aa r X i v : . [ c s . G L ] M a r Grasping Complexity
A. N. Gorban
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
G. S. Yablonsky
Parks College, Department of Chemistry, Saint Louis University, Saint Louis, MO 63103, USA
The century of complexity has come. Many people write and speak about complexity. The statement ofthe great physicist Stephen Hawking, “I think the next century will be the century of complexity,” in his‘millennium’ interview on January 23, 2000 (San Jose Mercury News) became a widely cited prophecy.The face of science has changed (see cartoon in Fig. 1). Surprisingly, when we start asking about theessence of these changes and then critically analyze the answers, the result are mostly discouraging. Whydo we talk about complexity? Somebody might answer that now we have to study non-linear systems andtherefore they are complex. The answer seems to be plausible, nonlinearity results in non-additivity of partsand in the emergence of new phenomena: “The whole is more than the sum of its parts.” But objectionappears immediately: non-linearity has been in the focus of scientific research already for more than acentury. Poincar´e and Lyapunov have studied nonlinear systems more than a century ago. Boltzmann’sequation and Navier–Stokes equation, the great nonlinear equations are more than a century old. Manyideas have been created and many methods developed. The study of non-linearity is not a symptom of thechange of era. More than a thousand years ago Aristotle had written that “the whole is something besidesthe parts” (Metaphysics, Book 8, Chapter 6) and the Western culture had accepted this idea from the verybeginning. By the way, ‘besides’ in this translation of Aristotle sounds much more precise than the widelyspread ‘more’.
Figure 1: Change of era: The direction is changed dramatically and the history of our motion is like a hood behind ourshoulders. To describe our recent direction we need to understand our past. Graphics by
Mikhail Molibog . We need another idea to understand the recent change of era and some people add that we have tostudy large systems, both large and non-linear. Does the idea of large dimension give us the key for
Email addresses: [email protected] (A. N. Gorban), [email protected] (G. S. Yablonsky)
Preprint submitted to Elsevier March 18, 2013
Known
Laws, beautiful and simple A miracle: A phenomenon that contradicts the known laws, to the best of our knowledge. Figure 2: The flight from miracle: Einstein’s road. understanding of new era? Not precisely! The curse of dimensionality is now a well known problem andthe term was proposed by Bellman in 1950s [1]. Fifty years before, in 1900, David Hilbert in his address tothe International Congress of Mathematicians in Paris has described 23 major mathematical problems tobe studied in the coming century [2]. The title of one of these problems sounds very strange and too broad“Mathematical treatment of the axioms of physics” but if we read beyond the title then we immediatelyrealize what has been the main problem for Hilbert: “As to the axioms of the theory of probabilities, itseems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactorydevelopment of the method of mean values in mathematical physics, and in particular in the kinetic theory ofgases.” He continues: “Boltzmann’s work on the principles of mechanics suggests the problem of developingmathematically the limiting processes, there merely indicated, which lead from the atomistic view to thelaws of motion of continua.” In the modern scientific jargon, Hilbert had asked about the correct methodsof level jumping and model reduction, from large number of interacting particles to mechanics of continua.For this purpose, he proposed to develop the theory of probability and other related disciplines. This is thestruggle with complexity of large nonlinear systems recognized as one of the most important problems formathematics of the 20th century.It is normal when the change of epochs under close examination looks as a continuous development, notas a jump. But we talk about a century of complexity and suddenly find that it was started more than acentury ago. Perhaps, the idea ‘nonlinearity+large dimension’ cannot separate the new era in spite of itsattractiveness and clearness. To understand the essence of changes we have to ask not only what appearsbut also what has gone (Fig. 1).What have been the most important scientific achievements of the 20th century? The new great laws:the great parade of the great discoveries, from the relativity and quantum mechanics to genetics and DNA.One of the main players of this great period, Albert Einstein, has described the discovery of the new laws asa “flight from miracle”: “The development of this world of thought is in a certain sense a continuous flightfrom the ‘miracle’.” [“Die Entwicklung dieser Gedankenwelt ist in gewissem Sinn eine best¨andige Fluchtaus dem ‘Wunder’,” [3].] What does it mean? Let us imagine: we have the laws, beautiful and simple(the Newton mechanics, for example). We find a phenomenon that we cannot describe using these laws.This is a miracle , a phenomenon that contradicts the basis laws. We trust in these laws, we know thatthey are supported by the previous development of science, we like them and try to use them again andagain to describe the miracle. If we fail then we have to use another way. We like our laws but we like therationality more, therefore we fly from the miracle by inventing new laws, which are