Mathematical forecasting of the future

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FACULTY OF MATHEMATICS AND COMPUTER SCIENCES

 

Scientists from the Division of Computational Mathematics of the Faculty of Mathematics and Computer Sciences of the Jagiellonian University have been conducting a wide-scale research project devoted to rigorous numerical analysis of dynamical systems for nearly twenty years. Their advanced mathematical studies are related, among other things, to forecasting the future.

It is not easy to explain the nature and benefits of this research without getting into mathematics and its specific language. However, when attempting to do so, it is worth delving into the historical scientific background of the subject.

The power of differential equations

People have always wanted to foresee the future and be able to influence it. When Isaac Newton published his famous three laws of dynamics in the second half of the seventeenth century, the forecasting of future events bravely entered the field of scientific research. This was due to the discovery of differential and integral calculus by Newton and Leibniz, which enabled to state the laws and processes of physics in the form of differential equations.

Contrary to algebraic equations from school, the solution of a differential equation is not a number, but a function describing the course of a given physical process in time. Additionally, there are an infinite number of solutions. In order to select one specific solution, the value of the function at a given moment is set. In the case of forecasts, the needed function is selected with the use of the state of the process at the current moment, determined through measurement. Calculation of the value of the function for future moments results in the expected forecast. If we change relevant parameters of the process, and thus also the equation coefficients, we may control it, obtaining a predefined objective in the future. This method is the foundation of the whole scientific and technological revolution of the eighteenth and nineteenth century.

Unfortunately, most differential equations cannot be solved precisely, so traditional approximation methods, or so-called numerical methods, are used. They are affected by various types of errors, in particular flaws related to the necessity of rounding in calculations. As a consequence, one cannot be certain that a solution is acceptable. This is usually tested experimentally, and the accuracy of calculations is increased if necessary.

However, sometimes conducting an experiment can be costly and unproductive. The first flight of the Ariane 5 rocket in 1996 ended in a disaster precisely because of an error in numerical calculations. Four satellites that the Ariane was supposed to put into orbit were lost. Total losses amounted to 70 million dollars. This was the most expensive failure resulting from numerical errors in history, and it undermined trust in the European space program for many years.

Consequences of problems with weather forecasting

Adjusting the solution by means of conducting more accurate calculations is based on the assumption that a small difference in the initial data leads to negligible differences between solutions. The analysis of differential equations connected to weather forecasting has distorted this simple vision.

From a theoretical point of view, the method developed by Newton is perfectly suitable for weather forecasting. As we have differential equations describing the state of the atmosphere, it is enough to solve them to obtain a forecast. This approach, which, by the way, led to a miserable failure, was tested for the first time in 1922 by the English scientist Lewis Fry Richardson. He explained that his failure was a result of the inaccuracy of calculations.

The creation of the first computers revived hopes for successful weather forecasts based on numerical methods. However, success was limited: only short-term forecasts were accurate. Edward Lorenz, an American mathematician and meteorologist, suggested in the 1960s that there might exist differential equations with a sensitive dependence on solutions of initial conditions, which is today shortly described as chaos. In such equations, even the slightest change in the initial conditions leads to enormous changes in solutions.

For differential equations in which the phenomenon of chaos occurs, numerical analysis is significantly more difficult and, in the case of long time intervals, even impossible. Even worse, it is generally impossible to determine whether chaos occurs in an equation with the use of classical numerical methods. The realization of this fact initiated a search for alternative methods.

A tool which enabled the accurate estimation of rounding in numerical analysis, so-called interval arithmetic, was developed in the mid-twentieth century by Polish mathematician Mieczyslaw Warmus and, at the same time, by the American scientist Ramon E. Moore. The essence of interval arithmetic is conducting calculations on intervals in the form [a, b] so that the accurate, although unknown, result certainly lies within the calculated interval. Unfortunately, the size of the interval increases with the number of calculations performed, and thus the practical value of the estimation decreases. Interval arithmetic itself does not enable us to determine whether we are dealing with the phenomenon of chaos in a differential equation, either. However, it turns out that when it is connecting to topology, such a verdict is possible.


