Ivan Kryven

Formation of planets out of the interstellar dust, emergence of tiny crystals, or growth of your network of friends: regardless colossal difference in the scales and the worlds, hidden similarities stand behind evolution of large populations. As a postdoctoral scholar at Molecular Science institute (HIMS, University of Amsterdam), I develop mathematical tools and models helping scientists to understand, simulate, and foresee these phenomena.

(PO 94720)
Science Park 904
1090 GS Amsterdam
Room number: C2.219

Collaborator network [ + ] [ - ]


Publications


  1. I. Kryven, S. Rüoblitz, Ch. Schütte. Solution of the chemical master equation by radial basis functions approximation with interface tracking, BMC System Biology 2015 [MORE] [PDF]
    Background: The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.
    Results: We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space. In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.
    Conclusions: The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models
  2. I. Kryven, PD. Iedema. Deterministic modelling of copolymer microstructure: composition drift and sequence patterns,
    Macromolecular Reaction Engineering [Special Issue: Statistical Modeling Tools and Approaches for Polymerization Reaction Engineering], Volume 9, Issue 3, pages 285–306, 2015; [MORE] [PDF]
    A concept of a population balance is applied to study evolving structures of propagating linear copolymer chains. As a result, a general numerical toolbox was developed capable to handle various copolymerization mechanisms utilizing two-dimensional distributions as means of system description. The method was applied to free radical copolymerization of styrene–acrylonitrile explaining peculiar bimodality in chain length distribution observed for asymmetrical initial monomer ratios.
  3. I. Kryven, PD. Iedema. Transition into the gel regime for crosslinking radical polymerisation in a continuously stirred tank reactor,
    Chemical Engineering Science, Volume 126, Issue 14, Pages 296–308, 2015 [MORE] [PDF]
    Crosslinking radical polymerisation in a continuously stirred tank reactor has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model covers both pre-gel and gel regimes in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The comparison with Monte Carlo simulations shows interesting and unexpected features.
  4. X. Cao, A. van Dam, B. de Leeuw, C. Geldhauser, J. Grasman, I. Kryven, D. Lahaye, L. Morelli, V. Rottschäfer, H. Zhou. Always Nice Weather in Europe. Proceedings SWI 2015 [MORE] [PDF]
    Weather forecasting relies on mathematical models that exhibit chaotic behavior. This renders the solution of these models very sensitive to errors in the model, to choices of the initial conditions and to rounding errors in the numerical solution procedure. Over the course of the past decade, various meteorological institutes in Europe have developed different at- mosphere models. Each of these models has its strengths and weaknesses. The principle behind the so-called Super Modeling approach is to merge these existing models into a single larger model to combine common strengths while overcoming individual weaknesses. This approach was initially proposed and developed by the KNMI in the Netherlands to improve the reliability of its weather forecasts. The task formulated for this Study Group problem was to reevaluate the Super Modeling approach and to formulate recommendations for its future development.
  5. I. Kryven, Topology evolution in macromolecular networks,
    Univeristy of Amsterdam, PhD Thessis, ISBN: 978-90-9028519-1, 2014; [MORE] [PDF]
    Governed by various intermolecular forces, molecular networks tend to evolve from simple to very complex formations that have random structure. This randomness in the connectivity of the basic units can still be captured employing distributional description of the state of the system; the evolution itself by particular stochastic processes, for instance Smoluchowski coagulation. The Smoluchowski coagulation equation can be extended to include collisions of orders distinct from 2, which allows developing a framework that covers various special cases being far beyond the scope of the original application of the coagulation equation. The combination of the population balance equation based on a generalization of the Smoluchowski coagulation and a meshless projection method based on Gaussian basis functions has been found to be a powerful tool allowing to solve many problems of prior unmanageable complexity. For instance, in the case of cross-linking polymerisation in bulk, the population balance model describing a particular set of reaction mechanisms has been present in literature for decades before it has been successfully and to a full extent solved by the numerical method. The other case studies address formation of various molecular networks in a few important cases: polymerisation of AB2 monomers, modification of linear chains into a branched polymer, copolymerisation with a composition drift, and coagulation/ coalescence of colloids of variable fractal dimension.