beautiful, simple and,at the same time, allow us to describe the phenomenon. After that, the miracle disappears and we havenew laws, beautiful and simple (Fig. 2)This scheme can be explained much deeper with more historical details and examples, but the main2teps are clear: we look for a miracle and find a phenomenon that seems to be in contradiction with thebasic laws; we try to demystify this miracle by rational explanations and models based on these laws; afterseveral attempts and failings we decide that new laws are needed and try to find new beautiful and simplelaws that demystify the new phenomenon and still can explain the other known phenomena not worse thanthe old laws.A new scheme of actions became dominant in the struggle with complexity. The complexity is recognizedas the gap between the laws and the phenomena. We assume that the laws are true. We can imagine a‘detailed’ model for a phenomenon but because of complexity, we cannot work with this detailed model. Forexample, we can write the Schr¨odinger equation for nuclei and electrons (formally, using indexes and signs ofsummation) but we cannot use them directly for modeling of materials or large molecules. We can imaginea detailed kinetic equation for a reaction network but cannot find reaction rate constants and cannot workwith this large system even if it is true.In some cases, bridging this gap between the laws and the phenomena can be achieved in model engi-neering by the special interaction between theoretical and experimental studies, and real engineering as well.Both the basic theory and the experiment will support the process of modelling. They may substitute foreach other. For example, we can make experiments instead of solving the extremely complicated equations.We are sure that the answer should be the same after filtering the noise for experimental errors. We also canorganise computational experiments instead of real ones. Again, we are sure that the answer should be thesame after cleaning the results from the errors. In the background of this belief the fundamental assumptionthat the possible world of the theory coincides with the real world of our experiments and practice (withsufficient accuracy). We can believe that somewhere else, for high energies, very small distances, or verylarge distances we need new laws, but not now.The interaction between theory and experiment in the model engineering may generate not only mathe-matical models but new experimental technics as well. For example, in chemistry, non-steady-state activityscreening can be based on the technique of Temporal Analysis of Products (TAP), invented by John Gleavesin 1988. The main idea of TAP is to treat the catalyst by a series of pulses of very small intensity relativeto the amount of catalyst [4]. This infinitesimal approach can be termed ‘chemical calculus’.The result of the struggle with complexity is a model that works. This is a sort of engineering: a modelis a device and this device should be functional. Applied mathematics and mathematical modelling becomea sort of engineering and instead of Einstein’s flight from miracle (Fig. 2) another scheme arises (Fig. 3):We know the laws and we have a phenomenon. We need a model for work. For different work need differentmodels are needed. We may combine the first principles, the empirical data and even the active experimentto create the model. There exist special technologies for testing and validation of models. The structureof the whole process seems to be similar to the design of machines and it might be reasonable to teach thestudents in applied mathematics the module of ‘Systems Engineering’, as a guide the engineering of complexsystems [5].The focus has moved from the revolution in laws to the production of intellectual devices. In the contextof the natural sciences this is model making under given basic laws. On the other hand, the systems underconsideration may be artificial and instead of the basic laws we deal with the man-made plans, projects andscenarios. Such systems as the Internet, social institutions, large plants, financial system and many othersystems are now in the focus of attention together with natural phenomena. The hybrid systems, that obeythe natural laws but experience significant influence of human activity and man-made projects are of greatinterest too, like climate or biosphere.The nature does not change and there will be many new laws to discover. The application of sciencealways exists too. The era of complexity is in the change of the focus of the research activity. From the epochof the great scientific revolutions we have moved to the epoch of the intellectual devices, from the revealingthe God’s or Nature plan to the intellectual engineering at various scales that is necessary to provide toolsfor prediction of the results of human activity. The new epoch may be ended some day but this is difficultto predict.The milestones of development rarely coincide with the ends of calendar centuries. We believe thatthe ‘century of scientific revolutions’ is situated between two giants, from L. Boltzmann to R.P. Feynman.Surprisingly, their contribution in the era of complexity is also huge. We can just recall Boltzmann’s entropy3 A phenomenon The
Basic
Laws,beautiful and simpleA complex modelthat follows the basic laws, but does not work , and we believe it is true A model that works A model is a device that works, Applied mathematics becomes
MODEL
ENGINEERING
