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Extreme events (henceforth Xevents) occur in natural, technical and societal environments. They may be natural or anthropogenic in origin, or they can arise simply from “chance”. They often entail loss of life and/or materials. They usually occur “by surprise” and therefore often only become the focus of scientific attention after their onset. Knowledge of Xevents is often rather fragmentary, and recorded experience is limited. Indeed, scientists do not really understand causes extreme events, they develop, and and they occur. In addition, we are rarely able to cope with their consequences, due to lack of anticipation and preparedness. All this has motivated us, the editors of this volume, to bring together specialists from a variety of fields of expertise, all of whom have a common background in mathematics and physics. We asked them to write their views about Xevents. The result is the present book of essays that will (hopefully) enable the reader to unlock the mysteries surrounding Xevents.

The urgency explicit in soliciting scientists to address the prediction of Xevents is understandable, but not really conducive to a foundational perspective. In the following methodological considerations, a perspective is submitted that builds upon the necessary representation of Xevents, either in mathematical or in computational terms. While only of limited functional nature, the semiotic methodology suggested is conducive to the basic questions associated with Xevent prediction: the dynamics of unfolding Xevents; the distinction between Xevents in the deterministic realm of physics and the nondeterministic realm of the living; the foundation of anticipation and the possibility of anticipatory computing; the holistic perspective. As opposed to case studies, this contribution is geared towards a model-based description that corresponds to the nonrepetitive nature of Xevents. Therefore, it advances a complementary model of science focused on singularity, providing a nondeterministic understanding of high-complexity phenomena.

Mathematical tools for the analysis of Xevents, maxima of processes and rare events are presented. Methods and concepts of classical statistical extreme value theory are described, as well as those of large deviation theory. Techniques from other areas such as statistical mechanics, the theory of dynamical systems and the theory of singularities are also briefly discussed.

Due to their great impact on human life, Xevents require prediction. We discuss scenarios and recent results on predictions and the predictability of Xevents, focusing on nonlinear stochastic processes since they are assumed to provide the basis for extremes. These predictions are usually of a probabilistic nature, so the benefit of this type of uncertain prediction is an additional issue. As a specific example, we report on the prediction of turbulent wind gusts in surface wind.

Are large biological extinctions such as the Cretaceous/Tertiary KT boundary due to a meteorite, extreme volcanic activity or self-organized critical extinction cascades? Are commercial successes due to a progressive reputation cascade or the result of a well orchestrated advertisement? Determining the chain of causality for Xevents in complex systems requires disentangling interwoven exogenous and endogenous contributions with either no clear signature or too many signatures. Here, I review several efforts carried out with collaborators which suggest a general strategy for understanding the organizations of several complex systems under the dual effect of endogenous and exogenous fluctuations. The studied examples are: internet download shocks, book sale shocks, social shocks, financial volatility shocks, and financial crashes. Simple models are offered to quantitatively relate the endogenous organization to the exogenous response of the system. Suggestions for applications of these ideas to many other systems are offered.

The analysis of Xevents arising in dynamical systems with many degrees of freedom represents a challenge for many scientific fields. This is especially true for the open, dissipative, and adaptive system known as the . Due to its complex structure, its immense functionality, and — as in the case of epilepsy — due to the coexistence of normal and abnormal functions, the brain can be regarded as one of the most complex and fascinating systems in nature. Data gathered so far show that the epileptic process exhibits a high spatial and temporal variability. Small, specific, regions of the brain are responsible for the generation of focal epileptic seizures, and the amount of time a patient spends actually having seizures is only a small fraction of his/her lifetime. In between these Xevents large parts of the brain exhibit normal functioning. Since the occurrence of seizures usually can not be explained by exogenous factors, and since the brain recovers its normal state after a seizure in the majority of cases, this might indicate that endogenous nonlinear (deterministic and/or stochastic) properties are involved in the control of these Xevents. In fact, converging evidence now indicates that (particularly) nonlinear approaches to the analysis of brain activity allow us to define precursors which, provided sufficient sensitivity and specificity can be obtained, might lead to the development of patient-specific seizure anticipation and seizure prevention strategies.