  6. I. Kryven, PD. Iedema. Transition into the gel regime for free radical crosslinking polymerisation in a batch reactor,
    Polymer, Volume 55, Issue 16, Pages 3475–3489, 2014; [MORE] [PDF]
    Crosslinking polymerization has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model, for the first time and to a full extent resolves the crosslinking problem as formulated by Zhu et al. [1] and covers both pre-gel and gel regimes, in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The model has been validated with data from an experimental crosslinking polymerization, Methyl Methacrylate with Ethylene Glycol Dimethacrylate. Non-trivial patterns in the time evolution of average quantities like crosslink densities, partly observed in prior studies [23], are naturally emerging from the model by computing marginals of the four-dimensional distribution possessing an interesting multimodal structure.
  7. I. Kryven, S. Lazzari, G. Storti. Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems,
    Macromolecular Theory and Simulations, Volume 55, Issue 16, Pages 3475–3489, 2014; [MORE] [PDF]
    A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion-limited-cluster aggregation (DLCA) and reaction-limited-cluster-aggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive
  8. I. Kryven, PD. Iedema. Topology Evolution in Polymer Modification,
    Macromolecular Theory and Simulations, Volume 23, Issue 1, pages 7–14, 2014 [MORE] [PDF] [JOURNAL COVER]
    A recent numerical method has opened new opportunities in multidimensional population balance modeling. Here, this method is applied to a full three-dimensional population balance model (PBM) describing branching topology evolution driven by chain end to backbone coupling. This process is typical for polymer modification reactions, e.g., in polyethylene, where initially linear polymer chains undergo hydrogen abstraction, and subsequent branching or scission. Topologies are distinguished by chain ends, number of branches, and number of reactive ends. The resulting time dependent trivariate distribution is utilized to extract various distributive properties of the polymer. The results exhibit excellent agreement with data from Monte Carlo simulations
  9. I. Kryven, PD. Iedema. Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding, Volume 54, Issue 14, Pages 3472–3484, 2013 [MORE] [PDF]
    A deterministic mathematical model for the polymerization of hyperbranched molecules accounting for substitution, cyclization, and shielding effect has been developed as a system of nonlinear population balances. The solution obtained by a novel approximation method shows perfect agreement with the analytical solution in limiting cases and provides, for the first time in this class of polymerization problems, full multidimensional results.
  10. I. Kryven, A. Berkenbos, P. Melo, DM. Kim. PD. Iedema. Modeling Crosslinking Polymerization in Batch and Continuous Reactors, Volume 7, Issue 5, pages 205–220, 2013 [MORE] [PDF]
    A new pseudo-distribution approach is applied to the modeling of crosslinking copolymerization of vinyl and divinyl monomer and compared to Monte Carlo (MC) simulations. With the number of free pending double bonds as the main distribution variable, a rigorous solution of the three leading moments of the molecular size distribution becomes possible. Validation takes place with data of methyl methacrylate with ethylene glycol dimethacrylate. Well within the sol regime perfect agreement is found, but near the gelpoint larger discrepancies do appear. This is probably due to the existence of multiradicals that are not taken into account in the population balance approaches.
  11. T. van der Aalst, D. Denteneer, H. Döring, MH. Duong, RJ. Kang, M. Keane, J. Kool, I. Kryven, T. Meyfroyt, T. Müller, G. Regts, J. Tomczyk The random disc thrower problem, Proceedings SWI 2013, Pages 59-78, 2013 [MORE] [PDF]