Figure 3: Struggle with complexity: the life battle of the model engineers. [6] and Feynman’s inventions of nanotechnology [7] or quantum computers [8].In the struggle with complexity there are many specific problems and tools. This issue presents severalslices of this activity:1. Measuring complexity: the curses and blessings of dimensionality;2. Model reduction and invariant manifolds;3. Fingerprinting, criteria, and interpretation of experiments;4. Modelling of classes of complex systems.In the first part,
Measuring complexity: the curses and blessings of dimensionality , the general problemsare approached. The first general problem is the curse of dimensionality . V. Pestov [9] demonstrates howthe curse of dimensionality affects the nearest neighbor search and the widely used kNN classifiers. Hedemonstrates how the performance of the kNN classifier in very high dimensions can become unstable.Then, he develops a procedure for the reduction of the multidimensional statistical learning problems to aone-dimensional problem by a Borel isomorphism of the spaces with measure.High dimensional problems are not always complex. From a certain point of view, they look much simpler:the central limit theorem in probability and the advanced results about measure concentration [10, 11, 12]demonstrate how convex sets in high dimension become ‘almost spheres’, and typical distribution functionslook like Gaussians. This phenomenon (we call it the blessing of dimensionality ) was recognised first instatistical physics by Maxwell and Gibbs [13]. For multiparticle systems (under some technical assumptions)the microcanonical ensemble with the given values of energy is equivalent to the canonical one which can berepresented by the entropy maximum with the same average energy. The Maximum of Entropy (MaxEnt)approach naturally appears in the limit of high dimension.In the middle of the 20th century, after C. Shannon’s works [14] and E.T. Jaynes papers [15], the MaxEntapproach became very popular as a maximization of the subjective uncertainty measured by the Boltzmann–Gibbs–Shannon entropy. In 1960, A. R´enyi invented non-classical entropies [16]. Csisz´ar, Morimoto, Tsallisand many other researchers developed this idea further, and now we have the rich choice of the entropies formany problems. This rich choice leads to the ‘uncertainty of uncertainty problem’: which entropy to use forthe uncertainty measurement? A.N. Gorban proposes to use all the entropies together [18]. This approachresults in a set of conditionally ”most random” distributions. Surprisingly, this set allows constructivedescription. This new ‘Maxallent’ (Maximizers of all Entropies) method is based on the understanding ofentropy as a measure of uncertainty which increases in Markov processes [17].In the work of M. Grmela [19], the Dynamical Maximum Entropy Principle is elaborated. It coversequilibrium and non-equilibrium thermodynamics and gives new approaches to some classical problems. Inparticular, the classical Chapman–Enskog expansion in the theory of Boltzmann’s equation [20] is describedby the entropy deformation. 4. Zinovyev and E. Mirkes develop the data approximation approach to measure the complexity ofdatasets [21]. They utilize the universal approximators, principal cubic complexes, and generalize the notionof principal manifolds and graphs [22] for datasets with nontrivial topologies and are constructed with agrammar of elementary graph transformations. Three natural types of data complexity are used and testedin the case studies: the geometric, structural and construction complexity.Idempotent and tropical mathematics provide asymptotic versions of the classical mathematics producedby the ‘dequantization’ procedure [23]. G.L. Litvinov evaluates the complexity of the algorithms for theidempotent problems and their interval versions and demonstrated that they may be much simpler than inthe classical mathematics [24].Model reduction is one of the major procedures in the struggle with complexity and section
Modelreduction and invariant manifolds in the issue includes papers about reduction of dynamical models. M.Slemrod [25] revisits the sixth Hilbert problem and demonstrates that the solution has to be negative forcompressible gas dynamics: the hydrodynamic limit does not lead to the classical compressible Euler orNavier–Stokes equations. This situation differs from the incompressible limit [26]. The key to this analysisis provided by the exactly solvable reduction models discovered by A.N. Gorban and I.V. Karlin [27, 28].Slow invariant manifolds are the main tools for model reduction in dissipative systems [29, 30]. The fastmanifold traditionally attracts less attention and plays an auxiliary role. It is used mostly for projection ofa motion on an approximate invariant manifold. V. Bykov and V. Gol’dshtein [31] demonstrate how to startmodel reduction procedures from fast manifolds and develop a theory of Singularly Perturbed Vector Fields(SPVF) with the main emphasis on fast invariant manifolds. The slow manifold appears as a by-product ofthis approach. The new approach is illustrated by the examples from chemical kinetics.The Lam and Gousis Computational Singular Perturbation (CSP) approach aims to find both fast andslow manifolds for a system of differential equations [32]. It was developed for application in chemicalkinetics. In their paper [33], P.D. Kourdis, A.G. Palasantz, and D.A. Goussis develop the algorithmicrealization of CSP and apply it to important biochemical systems with oscillations, the NF- κ B