number of high-fidelity simulations needed for UQ and SA in computationally expensive, realistic simulations for which standard approaches would be infeasible. This is done by exploiting, via adaptive refinement, the fact that in most problems, only a subset of the uncertain inputs are important and, moreover, these inputs interact anisotropically. In other words, only a subset of all inputs and input interactions generally have non-negligible (and non-equal) sensitivities. The goal of the adaptive refinement procedure is to exploit this structure and to therefore preferentially refine the directions corresponding to important inputs and input interactions. Figure 1 plots a visual summary of the sensitivity-driven approach through an example with d = 3 uncertain inputs (θ1, θ2, θ3) in which θ3 is the most important parameter, θ1 is the second most important, θ1 and θ3 interact strongly, and θ2 is the least important parameter. Observe that the sensitivity-driven approach exploits this structure.

Visual illustration of the sensitivity-driven dimension-adaptive sparse grid framework for uncertainty quantification and sensitivity analysis at scale. The framework is demonstrated in an example with d = 3 uncertain inputs (θ1, θ2, θ3). The goal of the sensitivity-driven approach is to explore and exploit the fact that in real-world simulations only a subset of the uncertain inputs are important and that these inputs interact anisotropically.

To this end, we employed the sensitivity-driven approach in the context of simulations of nonlinear turbulent transport in tokamaks. These simulations represent a paradigmatic example in which UQ and SA are needed but in which most standard approaches are computationally infeasible due to the large cost of these simulations. The experimental error bars of various input parameters (such as the spatial gradients of the density and temperature of a given plasma species) can be relatively large, on the order of a few ten percent, which makes the SA task especially valuable since understanding the impact of these uncertainties is critical. Moreover, ascertaining the impact of variations in parameters that characterize the confining magnetic field is crucial as well for, e.g., design optimization. As a practically relevant example of such simulations, we focused on the near-edge region of tokamaks, which is recognized as crucial for setting the overall performance of these devices. A visual summary of these simulations is depicted in Figure 2.

Turbulent transport in the near-edge of tokamaks. From left to right: in the tokamak design, a hot hydrogen plasma is confined in a doughnut-like shape with the aid of strong magnetic fields (figure courtesy of EUROfusion). However, the magnetic confinement is not perfect: turbulent fluctuations driven by micro-instabilities cause heat losses from the hot core towards the colder edge. In so-called high-confinement (H-mode) discharges, one can induce the formation of a near-edge region characterized by reduced transport and steep gradients.  The properties of this pedestal are influenced by the residual turbulent transport.

In our numerical experiments, we considered a specific near-edge simulation scenario with eight uncertain inputs. The output of interest was the time-averaged electron heat flux. Simulating turbulent transport in the edge of fusion devices is, however, nontrivial. Prior to performing UQ and SA, we therefore performed preliminary runs to determine the grid resolution that ensures that the underlying high-fidelity simulations - including runs for the extrema of the parameter space, which, in turn, yield the extrema turbulent transport levels - are sufficiently accurate. The grid that provided the desired accuracy comprised more than 264 million degrees of freedom, for which the respective high-fidelity simulations performed on 16 nodes (896 compute cores in total) on a supercomputer required between 4000 and 14,000 core-hours; the average run time exceeded 8000 core-hours. We then started the sensitivity-driven dimension-adaptive refinement procedure using the aforementioned grid resolution for the high-fidelity simulations. In addition, the employed tolerances that guided the adaptive refinement procedure were 10-4. One of the strengths of the sensitivity-driven approach is that it can be easily coupled with the underlying simulation code since it only requires prescribing the simulation inputs (these are the grid points living on the sparse grid constructed via adaptive refinement) and the value of the corresponding output of interest. It was therefore trivial to couple the sensitivity-driven approach with the considered simulation scenario. The employed tolerances were reached after a mere total of 57 simulations. That is, our approach required a total of only 57 high-fidelity simulations for UQ and SA in a complex, real-world simulation with eight uncertain parameters. This was possible because out of the eight parameters, only four had non-negligible sensitivities - with two parameters being significantly more important than the other six - and, moreover, the only non-negligible interactions occurred between the important parameters. This low cost in terms of required high-fidelity simulations was a tremendous result. Moreover, the interpolation-based surrogate model for the parameter-to-output of interest mapping intrinsically provided by our method was accurate and nine orders of magnitude cheaper to evaluate than the high-fidelity model. Our goal to show that our method can allow UQ and SA at scale was therefore achieved.

Overall, this work highlighted the need and importance of interdisciplinary research and collaborations between computational scientists and domain experts in solving real-word problems through computing. We employed tools from applied mathematics such as approximation theory, tools from probability theory and statistics, but also from computer science and high-performance computing, as well as tools from (computational) plasma physics. Moreover, computational scientists collaborated with plasma physicists to address the computational challenges of performing UQ and SA in simulations of nonlinear turbulent transport in fusion devices. 

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