A very small cause which escapes our notice
determines a considerable effect that
we cannot fail to see, and then we say that
the effect is due to chance.
(Henri Poincare ,  1854–1912)

Stochastic models, complex systems and networks, uncertainty

In modeling many phenomena and processes in Nature as well as in man-made systems there is often some indeterminacy in the future evolution of such systems, phenomena, and processes that, by the language of mathematics, can be described by probability distributions. This includes both traditional areas of science & engineering as well as such areas as finance, business, social sciences and economics where in a number of cases tools of applied probability become very important.

In this focus area, our research interests have included the following topics:

• Stochastic models in biological and environmental sciences;
• Monte Carlo based non-perturbative approaches, critical phenomena and phase transitions in large lattices;
• First passage time problems in finance, science & engineering, risk assessment and uncertainty management, human factors, complex systems and networks;
• Markov Chain approximation and evolution.

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Stochastic models in biological and environmental sciences

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in [ EJ-130 ] we developed a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. We considered the influence of stochasticity in the analysis of cell cycle dynamics by accounting fluctuations of relative concentrations of proteins during special events of cell cycles and demonstrated that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

Most interesting problems in biological, environmental and materials sciences are multiscale with a range of spatial and temporal scales as discussed in [ EJ-58 ]. In solving such problems with atomistic and Molecular Dynamics (MD) tools, an initial topological optimization can lead to a substantial speed up of subsequent computations. New efficient algorithms for such an optimization were discussed in [ EJ-58 ] based on several distance geometry ideas. In applying MD tools, we often have to choose between different schemes of stochastic thermostats, e.g. Langevin, Berendsen, etc in such a way that the `fluctuation-dissipation' theorem is obeyed to guarantee the chosen statistics. These tools provide a powerful technique for the analysis of complex biological structures. In [ EJ-137 ] we applied these tools for the analysis of RNA nanostructures where we discovered a phenomenon of self-stabilization. Ribonucleic acid (RNA) has been proposed as a promising alternative to DNA and proteins for the design of the artificial self-assembled materials at nanoscale (e.g. [1] and references therein). As compared to DNA and some other bio-molecules, RNA offers not only a much greater variety of interactions but also such an important feature as great conformational flexibility of RNA structures. This feature is used by Nature via the ubiquitous catalytic function of RNA, and now it also makes RNA a very promising functional nano-material for the design of the man-made molecular devices and systems. We presented several promising multiscale methodologies for simulation of such biomolecular systems as RNA nanostructures, focusing on the development of a hierarchy of simplified coarse-grained mesoscopic models suitable for the description of such biological nanostructures and details of this can be found in [ EJ-145 ].

The coarse-graining approach is one of the most important modeling methods in research of long-chain polymers such as DNA molecules.  We applied several such models for the analysis of polymeric fluids and relaxation of DNA bio-molecules [ EJ-105 , EJ-109 , EJ-126 ]. We analyzed the behavour of these systems under different shear rates [ EJ-109 ], extensional flows [ EJ-126 ], and provided new results of Brownian dynamics simulations of single DNA molecules in shear flows taking into account the effect of internal viscosity [ EJ-105 ] where our model included the Smoluchowski equation.

Climate systems models have been discussed by us earlier in [ EJ-23 ] with the error analysis discussion found in[ EP-3 ]. Other topics of interest in biological, medical and environmental sciences can be found here.

Monte Carlo based non-perturbative approaches, critical phenomena and phase transitions in large lattices