signalingsystem.The problem of model reduction for systems with symmetries is analyzed by B. Sonday, A. Singer andI.G. Kevrekidis [34]. They use the Kuramoto-Sivashinsky equation with periodic boundary conditions anda stochastic simulation of nematic liquid crystals as examples, and apply the eigenvector-based techniquesfor model reduction. They also use a new technic, Vector Diffusion Maps [35], that combines, in a singleformulation, the symmetry removal step and the dimensionality reduction step.B.R. Noack, R.K. Niven [36] develop further a MaxEnt closure strategy for Galerkin systems arisingfrom a projection of the incompressible Navier-Stokes equation onto orthonormal expansion modes. Theyaim to discover and demonstrate a new face of the turbulence closure problem.R. Hannemann-Tamas, A. Gabor, G. Szederkenyi, and K.M. Hangos formulate the model reductionproblem for chemical kinetics as a quadratic programming problem [37]. The objective function is derivedfrom the parametric sensitivity matrix. The method eliminates unnecessary reactions for a given level oftolerance and adjusts the rate constants of the remaining reactions for error minimization. The efficiency ofthe approach is demonstrated on the known benchmarks.The transition from dynamics to thermodynamics is the most complicated step on the stair of reduction[30]. In the paper by T. Chumley, S. Cook, and R. Feres [38] this step is analyzed for billiard-like randomsystems. These systems exhibit irreversible thermodynamics behavior, indeed.The ideal model reduction technology starts from the detailed system and produces the reduced one.This picture may be oversimplified. Indeed, in many practically important cases the mathematical modelcannot be produced without simplifications and model reduction becomes a tool for model constructionfrom scratch. It may be also used for construction of semi-empirical methods and active theory-drivenexperiments. In engineering, many semi-empirical criteria were invented to separate regimes: laminar fromturbulent, shocks from smooth incompressible flows and many others. The modern fingerprinting idea mayfind its logical roots in the semi-empirical criteria. “The goal of the fingerprint analysis is to find features andcharacteristics of observed complex behavior, based on which it is possible to find out the model, its class orits family, and to determine its characteristics” [39]. The fingerprints, patterns, signatures or motifs allowus to work with complex systems without extraction of deep and expensive information. Kinetic signatures5n biochemical reactions [40], motifs of genetic sequences [41] patterns in time series [42] (cardiogramms andencephalogramms, for example) give us nice examples of fingeprinting.The paper by D. Constales, G.S. Yablonsky, and G.B. Marin [43] opens Section
Fingerprinting, criteria,and interpretation of experiments . They study the basic patterns in simple reaction networks. This workaims to analyze appearance of some basic patterns in chemical kinetics, to review and extend the previousfindings [44]. Authors supplement the classical notion of complexity by ‘simplexity’ to reflect the richdiversity of patterns which can be produced even by simple systems.A useful example of a criterion validation is given in the work by F. Xia and R.L. Axelbaum [45]. Theypropose to use the local ratio C/O to classify various regimes and zones of diffusion flames. Radical pooland soot precursor zones are shown to be clearly delineated in C/O ratio space. This ratio is validated as acriterion for interpreting flame structure.M.J. Hankins, T. Nagy, and I.Z. Kiss [46] develop an original technology for active experiment forconstruction of nullcline-based models and demonstrates its efficiency on the modelling of the electrochemicalreaction. Perhaps, the first author who proposed to use the nullcline-based models instead of detaileddifferential equations was A.N. Kolmogorov [47, 48]. M.J. Hankins et al use the nullcline-based models withthe singular pertirbation assumption (time scale separation). Under this assumption, the nullclines may beextracted from the control experiment with a combination of active and proportional controllers acting onthe fast and the slow variables.The section
Modelling of classes of complex systems includes four papers about four classes of systems:networks, finance, catalysis (in chemical engineering) and bioreactors. The new tools and case studies arepresented. H. Sayama, I. Pestov, J. Schmidt, B.J. Bush, C. Wong, J. Yamanoi, and T. Gross describe themethods based on adaptive networks with self-organization of structure for modeling of complex networks likesocial, transportation, neural and biological networks [49]. B.E. Baaquie describes a quantum mathematicsapproach to financial modelling [50]. F.J. Keil presents a thorough review about modelling in catalysis,from quantum chemical methods for calculating reactions on the active centers to transport in porous media[51]. I. Iliuta and F. Larachi study dynamics of bacterial cells in trickle-bed bioreactors. They model thebasic processes, fluxes in multiface flows, population balance for cells and agglomerates, biomass dynamics,dynamics of agglomeration and filtration [52].Neither one issue of a journal, nor a large encyclopedia can capture everything about such a broad anddynamic subject as grasping complexity, but we hope that various faces of the modern era of complexity arepresented here.
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