When Professor Maciej Paszyński from the AGH UST Faculty of Computer Science, Electronics and Telecommunications, together with his team, published an article describing a method for fast computer simulations, he could not have anticipated how this story will unfold. His text was recognised as noteworthy in the context of battling the pandemic by an algorithm that had been sweeping through published research papers. This, in turn, inspired our researcher to create a model dedicated to the examination of the pathogen spreading in the air.
It all started last year, in June, when Professor Maciej Paszyński together with his team, during the International Conference for Computational Science, published an article in which they put forward a method for computer modelling of the advection-diffusion phenomenon, i.e., the flow and transport of a substance. Such physical phenomena are described in mathematical models by means of complex simultaneous equations, where every important variable is expressed with numerical parameters. It is clear that modelling complex processes requires an enormous amount of calculations. If not for adequate mathematical methods, even modern supercomputers would not be able to cope with the calculations in an acceptable time span. In addition, it is advised against overloading such computers, as they consume a lot of energy, the production of which is not only expensive but affects the environment.
The first article
Perhaps we should return to the aforementioned article for a moment, in which Professor Paszyński put forward an elaboration on the ADI method, used since the 1960s to solve time-dependent problems. It allows for the splitting of one calculation process into several smaller processes that happen simultaneously. The totality of the processes aims to achieve one common solution but much more efficiently than in the case of not splitting the calculations. To date, the ADI method has been used to solve two-dimensional problems. Professor Paszyński’s contribution, however, showed a solution which facilitates the use of the method for three-dimensional problems. “My contribution was the generalisation of the ADI method so that it could deal with B-spline basis functions. These kinds of three-dimensional functions, which approximate the sought values in numerous places in a given space, allow us to run more precise simulations”, explains the researcher.
As it turned out, the effort was soon appreciated. Firstly, by a computer programme which operates on AI, and which the Springer Nature publishing house, the publisher of the article, uses to sift through its publications in order to identify materials which would potentially benefit the fight against the spread of COVID-19. Subsequently, by the publisher’s head who wrote a personal letter to Professor Paszyński asking permission to publish the article in an open-access formula. This was the way of answering the global motion to share such publications with the World Health Organization (WHO) and global research paper libraries.
The model of COVID-19 propagation by coughing
The distinction became an inspiration to conduct further research. Not long after, relying on the method described earlier, Professor Paszyński in collaboration with Professor Ignacio Muga from the Pontificia Universidad Católica de Valparaíso in Chile, proposed a way of modelling the phenomenon of COVID-19 propagation by coughing. The researchers described it in an article published in the Bulletin of the Polish Academy of Sciences. The simulation run with the use of the model showed that the range of the virus spreading in air is nearly 4 times smaller when a person is wearing a face mask. The cloud of pathogens resulting from coughing is also significantly reduced and lingers shorter in the air.
“We’ve modelled it using advection parameters, measuring the “wind” force going through the fabric. It is obviously a prototype. In the future, for instance, we may think about how a face mask should be made so that the spread of the virus is minimal. Or, for example, launch simulations showing how the geometry of the surroundings affects what’s happening with the cloud of pathogens”, says Professor Paszyński.
Due to the optimisation carried out by the researchers, performing the aforementioned simulation does not require the use of computers with extraordinary processing powers. An ordinary laptop with an average processor would suffice, and we all have such gear at home.
A2S research team
Methods for computer simulations, on which Professor Paszyński and his A2S team work, are based on transformations of functions. The researchers divide the space into an extensive number of geometric shapes which are described by means of functions. The models created by the researchers have adaptive capability, i.e. they automatically conform to the problem that is being solved. The team worked on, among other things, models that allowed examination of the progression and regression of cancer under various types of therapy; the impact of the mining industry on the local environment; or the course of atmospheric phenomena. “We want to do something that is important for the society”, says the team’s head.
During the implementation of the models in the computer environment and the execution of the simulations, Professor Paszyński’s team has been working in collaboration with Professor Keshav Pingali from the University of Texas at Austin, who created the GALOIS library dedicated to such endeavours.
“Laptops or work stations have multiple cores, sometimes several dozen of them. The GALOIS library can be used, in turn, to simultaneously process functions; i.e., it allows us to perform transformations of functions in various places of the said function concurrently, using multiple cores at the same time. It is necessary if we want our calculations to be ready several dozen times faster”, explains Professor Paszyński.
The researcher emphasises that the advantages of working with Professor Pingali are reciprocal: “He and his team are creating the library and we accelerate the simulations, which makes him very happy because we give him interesting applications which, at the same time, verify if his library works well”.
Fast and stable simulations
The simulation methods described by Professor Paszyński’s team are characterised by linear computational cost. This means that the amount of time required to solve a problem increases proportionally to the complexity of the problem. For example, when the solution approximates a million functions, the amount of time necessary to calculate them is 10 minutes; in the case of 10 million functions, it is 100 minutes. If the computational cost was quadratic, then the amount of time needed to calculate those 10 million functions would not be 100, but 10,000 minutes, etc.
“Computers are getting bigger and faster, enabling us to launch even greater simulations. But if the computational complexity was not linear, then even the biggest computers would not be able to run such simulations. Only algorithms that scale themselves linearly will survive the development of computers”, explains the researcher.
One of many challenges that the creators of models face is designing them to work stably. Professor Paszyński explains: “We try to transfer reality to a computer which has its limits. The representation of figures is only approximated, which means that their values are stored up to the 16th decimal place. If something occurred in the 20th decimal place, the computer would not be able to process that. If it was something significant, the simulation would cease to work. We have to create extremely sophisticated mathematical methods so that our simulations do not ‘explode’”.
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