Visualization of the differential equations discovered by Lorenz.
It demonstrates the sensitive dependence of solutions
on initial conditions, so-called deterministic chaos

When a cup and a donut are indistinguishable

Topology is a separate area of mathematics that emerged at the beginning of the twentieth century in the works of the French mathematician Henri Poincaré on the stability of differential equations describing the dynamics of our planetary system. This field of mathematical studies, which has already become independent, concerns the properties of space (e.g. geometric figures) that are preserved under continuous deformations including stretching and bending, but not tearing or gluing.

Such properties include the number of holes in space. From the point of view of topology, a cup with a handle and a donut are indistinguishable, as the cup can be transformed into a donut without tearing or gluing. In this process, the opening in the handle of the cup is transformed into the hole in the donut. Both of these objects are topologically distinguishable from a glass ball, which is devoid of any holes.

In the mid-1990s, Konstatntin Mischaikow and Marian Mrozek referred to topological methods demonstrating a new method, which practically documented the existence of chaos in a specific differential equation. Encyclopedia Britannica considered the result of their work to be one of the four most important achievements in mathematics in 1995. On one hand, it proved that rigorous numerical analysis of chaotic differential equations is possible. On the other hand, it posed a series of questions about performing it, as the original proof of chaos in the Lorenz equations required performing eighty hours of calculations.

Further tasks

This started a multi-dimensional project that has been in process since that time at the Division of Computational Mathematics, with the objective to achieve a rigorous and automated numerical analysis of a wide class of differential equations. "Our work covers, among others: the development of theoretical methods enabling a detailed analysis of solutions of differential equations basing on rigorous numerical methods, topological and analytical tools, but also the development of effective interval algorithms for differential equations, algorithms for the determination of socalled topological invariants, and algorithms enabling an automated and rigorous numerical analysis of differential equations," explains Professor Marian Mrozek, who coordinates the research carried out at the Jagiellonian University.

Within the realization of the main project, a surprisingly new and important parallel project emerged. It turned out that in recent years the number of applications of topology outside mathematics has rapidly increased. It is used in image recognition, the classification of texts, the analysis of experimental data, robotics, and sensor networks. "Our experiences related to topological algorithms proved very useful. They are now applied also outside the original designation which was aimed at the numerical analysis of differential equations," Professor Mrozek informs.

So far, scientists have succeeded in developing a CAPD library consisting of over 60,000 lines of code, which is used for rigorous numerical analysis of differential equations, and a sub-library CAPD::REDHOM, which encompasses the implementation of topological algorithms. The library contains computer programs that enable the practical application of the discussed methods. It is successfully used by numerous scientists, both in Poland and throughout the world.

Sampled dynamical systems are another important area of research. "The topological method of analysis of differential equations may be adapted for the analysis of processes for which we do not have a ready differential equation yet. However, the verification of such analysis must be based on a tool other than interval arithmetic. We are planning to use the method of so-called persistent homology for this purpose. If we succeed, we will have a very powerful instrument for automatic analysis of a wide class of dynamical processes — from differential equations through iterated function systems, to time series and parameterized dynamical systems," Professor Mrozek concludes.


Research team: Paweł Dłotko, PhD; Marc Ethier, PhD; Paweł Gniadek, PhD; Mateusz Juda, PhD; Tomasz Kapela, PhD; Małgorzata Moczurad, PhD; Professor Marian Mrozek; Daniel Wilczak, PhD; Professor Piotr Zgliczyński; Marcin Żelawski, PhD; Piotr Brendel, MSc; Aleksander Czechowski, MSc; Grzegorz Jabłoński, MSc; Piotr Kamieński, MSc; Łukasz Piękoś, MSc; Piotr Rytko, MSc; Robert Szczelina, MSc; Hubert Wagner, MSc; Irmina Walawska, MSc; Frank Weilandt, MSc

Major research partners: the Institute of Science and Technology, Austria and Rutgers University, New Jersey, USA.