It is well-known that restricting geometry of a many particle system can have fundamental effects on phase transition phenomena.  Even for relatively simple ideal geometries, we frequently observe rich behaviors due to a competition between different types of effects, e.g. surface effects and finite size effects.  When the transition changes its character (e.g., form three-dimensional to two-dimensional as well as in other cases), critical exponents will also change. It is very important to be able to predict such exponents with a sufficient degree of accuracy in a range of physically relevant situations. In a series of publications [ EJ-103 , EJ-115 , EJ-143 ] we focused on the study of a number of lattice spin models near the critical (phase transition) point by using non-perturbative approaches based on Monte-Carlo methods.  We started with the XY models and demonstrated how more accurate (than previously known) estimates of critical exponents can be obtained [ EJ-103 , EJ-115 ] . We also applied our methodology to the analysis of Goldstone singularity in such models by using the efficient procedures we initiated [ EJ-103 ], and applied the three-dimensional O(4) model in the ordered phase to study the Goldstone mode effects [ EJ-143 ]. The Wolff cluster algorithm provides one of the most effective tools to simulate important properties of many particle systems near the critical point. However, a major drawback in the application of this algorithm were related to difficulties in its parallelization, in particular in the application to large lattices. On the example of three-dimensional Ising models, in [ EJ-142 ] we demonstrated for the first time how parallelization of the Wolff single cluster algorithm can be achieved. Our procedure is generalizable to other lattice spin models and provides an important new tool for the simulation of very large lattices near the critical phase transition point.

First passage time problems in finance, science & engineering, risk assessment and uncertainty management, human factors, complex systems and networks

We are interested in modelling financial and socio-economic phenomena where uncertainty presence may lead to qualitatively different dynamics. Our major interests here focus on the First Passage Time problems and the development of efficient numerical methodologies for their solution, but our interests include also inverse problems in financial mathematics and several other areas. Mathematically, FPT problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution.  We developed a fast Monte Carlo method in the case of multivariate (and correlated) jump-diffusion processes  ( JDP). This development and its generalization [ EJ-108 , EJ-128 ] allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises.

Uncertainly management is another area of interest. We were analyzing uncertainty management in the presence of human factors. It is known that with the increasing complexity of technological systems that operate in dynamically changing environments and require human supervision or a human operator, the relative share of human errors is increasing across all modern applications. This indicates that in the analysis and control of such systems, human factors should not be eliminated by conventional formal mathematical methodologies. Instead, they must be incorporated into the modelling framework. These issues were addressed in [ EJ-129 ]. We demonstrated that the problem can be reduced to a set of Hamilton-Jacobi-Bellman equations where human factors are incorporated via estimations of the system Hamiltonian. The proposed methodology provides a way to integrate human factors into the solving process of the models for complex dynamic systems.

In a number of applied problems we have to deal with highly oscillatory functions characterizing certain processes or phenomena and given with incomplete information. The development of efficient formulae for numerical integration of such functions is a well-known challenge, and we developed and theoretically justified several classes of optimal formulae for numerical integration of highly oscillatory functions in one- and multi-dimensional cases [ EJ-27 , EJ-43 , EJ-48 , EJ-66 ], focusing on the cases with incomplete information available.

Markov Chain approximation and evolution

Many concepts and tools used in information theory, control, theory of approximation, dynamical systems and artificial intelligence are closely linked. Among such tools are neural networks. We proposed a general framework for the analysis of a connection between the training of artificial neural networks via the dynamics of Markov chains and the approximation of conservation law equations [ EJ-120 ]. This framework allows us to demonstrate an intrinsic link between microscopic and macroscopic models for evolution via the concept of perturbed generalized dynamic systems introduced earlier in [ EJ-25 ].  A number of related theoretical questions were analyzed in [ EJ-21 , EJ-60 ]. In particular, in [ EJ-60 ] we analyzed in detail a theoretical framework for deterministic and stochastic dynamics with hyperbolic Hamilton-Jacobi-Bellman equations. It was proposed to approximate the original controlled dynamics with a sequence of PDEs. By applying the Steklov-Poincare operator technique the general form of equations in such a sequence was established. In [ EJ-21 ] we dealt with nonsmooth optimal control problems in the case when the control was allowed to be a discontinuous function. We analyzed smoothness assumptions on an adjoint process in deterministic and stochastic cases. Possibilities of steep generalized space-gradients of the adjoint function implied the necessity of an approximation of the Hamiltonian. The key question of such an approximation is a relationship between the control and the value function. Under quite general assumptions it was proved that the performance measure for the original process is determined by the control function with possible discontinuities.

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