In this blog post, we share an example for using the QMPCy package to accelerate rare-event Monte Carlo (MC) simulations in AvailSim4 [1]. The effort is part of a more general study of advanced MC methods for reliability studies of the CERN Machine Protection group.

#### Introduction

The European Organization for Nuclear Research, CERN, is home to the largest particle accelerator in the world, the Large Hadron Collider (LHC). The machine is producing and recording data from high-energy collisions of proton beams, allowing scientists all over the world to test theoretical models and hypotheses. The unprecedented beam energies create potential for pushing further the very boundaries of human knowledge and answering the most fundamental questions regarding the origins of the universe.

The LHC consists of many sophisticated systems with responsibilities such as injecting the particles into the beam orbit, maintaining beams on very precise tracks, or cooling down the superconducting magnets and radio frequency cavities – to name just a few. The Machine Protection group is primarily focused on two types of systems: those protecting the superconducting magnets and their circuits and those protecting the accelerator equipment from damage due to the circulating high-energy beams. These systems are critical as their failures may cause severe damage to the accelerator.

Reliability engineering offers a wide range of methods to quantify potential risks and their consequences. Probabilistic methods such as Monte Carlo (MC) simulations are used to assessing risks of complex systems for which no analytic solution can be derived. However, MC simulation may come at a significant computational cost. In this short post, we present the AvailSim4 tool in which we have implemented a Quasi-Monte Carlo extension to make the computations more efficient. The open-source implementation of the framework utilizes the QMCPy package.

#### AvailSim4

AvailSim4 is an open-source framework developed in the Machine Protection group of CERN’s Technology Department. AvailSim4 provides an environment for availability simulations with several features specifically developed for particle accelerator applications.

An AvailSim4 model can include several sub-systems, each described by a failure probability distribution, recovery distribution and functional dependencies. The structure of those dependencies forms a tree, which models the entire system. A simplified example of such a system is presented in the picture above. The overall component **System** consists of two children: basic **A** (without children) and compound **B** – which further splits into two components **B1** and **B2**. The relation between the two is defined to be guided by **OR** logic: meaning, component B is working if either of its two children is operational.

AvailSim4 combines MC with Discrete Event Simulation (DES) approach. This means that each MC iteration is a realization of a random timeline of events. A visualization of a timeline (with supporting timelines of individual components) is displayed in the plot above. The overall analysis is the result of how often and for how long individual components fail. In this sense, the framework is a standard example of a MC simulation: it runs multiple instances of the experiment with different random numbers and obtains estimations of quantities of interest by calculating their averages.

The main difficulty is the shift to a rare-event regime, where events of interest occur in only a small fraction of iterations. Computation efficiency can be improved on two separate levels: code optimizations (e.g. through profiling, distributed computing, precompiling critical functions, etc.) and algorithm enhancements. Applying Quasi-Monte Carlo is a member of the latter group.

#### Challenges of Quasi-Monte Carlo in AvailSim4

Using Quasi-Monte Carlo did not require substantial changes in the Monte Carlo implementation in the case of AvailSim4. The bigger change is on the conceptual level. Understanding the differences is essential for interpretation of our results.

However, the use of QMC methods comes with a dimensionality limitation, which is important in the studied use case. First, the computational efficiency improvement comes from using samples covering the problem space more evenly. In our use case, that space should be viewed in the MC-sense rather than a single DES-sense. This means that consecutive samples will be employed across different DES iterations – and the need for more random values in an individual iteration will be addressed by generating samples of multiple dimensions. This aspect will be further discussed below.

#### Results

The test case is made of a very simple system, consisting of a few redundant components that have the same properties and fail at times drawn from an exponential probability distribution. The aspect that changes between the three presented cases is the number of those components. In such a scenario, the more components are needed, the less likely a critical failure is.

In the left-hand side plots above, we see the progression of the accuracy as the number of DES iterations increases. Orange lines represent QMC results, while blue ones are results of the MC mode. In only one case (5 components, 100 iterations), the QMC mode is less accurate. All remaining test cases show that the orange line is closer to the value to which both lines eventually converge. The execution time comparison is featured on the right-side. Results are also relatively stable: QMC adds a small overhead at the beginning (to generate a large matrix of random numbers before iteration rather than in it), however it is visible only in the case of small numbers of iterations. This overhead does not eliminate the advantage of the method, which is coming from using less iterations required to obtain certain accuracy. Also, the more iterations are completed, the smaller the relative difference, as generating random values takes place only once.

All results presented in this section are further discussed in [2]. This includes additional test cases and a comparison of QMC with Importance Splitting, another method to significantly speed up MC simulations.

Another aspect are limitations of the approach and using QMC in general. It has already been said that samples are multidimensional, so that each one contains enough random values for all components in the DES. However, there is an additional complication. A significant element of all availability and reliability simulations is that components may be repaired and returned to their fully operational state indefinite number of times. Assigning each component only a single failure time is a solution that falls short in those terms. Instead, the exiting implementation provisions more random numbers (by assigning more dimensions) for each component – whenever a given element fails, the next failure time is taken as a value of the next dimension assigned to it.

This fact brings the most serious limitation of the approach. The number of available random failure time values needs to be decided prior to commencing simulations and must assume the worst case, so that no component runs out of failure times before finishing its lifetime. When some of the components fail relatively often, the total number of dimensions will often end up close to the current limits of low-discrepancy sequence generators. This also adds to the overhead at the beginning – we need to pre-emptively generate many more values than when generating them only when needed (i.e., as it is done with standard pseudo-random number generator).

#### Conclusions

During this study, the QMC method evolved from a proof-of-concept addition to a fully implemented feature of AvailSim4. The most important advantage of QMC methods is that the change from crude MC is almost transparent for the users: no further information or inputs are required. It would be fully transparent had there not be the limitation caused by the number of dimensions of each sample, which is something to which users need to pay attention to.

In our tests of the rare-events scenarios, the gains are visible. The results’ accuracy increased as their variance diminished. However, getting orders of magnitude improvements is strictly impossible, as the method of obtaining results is still based on simple calculation of averages, such as the numbers of events of interest.

#### References

- AvailSim4 GitLab repository, https://gitlab.cern.ch/availsim4/availsim4
- M. Blaszkiewicz, “Methods to optimize rare-event Monte Carlo reliability simulations for Large Hadron Collider Protection Systems”, MSc Thesis, https://cds.cern.ch/record/2808520

##### Milosz Blaszkiewicz

Junior fellow at European Organization for Nuclear Research (CERN). Interested in rare-event simulation, high performance and distributed computing.