Research

Diffusion in complex media

Analysing the movements and interactions of particles in media such as nucleus and cytoplasm of biological cells or their membranes, polymer solutions, and other crowded environments, is crucial for the study of these systems as a whole and can deeply affect our understanding of the biochemical and transport phenomena. This, in turn, has further implications for biology and medicine. Recent rapid progress in the area has been facilitated by new experimental techniques such as single particle tracking microscopy, which give new and more detailed insight into microscopic world. However, this influx of data causes statistical challenges: the delicate features of dynamics can be easily lost if the properties of the estimators and experimental noise are not taken into account. We try to manage these difficulties using mixture of time series techniques, spectrum analysis, hypothesis testing and regression/machine learning tools.

nature

Left: protein trajectories over the microscopic image of their intercellular environment. Right: one trajectory with statistically determined 4 diffusion states. These correspond to various protein interactions. Adapted from [1].


The mathematical modelling of such trajectories started with the groundbreaking random walk model proposed by Einstein which was later formalised into the modern theory of Brownian motion and stochastic processes. Important tools include stochastic differential equations, fractional Brownian motion and continuous time random walks. This framework is currently very universal and the same models reappear in many different branches of natural science, engineering and finance, making the transfer of knowledge an interesting and important aspect of progress. Even so, complex media have its own set of challenges; in particular experimental observations such as anomalous power-law diffusion and non-Gaussian distributions and their link to the heterogeneity of the environment are still under active theoretical investigation.

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  1. T. Sungkaworn, M.L. Jobin, K. Burnecki, A. Weron, M.J. Lohse, D. Calebiro; "Single-molecule imaging reveals receptor-G protein interactions at cell surface hot spots", Nature vol. 550 no. 7677 pp. 543-547, 2017, doi: 10.1038/nature24264
  2. C.D. Bello, A. Chechkin, A.K. Hartmann, Z. Palmowski, R. Metzler; "Time-dependent probability density function for partial resetting dynamics", New Journal of Physics vol. 25 no. 8 pp. 1-15, 2023, doi: 10.1088/1367-2630/aced1d
  3. M. Balcerek, K. Burnecki, S. Thapa, A. Wyłomańska, A. Chechkin; "Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions", Chaos vol. 32 no. 9 pp. 1-16, 2022, doi: 10.1063/5.0101913
  4. J. Ślęzak, R. Metzler, M. Magdziarz; "Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion", New Journal of Physics vol. 20 pp. 1-26, 2018, doi: 10.1088/1367-2630/aaa3d4
  5. K. Burnecki, G. Sikora, A. Weron, M.M. Tamkun, D. Krapf; "Identifying diffusive motions in single-particle trajectories on the plasma membrane via fractional time-series models", Physical Review E vol. 99 no. 1 pp. 1-10, 2019, doi: 10.1103/PhysRevE.99.012101

Partial differential equations and dynamical systems

Our deterministic modelling group is dedicated to applying mathematical principles to real-world problems across diverse fields. We are particularly interested in problems that involve complex physical phenomena and require sophisticated mathematical modeling and numerical simulation techniques. Our research spans a wide range of topics, including:

  • Dynamics of non-smooth systems: We study analytically and numerically the dynamics of systems with macroscopic sudden state changes and discontinuous nonlinearities, e.g. Coulomb's friction law, Newton's impact law, or models of switching elements such as transitors.
  • Biomechanical modeling: We investigate the mechanics of biological tissues, focusing on applications in ophthalmology. These models help us understand the underlying mechanisms of various eye diseases and injuries, as well as optimize diagnostic procedures.
  • Hemodynamics: We study the flow of blood through arteries and veins, with a focus on understanding the impact of various factors, such as blood pressure, vessel geometry, and blood properties, on blood flow patterns and the development of cardiovascular diseases.
  • Porous media flows: We explore the complex behavior of fluids flowing through porous media, such as soil, rocks, and biological tissues. Our research focuses on anomalous diffusion, a phenomenon characterized by non-standard diffusion behavior, which arises in many natural and engineered systems.
  • Capillary phenomena: We investigate the physics of fluid behavior at the microscale, particularly capillary forces, which play a crucial role in many natural and technological processes.
  • Climate dynamics: We explore the complex interactions between the atmosphere, oceans, and land surface that drive climate variability and change.
  • Theoretical studies: We delve into the theoretical foundations of partial differential equations and numerical methods, developing new mathematical techniques to analyze and solve complex problems.
3D model of the brain cardiovascular system of a neurological patient. Simulations allow for predicting the outcome of the added bypass. From joint project with Dariusz Szarek.