    We describe a number of approaches to a question posed by Philips Research, described as the "random disc thrower" problem. Given a square grid of points in the plane, we cover the points by equal-sized planar discs according to the following random process. At each step, a random point of the grid is chosen from the set of uncovered points as the centre of a new disc. This is an abstract model of spatial reuse in wireless networks. A question of Philips Research asks what, as a function of the grid length, is the expected number of discs chosen before the process can no longer continue? Our main results concern the one-dimensional variant of this problem, which can be solved reasonably well, though we also provide a number of approaches towards an approximate solution of the original two-dimensional problem. The two-dimensional problem is related to an old, unresolved conjecture ([6]) that has been the object of close study in both probability theory and statistical physics. Keywords: generating functions, Markov random fields, random sequential adsorption, Rényi’s parking problem, wireless networks
  12. V. Kukharskyy, Ya. Savula, I. Kryven Modified method of residual-free bubbles for solving the advection-diffusion problem with high Peclet number, Series Appl. Math. and Informatics. Visnyk of the Lviv University, Volum 20, Pages 85-94, 2013 [MORE]
    The numerical scheme based on the residual free bubbles approach and the hierarchy basis discretization for advection-diffusion problems with dominated advection is introduced. Among properties of the developed method are better stability to oscillations for hight Peclet numbers, better time performance, and simplicity in parallel computing application. The computational analysis of convergence and time performance on an example of a 2D symmetric advection-diffusion problem for 10e10 Peclet number is performed.
  13. I. Kryven, PD. Iedema. A Novel Approach to Population Balance Modeling of Reactive Polymer Modification Leading to Branching, Volume 22, Issue 2, pages 89–106, 2013 [MORE] [PDF] [JOURNAL COVER]
    The mathematical treatment of polymer modification systems, described by population balances containing convolution is discussed. The two-dimensional case (molecular weight vs. number of branch points) was considered by utilizing approximations of distributions, expanding them in terms of Gaussian basis functions. Three branching reactions were addressed: chain backbone to chain end point coupling; three-functional coupling of chain ends; and crosslinking. The results were compared to those of Monte Carlo (MC) simulations. Good agreement was observed, although the quality of a distribution as generated by the numerical approach is much better in view of the strong scatter in the MC data.
  14. J.B. van den Berg, R. Castro, J. Draisma, J. Evers, M. Hendriks, O. Krehel, I. Kryven, K. Mora, B. Szabó, P. Zwiernik. Non-imaging Optics for LED-Lighting, Proceedings of SWI 2012, Editor MAA. Boon, ISBN 978-90-6464-630-0, Pages 70-103, 2013 [MORE] [PDF]
    In this report, several methods are investigated to rapidly compute the light intensity function, either in the far field or on a finite-distance screen, of light emanating from a light fixture with a given shape. Different shapes are considered, namely polygonal and (piecewise) smooth. In the first case, analytic methods are sought to circumvent the use of Monte Carlo methods and ray-tracing with large sample size. In the second case, refinements of the Monte Carlo method (notably using a bootstrap procedure) are devised to minimize the number of samples needed for a good approximation of the intensity function.
  15. W. Kager, I. Kryven, K. Myerscough, T. van Opstal, T. Rot. Statistical Modelling of Pre-Impact Velocities in Car Crashes, Proceedings SWI 2011, Pages 133-148, 2013 [MORE] [PDF]
    The law wants to determine if any party involved in a car crash is guilty. The Dutch court invokes the expertise of the Netherlands Forensic Institute (NFI) to answer this question. We discuss the present method of the NFI to determine probabilities on pre-impact car velocities, given the evidence from the crash scene. A disadvantage of this method is that it requires a prior distribution on the velocities of the cars involved in the crash. We suggest a different approach, that of statistical significance testing, which can be carried out without a prior. We explain this method, and apply it to a toy model. Finally, a sensitivity analysis is performed on a simple two-car collision model.

