**The numerical analysis and applied mathematics research group NUMA at KU Leuven has successfully developed a new Uncertainty Quantification (UQ) software package allowing for an efficient treatment of problems that depend on many uncertain parameters.**

The research group had been part of the EUFORIA research project, an SBO project (*Strategisch BasisOnderzoek* or Strategic Basic Research) that ran from February 2015 through April 2019. Funded by IWT (now VLAIO), SBO projects support “high-quality level basic research with a pronounced focus on high-risk, inventive and original research and with a high and strategic valorization potential of the results in Flanders.” In this case, the project brought together academics from five institutions and worked with a “Users Committee” of Flanders-based companies to assess the project’s outcomes. The Users Committee members’ activities include domains as diverse as heat exchangers, computational physics (CFD), and software and energy engineering, ensuring that the project output would be widely applicable. One of the more extreme uses for the software, and one that eloquently illustrates how it works, involves designing earthquake-resistant structures. While certainly not the only application, it is an exciting advancement in civil engineering.

**Uncertainty Quantification in earthquake-resistant structures**

Inside Taiwan’s Taipei 101 building, one of the world’s tallest skyscrapers, is a large metal sphere weighing 660 tons, mounted between the 88th and 92nd floor. This sphere is not an architectural frivolity but is one of the buildings’ key features. As earthquakes are common in east Taiwan, tall earthbound structures must be designed so that they can withstand the additional stresses produced inside their structural components. If those stresses become too large, the building might experience permanent damage or even collapse. The large sphere on top of the Taipei 101 is designed precisely to prevent such catastrophic events from happening. To that end, structural engineers who design such buildings must take into account the inherently uncertain nature of earthquakes, since no two earthquakes are ever identical and all differ in strength, vibration pattern and duration. To do so, engineers can resort to UQ, the study of the impact of imperfect information on the design of products or processes.

**Earthquake modeling**

To perform uncertainty quantification, one must have a *mathematical model* describing the impact of an earthquake on the horizontal motion of the building. In civil engineering, this motion is typically simulated using rigid body models for the building. Seismologists then describe the ground acceleration caused by earthquakes as a combination of many high-frequency components. Subsequently, the ground acceleration is used as input to the rigid body model. Together, the ground acceleration and building model form a mathematical model for the effect that an earthquake has on the building’s motion. In this mathematical model, the high-frequency components form the set of input parameters, whereas the model output – also termed the *quantity of interest* – is given by the maximum stress inside the building.

However, as every earthquake is unique, the maximum stress should be computed for a wide variety of possible model inputs. Consequently, these inputs now become *uncertain parameters*, and, as a result, the predicted output quantity (the maximum stress) also becomes uncertain. Instead of a single output value, the result is now specified as a distribution of potential output values. Determining the effect the uncertain parameters have on the distribution of the output quantity of interest is the main task of UQ.

It is an exciting advancement in civil engineering.

**UQ – Monte Carlo methods to the rescue**

Once the mathematical model is set up, the uncertainty quantification task can start. A very popular UQ technique is the brute-force *Monte Carlo method*, which repeatedly chooses a random value for each of the uncertain parameters. For each set of random input parameters, one computes the quantity of interest using the mathematical model. The obtained set of output values – sometimes called the *ensemble* – is then used to approximate the distribution of the quantity of interest. Unfortunately, it is well known that the Monte Carlo method suffers from a low efficiency: to get an additional digit of accuracy, the number of model evaluations must increase a hundredfold!

To remedy the inefficiency of the Monte Carlo method, about 10 years ago, *multilevel* methods were invented. Instead of only computing with the high accuracy model as in Monte Carlo, multilevel methods use many cheap-to-compute model evaluations with low accuracy, and subsequently add corrections from a hierarchy of models with increasing levels of accuracy – but also increasing computational cost. Each model in the hierarchy is referred to as a *level*, and, when combined with Monte Carlo, the method is called Multilevel Monte Carlo. In the context of earthquake-resistant buildings, a hierarchy of cheaper models can be obtained by varying, for example, the time increment used in the simulation of the earthquake-building model. Simulations with finer time steps yield more accurate results, but are also more expensive to compute.

It is important to stress that a multilevel method yields the same accuracy as the Monte Carlo method, but its computational cost is heavily reduced. For a simple one-story building model, for example, the distribution of the maximum stress inside the building subject to an earthquake-input with 1000 high-frequency components is computed almost 10 times faster with a 6-level method than with the original Monte Carlo method.

**Open-source UQ software – MultilevelEstimators.jl**

The multilevel Monte Carlo methodology for uncertainty quantification is implemented in an all-purpose Julia software library MultilevelEstimators.jl, created by NUMA. This freely available generic library focuses on the efficient computation of the distribution of a quantity of interest using multilevel methods.

Rightly so, the research group is proud of their achievement. They hope more applications for it will come to light as more audiences begin to use it. The software is both lightweight and flexible, with the major advantage that it wraps around an existing user code that is considered as a black box; no modifications to the user code are required! The software has proven to be significantly more efficient in several industrial cases and is now ready to be used in your application! If you want to try out the software in your own domain, have questions, or just want more information, feel free to get in touch with Ward.Melis@kuleuven.be.