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  1. M. Desroches, P. Kowalczyk, S. Rodrigues; "Spike-adding and reset-induced canard cycles in adaptive integrate and fire models", Nonlinear Dynamics 104 pp. 2451–2470, 2021, doi: 10.1007/s11071-021-06441-z
  2. B. López, H. Okrasińska-Płociniczak, Ł.S. Płociniczak, J. Rocha; "Time-fractional porous medium equation: Erdélyi-Kober integral equations, compactly supported solutions, and numerical methods", Communications in Nonlinear Science and Numerical Simulation vol. 128 pp. 1-14, 2024, doi: 10.1016/j.cnsns.2023.107692
  3. Z. Wróblewska, P.S. Kowalczyk, Ł.S. Płociniczak; "Stability of fixed points in an approximate solution of the spring-mass running model", IMA Journal of Applied Mathematics vol. 88 no. 3 pp. 429-454, 2023, doi: 10.1093/imamat/hxad014
  4. Ł.S. Płociniczak; "Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data", Computers & Mathematics with Applications vol. 127 pp. 181-191, 2022, doi: 10.1016/j.camwa.2022.09.028
  5. Ł.S. Płociniczak; "Asymptotic analysis of internal relaxation oscillations in a conceptual climate model", IMA Journal of Applied Mathematics vol. 85 no. 3 pp. 467-494, 2020, doi: 10.1093/imamat/hxaa014
  6. Ł.S. Płociniczak, M. Świtała; "Monotonicity, oscillations and stability of a solution to a nonlinear equation modelling the capillary rise", Physica D, Nonlinear Phenomena vol. 362 pp. 1-8, 2018, doi: 10.1016/j.physd.2017.10.008
  7. Ł.S. Płociniczak; "Numerical method for the time-fractional porous medium equation", SIAM Journal on Numerical Analysis vol. 57 no. 2 pp. 638-656, 2019, doi: 10.1137/18M1192561
  8. Ł.S. Płociniczak, W. Okrasiński, J.J. Nieto, O. Domínguez; "On a nonlinear boundary value problem modeling corneal shape", Journal of Mathematical Analysis and Applications vol. 414 pp. 461-471, 2014, doi: 10.1016/j.jmaa.2014.01.010

Financial and actuarial mathematics, risk theory and management

The financial and actuarial fields demand sophisticated techniques to navigate the complexities of the modern markets. Advanced statistical modelling, encompassing considerations like climate change and extreme events, provides a robust foundation for risk assessment. This includes analysing ruin probabilities, optimizing dividend strategies, and incorporating the impact of catastrophic events (CAT bands) into risk models. These advancements empower financial institutions and insurance companies to make informed decisions and enhance their resilience in an increasingly uncertain environment. Some of our recent results include:

  • Natural catastrophe risk management: From building multidimensional loss models to novel CAT securities, this research includes the development of novel insurance-linked securities that help mitigate the risk related to natural catastrophes. We focus on creating innovative multi-peril and multi-risk catastrophe bonds (CAT bonds) that incorporate peril type and region type aspects into their construction. Using tools such as martingales, exponential changes of measure, and randomly stopped processes, we derive analytical prices of these instruments. The analysis is supplemented by various advanced statistical and machine learning techniques that help in calibrating the models to real-world data.
  • Exact and approximated results for the ruin probabilities: This includes the derivation of exact and approximated results for ruin probabilities for one-dimensional and multidimensional risk processes including bivariate insurer-reinsurer model, risk processes perturbed by Brownian and Lévy motions, and risk processes in regime-switching environments as well as the derivation of optimal dividend strategies. Our analysis considers a wide range of claim severity distributions ranging from light- to heavy-tailed and explores various claim arrival processes including general renewal processes, as well as the classical and Parisian-type ruin.
  • Pricing and optimal exercise of American options under complex market models: This research focuses on pricing and optimally exercising American options in complex market environments. We employ advanced mathematical and numerical techniques like scale functions and the least squares Monte Carlo method to derive optimal exercise strategies and analyse the impact of market complexities on option pricing.
  • Exploring the theory of financial stochastic processes: This includes work on the fluctuation theory for Lévy processes, branching processes, Markov additive processes, and queueing systems, providing a deeper understanding of the underlying dynamics of financial and insurance markets.
Left: A sample trajectory of a classical risk process with moments of the classical ruin τ and the Parisian ruin τd marked. Right: Price of two-region CoCo CAT Bond obtained from model with independent loss amounts for D₁ = 5 billion dollars, with respect to varying D₂ (in billion dollars) and ν.