Research

Deep Background Layer

Numerical Methods for Chemical Master Equations

The chemical master equation is the fundamental equation describing stochastics of chemical/biological systems. This differential-difference equation describes temporal evolution of the probability density function for system states. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level - the essential support.

I study projection methods that represent the probability density function with a small number of basis functions. Furthermore, the system of basis functions is reconsidered on every time step in such a way that the solution is approximated only on the essential support of the probability density function instead of the whole state space.

The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability or supported on domains with complex topology. The figure on the right illustrates application of the method to the chemical master equation describing the process of binary cell differentiation. [MORE]


[FULL IMAGE]             [ANIMATION ~ 20Mb]

Random graph models explaining molecular networks

Graph theory has always been a source of inspiration for describing many chemical systems. When it comes to large molecular structures, e.g. hyperbranched or cross-linked polymers, random graph models have been right in the center of attention. These models allow to capture the essence of the polymerisation: temporal assemblage of monomers (vertexes) into polymer (the network). However, existing random graph models do not capture all nuances that are so common in the molecular world. For instance, a node A may prefer to obtain a new connection with a node B that is already linked to A by a short path. As a direct consequence of such a rule, the system obtains many triangles, that are almost not observable in the standard Erdős–Rényi model. A reverse effect also may take place: a new connection with a node A may be less probable if A is surrounded by a 'dense' network, according to a certain centrality measure.

One of the remarkable features of such models is their tendency to exhibit high modularity, i.e nodes can be partitioned into clusters, highly interconnected inside, but with few inter-cluster edges. The figure on the left shows the cluster analysis on a network with a local connection preference. The colors indicates cluster membership. The node sizes denote centrality: the degree of being surrounded by the neighbours that may obstruct new connections.

Besides chemical industries, one of the most exotic applications for this type of models, is the art restoration. The reason behind it is that polymerized oil paint is a strongly cross-linked network and observation of the topology on the molecular level is practically impossible. Nevertheless, understanding the structure of and patterns present in the network will shed light on processes taking place in the paint layers on the span of microseconds and centuries.

Composition drift and sequence patterns in linear polymers

Due to the stochastic nature of polymer systems, the distributional description became a widely established tool. From this perspective, a polymer molecule is viewed as a single entity having various properties (e.g. size, branching index, fraction of comonomers, etc.); while the distribution represents an ensemble of these molecules. However, a polymer molecule is a complex system in itself, and such an approach may often be too coarse.

This project breaks the paradigm down by introducing distributional description on multiple levels of scales: to represent an ensemble of polymer molecules and to represent a single molecule as an ensemble of monomers. Even information on sequential inclusion of comonomers is captured by considering positional distributions: a probability of inclusion of certain monomer type as a function of position or probability of having a sequence of a given pattern starting at given position. Analysis of these new types of distributional properties reveals a tendency for polymer composition being statistically dependent on the position - a phenomena referred to as a "composition drift".

[MORE]

Mathematical model for coagulation and coalesence of fractal colloids

A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation based on cluster mass and fractal dimension is solved, employing a discretisation method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion-limited-cluster aggregation (DLCA) and reaction-limited-cluster-aggregation (RLCA) kernels and different coalescence rates.

[MORE]

Phase transition in cross-linked molecular networks


Gelation is a drastic transformation of physical properties of the material that can be observed during crolss-linking polymerisation. This process is linked to one of the central phenomena in random graph theory - emergence of the giant component. The project aims in building a bridge between graph theory and polymer chemistry in order to combine the strong sides of the two worlds. Numerical approach allows here to consider the chemical system with least simplifications possible.

[MORE: POLYMER 2015] [MORE: CES 2015]

Effects spatial configuration in hyperbranched polymers


Steric hindrance occurs when some groups within a molecule shield the rest of groups: making the shilded groups unavailable for chemical reactions. Although steric hindrance is sometimes a problem it can also be a very useful tool, and is often exploited by chemists to change the reactivity pattern of a molecule. Population balance approach allows to construct a model accounting for this phenomenon. Since the geometric shape of a molecule is dictated by random processes, the effect of geometry can be obtained only in terms of propagating uncertainty rather than a precise number.

[MORE]

Meshless projection for population balance equation


The stochastic approach has brought a refreshing way to describe the physical reality of the World by mathematical language. However, the new models that account for stochasticity, have emerged as a very complex mathematical objects. In polymer chemistry stochastic models, namely population balance equations (PBE), provide a means to solve a non-trivial and frequently self-contradictory task: describing a random system compactly while preserving properties of interest at the same time. PBEs are difficult to solve exactly and one is forced to seek an approximation.

This project focuses on a new numerical method utilizing Gaussian radial basis functions to find approximation to the exact solution of PBEs. A particular attention is devoted to the PBEs that describe evolution of a random network, and thus contain convolutional terms.

[MORE: MTS 2014] [MORE: MTS 2013]