Many Xevents in the geological past exceeded the strengths and intensities observed for modern-day natural events. The number of extraordinary events that occurred in the geological past is of course much larger than the number we witness today because the geological timescale covers millions of years. This contribution focuses on these Xevents from earth’s geological history, including selected examples from plate tectonics, earth magnetism, ice age cycles, volcanism, earthquakes, meteorite impacts and floods. Events related to these processes occur on different timescales. For example, drastic modifications of atmospheric and oceanic circulation due to continental shift (which creates new mountain ranges and reshapes land masses and oceans) take millions of years, while meteorite impacts happen within seconds. However, any these processes can be the trigger for dramatic consequences, like mass extinctions of life, or global glaciations. An overview of a research program that considers historic and prehistoric flood events is given. Based on the water levels observed during floods, the palaeodischarge can be determined and used to improve the reliability of flood predictions. Investigations of Pleistocene ice-dammed lake outburst floods (the largest flood events in the Earth’s history) are useful when developing new methods and techniques that can be applied to younger events of a smaller scale in other environments.

This chapter presents an overview of some typical meteorological extreme events. For reasons of conciseness we restrict ourselves to wind and precipitation extremes. The major goal is to emphasize the fact that very different types of wind or precipitation extremes may occur on different scales in space and time. The main phenomenological presentation is supported by short descriptions of conceptual models, in order to help the reader to grasp some of the underlying physics. We show that it is debatable as to whether the concept of universality holds for extremes, even for those involving atmospheric motion alone.

Rogue or freak waves sink ships at an alarming rate — estimated at one large ship every few weeks worldwide. It is thought that vulnerable ships (light cargo ships) simply break in two when they plough into a 60 foot wave preceded by a 40 foot hole in the sea, as some sailors that have survived such experiences have called it. Wave refraction due to current eddies (which are ubiquitous in the oceans) has long been suspected to play a role in concentrating wave energy into rogue waves. Existing theories have been based on refraction of plane waves, not the stochastic Gaussian seas one finds in practice. Gaussian seas ruin the dramatic focal caustic concentration of energy, and this fact has discouraged further investigations. Although it was thought that chaos, or the extreme sensitivity to initial conditions displayed by individual ray trajectories would quickly wipe out all significant fluctuations, we show that this is incorrect, and the fluctuations are “structually stable” entities. Significant “lumps” of energy survive the averaging over wave directions and wavelengths. We furthermore demonstrate that the probability of freak waves increases dramatically in the presence of these lumps, even though most parameters, such as the significant wave height, are unchanged. We show here that a single dimensionless parameter determines the potential for freak waves; this is the “freak index” of the current eddies — a typical angular deflection in one focal distance, divided by the initial angular uncertainty of the incoming waveset. If the freak index is greater than 2 or so, truly spectacular enhancements of freak index waves can result, even though the caustics are washed out by the Gaussian nature of the impinging sea.

Even today, lifetime predictions of construction parts are still based on the Wöhler method, which is almost 150 years old. To construct a reliable Wöhler diagram, it is necessary to perform alternating load fatigue experiments on a huge number of equivalent samples for up to 10 or 10 load cycles. The lifetime under a specific applied load is then deduced from this diagram using statistical techniques.

Physically, the reason for fatigue and finally fracture is the accumulation of lattice defects like dislocations, vacancies and vacancy clusters, which are produced even when the load is significantly below the material’s yield strength. The progress of fatigue can be observed from its earliest stages — after only a few load cycles — up to the final state of fracture by employing positrons as extremely sensitive lattice defect probes. In situ experiments can be performed to study test samples or real construction parts under realistic conditions. In steels a critical defect density is reached just before fatigue failure occurs. The point of failure can therefore be extrapolated from the early stages of fatigue by monitoring the defect density.

Spatially resolved experiments performed on a simple carbon steel and employing the Bonn Positron Microprobe indicate significant variations in defect densities over the region under stress even after just a few load cycles. These inhomogenieties grow from a typical starting size of less than a millimeter to encompass the entire volume after further fatigue. With more experimental experience and a better theoretical understanding of this process, this new prediction method should lead to much simpler and more reliable predictions of the lifetimes of metallic materials in the near future.