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  1. J. Al-Hadad, Z. Palmowski; "Perpetual american options with asset-dependent discounting", Applied Mathematics and Optimization vol. 89 no. 1 pp. 1-35, 2024, doi: 10.1007/s00245-023-10084-4
  2. K. Burnecki, M.N. Giuricich, Z. Palmowski; "Valuation of contingent convertible catastrophe bonds - the case for equity conversion", Insurance. Mathematics and Economics vol. 88 pp. 238-254, 2019, doi: 10.1016/j.insmatheco.2019.07.006
  3. K. Burnecki, M.A. Teuerle, M. Zdeb; "Pricing of insurance-linked securities: a multi-peril approach"; Journal of Mathematics in Industry, 14(1), 2024, doi: 10.1186/s13362-024-00154-9 4
  4. K. Burnecki, M.A. Teuerle, A. Wilkowska; "Ruin probability for the insurer-reinsurer model for exponential claims: a probabilistic approach", Risks vol. 9 no. 5 pp. 1-10, 2021, doi: 10.3390/risks9050086
  5. S. Foss, D. Korshunov, Z. Palmowski; "Maxima sampling on random time intervals for heavy-tailed compound renewal and Lévy processes", Stochastic Processes and their Applications 76, 104422, 2024, doi:0.1016/j.spa.2024.104422
  6. Z. Palmowski, P. Stępniak; "Last-passage american cancelable option in Lévy models", Journal of Risk and Financial Management vol. 16 no. 2 pp. 1-14, 2023, doi: 10.3390/jrfm16020082

Time series techniques for the industrial data analysis

Industrial data analysis involves processing related to the operation of machinery, such as vibration signals, acoustic signals, and temperature measurements, as well as applications in telecommunications networks and medical data sets. Our research in this field emphasizes the development and application of novel statistical methods and time series techniques for anomaly detection, with particular attention to identifying local faults in machinery, detecting operational mode changes, or identifying configuration changes in network systems.

timeSeries

Left: test rig used in the experiment. Right: a) real vibration signal with Gaussian noise and (c) its spectrogram, (b) real acoustic signal with non-Gaussian noise and (d) its spectrogram. Adapted from [2].


Recent advancements have centered on the creation of methods tailored to data that exhibit outliers and non-Gaussian characteristics, in particular the presence of heavy, power-law tails. In such cases, traditional anomaly detection methods prove inadequate, prompting the research team to develop new approaches aimed at detecting regime changes and identifying cyclic patterns under the assumption of non-Gaussian signal distributions. We integrate the newly developed techniques into industrial applications through collaborative projects with industry partners such as KGHM, Nokia Solutions and Networks, etc. Hugo Steinhaus Center is also a part of European Consortium of Mathematics in Industry (ECMI).

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  1. J. Witulska, A. Zaleska, N. Kremzer-Osiadacz, A. Wyłomańska, I. Jabłoński; "Robust variance estimators in application to segmentation of measurement data distorted by impulsive and non-Gaussian noise", Measurement 239, 115472, 2025, doi: 10.1016/j.measurement.2024.115472
  2. M. Gabor, R. Zdunek, R. Zimroz, A. Wyłomańska; "Bearing damage detection with orthogonal and non-negative low-rank feature extraction", IEEE Transactions on Industrial Informatics 20(2), 2944-2955, 2024, doi: 10.1109/TII.2023.3300455
  3. D. Szarek, I. Jabłoński, R. Zimroz, A. Wyłomańska; "Non-Gaussian feature distribution forecasting based on ConvLSTM neural network and its application to robust machine condition prognosis", Expert Systems with Applications 230(15), 120588, 2023, doi: 10.1016/j.eswa.2023.120588
  4. J. Hebda-Sobkowicz, R. Zimroz, M. Pitera, A. Wyłomańska; "Informative frequency band selection in the presence of non-Gaussian noise - a novel approach based on the conditional variance statistic", Mechanical Systems and Signal Processing 145, 106971, 2020, doi: 10.1016/j.ymssp.2020.106971
  5. P. Kruczek, J. Obuchowski, A. Wyłomańska, R. Zimroz; "Cyclic sources extraction from complex multiple-component vibration signal via periodically time varying filter", Applied Acoustics 126, 170-181, 2017, doi: 10.1016/j.apacoust.2017.05.013