Mathematics

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Research Topic Summary.

The aim of this research is to develop and investigate adaptive classification methods that are effective under heteroskedastic conditions. Heteroskedasticity, where data variance changes depending on the characteristics of an object, is a common issue in various fields, such as economics, medicine, and social sciences. Traditional classification methods may be ineffective when data has a non-homogeneous structure, making it essential to create new algorithms that can accurately classify heterogeneous data. This study will focus on creating a mathematical model to describe object distribution and developing procedures based on this model to enable precise data classification. Additionally, methods will be proposed for integrating additional information about object positions and context into classification algorithms to further improve accuracy. At the conclusion of the study, a comparison of the developed procedures with alternative methods will be conducted using real data to assess their practical applicability. The expected outcomes include the development of classification procedures with higher accuracy and efficiency, tailored for heteroskedastic data. These results will contribute to advancements across fields—from finance to biomedicine—ensuring more precise data analysis and forecasting.

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Research Topic Summary.

Markov chain Monte Carlo (MCMC) methods, coupled with Bayesian theory and statistics, can be universally applied to generate a wide range of samples with respect to complex or high-dimensional distributions. Direct sample generation from such distributions is often inefficient or even impossible. It can be seen that an algorithm implementing MCMC methods can construct a Markov chain of the samples under consideration which converges to the distribution of interest. The objective of the research is to develop a common methodology for accurate and fast estimation of state-space model parameters and application of Bayesian statistics by combining particle filtering and Markov chain Monte Carlo methods. Uždaviniai susietai su laukiamais rezultatais: 1. Atlikti būsenų erdvės modeliavimo ir parametrų vertinimo su dalelių filtracijos ir MCMC metodais naujausios mokslinės literatūros analizę, sudaryti pradinę vertinimo metodologiją; 2. Apdoroti bandomuosius ir realius duomenis bei atlikti Bajeso statistikos taikymo tyrimus; 3. Išplėtoti dalelių MCMC algoritmą ir ištirti kitus tikslumą bei greitį lemiančius veiksnius; 4. Apžvelgti ir išbandyti MCMC algoritmo spartinimo galimybes, pvz. išlygiagretinimą; 5. Palyginti parametrų atnaujinimą su naujais ir įvairaus neapibrėžtumo duomenimis; 6. Mokslinių tyrimų rezultatų apibendrinimas ir galutinių pasiekimų viešinimas. For further information on the topic and relevant R&D work please contact the supervisor.

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Research Topic Summary.

This research focuses on the development of advanced mathematical and computational methods for accelerating subsurface flow simulations. The project will involve the formulation and analysis of numerical schemes for multiphase and reactive flow, including rigorous mathematical proofs of stability, convergence, and error bounds. Beyond theoretical development, the work will integrate high-performance computing techniques with reduced-order modeling to create efficient and scalable solvers for large-scale partial differential equation systems governing subsurface flow. The resulting methods will enable significantly faster and more accurate simulations, with provable numerical properties and applicability to real-world energy and environmental problems.

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Research Topic Summary.

New Bayesian multimodal survival models will be developed that fuse clinical, imaging, molecular data to predict human time-to-event outcomes. The central gap is that most multimodal survival work either concatenates features into Cox type models or uses end-to-end deep nets, leaving limited theory for identifiability and inference, weak uncertainty quantification, and ad-hoc handling of censoring, competing risks, and missing modalities. Methodological directions of the research project will include (1) multi-view sufficient dimension reduction for censored data to produce modality-specific low dimensional scores, combined through transformation or additive hazards models with flexible (e.g., spline or Gaussian process) baseline hazards; (2) hierarchical Bayesian models with structured sparsity (group lasso or horseshoe priors) to share information across modalities while controlling overfitting; (3) principled treatment of incomplete or informatively missing modalities using selection/pattern-mixture models with doubly robust or orthogonalized estimating equations. Additional direction which will be pursued during this research implementation is a calibrated uncertainty for individual survival curves (for example partial likelihood bootstrap or posterior credible bands) and extensions to competing risks and dynamic prediction via joint modeling with longitudinal biomarkers.

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Research Topic Summary.

The doctoral research focuses on human movement analysis by integrating image data, video recordings, heart activity signals (ECG), and electromyograms (EMG). Using advanced mathematical methods, statistical models, and machine learning algorithms, the goal is to develop a unified multimodal model for assessing human physical state and motion. The study will include data fusion, image and signal processing, and the development of new computational tools for interpreting physiological and biomechanical information.

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Research Topic Summary.

This research develops a unified framework for modeling complex, high-dimensional, and time-varying dependencies in multi-line insurance portfolios and for optimizing reinsurance decisions across multiple periods. It introduces adaptable dependence structures that overcome the limitations of traditional static copula models and formulates reinsurance optimization as a stochastic control problem using dynamic programming and robust optimization. The goal is to create theoretically sound and practically resilient reinsurance strategies that remain effective under heavy-tailed distribution.

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Research Topic Summary.

Coupled Map Lattices (CML) are widely used in the study of dynamical systems. The Coupled Map Lattice of Matrices (CMLM) represents a relatively new class of models and a promising research direction, first introduced in 2018 by the author of the proposed dissertation topic together with co-authors. Matrix iterative maps and their networks form an active and rapidly developing area of research, with results published in high-level international scientific journals. This growing body of work demonstrates both the scientific importance and the broad applicability of the CMLM approach. An increasing number of studies cite and apply CMLM models, reflecting growing international interest and encouraging collaboration among researchers who aim to integrate CMLM concepts into their own areas of investigation. The main goal of this research is to explore and apply networks of iterative matrix maps for matrices of order n. The extensive literature on scalar coupled map lattices highlights the vast potential for extending these models to matrix-based systems, offering new perspectives and challenges for theoretical and applied research. More specific research objectives include: • investigatingtransient processes (such as traveling and spiral waves, solitons, and chimeras) in coupled map lattices of matrices; • studying the theoretical aspects and applications of divergence processes in CMLMs; • and developing methods for finite-time stabilization of unstable solutions in matrix-based coupled map lattices. The proposed research aims to promote significant advances in these areas, contributing to scientific breakthroughs and expanding the theoretical foundations and applications of CMLM studies.