MMSC 2024: Papers with Abstracts

Abstract. Rapid developments in aerospace technologies demand reliable procedures to plan ro- bust missions with high safety. To increase safety under uncertainties in model parameters or environmental conditions, multi-objective robust optimization methods via sensitivity minimization can be used. An acceptable trade-off between a nominal operational cost (e.g., time, energy) and robustness is searched for to plan missions that are less prone to disturbances. The presented analysis considers open-loop and closed-loop sensitivity min- imization approaches and utilizes multi-objective optimization to assess the performance and the limitations of both approaches. To solve the multi-objective optimization prob- lems, scalarization techniques are employed using weighted sums and cost bounds. By varying weights and cost bounds, multiple optima can be calculated, resulting in an ap- proximate Pareto front and giving rise to an overview of the trade-off between optimality and robustness of the solutions. The analysis is performed for robust unmanned aerial vehicle (UAV) trajectory optimization minimizing positional sensitivities.

Abstract. Curvature plays an important role in the function of biological membranes, and is therefore a readout of interest in microscopy data. The PyCurv library established itself as a valuable tool for curvature estimation in 3D microscopy images. However, in noisy images, the method exhibits visible instabilities, which are not captured by the standard error measures. In this article, we investigate the source of these instabilities, provide ade- quate measures to detect them, and introduce a novel post-processing step which corrects the errors. We illustrate the robustness of our enhanced method over various noise regimes and demonstrate that with our orientation correcting post-processing step, the PyCurv library becomes a truly stable tool for curvature quantification.

Abstract. Algorithms for the numerical solution of contact quasi-static problems of deformable solid mechanics were constructed. The contact interaction of the body system was con- sidered using the mortar method (a variant of the Lagrange multiplier method). The de- veloped algorithms were used to simulate the thermomechanical state of a fuel element section, considering creep and cracking. The results of calculations for axisymmetric for- mulation of the problem for the operation mode of a fuel element with constant heat release in fuel pellets are presented.

Abstract. The problem of optimizing trajectory tracking algorithms is considered. Based on measurements of a moving object, such algorithms iteratively make estimates of its state. These algorithms contain parameters that affect the quality of their work, for example, the noise variances in a mathematical model of the object’s dynamics. A multicriteria evolutionary optimization algorithm for such parameters is proposed based on genetic procedures. We also elaborate a procedure for using this algorithm on real data in which random measurement errors are simulated along the real trajectory. The system of criteria is proposed that assesses both the total mean square deviation of the trajectory tracking algorithm’s output and the quality of its transition processes after a change of the object’s motion mode. The algorithm was tested on model and real air traffic data.

Abstract. DNA methylation is a modification of the biochemical environment of a nucleotide that can occur at so-called CpG sites in the DNA strand. Just as a genetic mutation, it can benefit or harm the organism, depending on where exactly it happens and to what ex- tent. This work focuses on two questions regarding the pattern evolution of methylation in certain DNA sequences, since the impact of methylation has been observed to depend on these patterns: does the size of (de-)methylated CpG clusters depend on reactions with other CpG sites? And can these reactions alter epigenetic variation, i.e. population-wide methylation patterns? To describe the methylome evolution within one individual (on a single cell basis), but also inter-generational developments, we formulate two mathemati- cal models and corresponding master equations: one considering the influence of a single neighboring CpG site and one regarding both nearest neighbors. As the master equations can only be solved for certain parameter values, we use numerical simulations for further analysis. The simulation is compared to the analytical solution for validation, and then it is used for the investigation of the aforementioned questions. We find that for the chosen parameters, the cluster size increases if neighboring interactions are involved, indepen- dently of methylation status. Our results suggest that the epigenetic variation is larger in the case of the models which include neighboring interactions.

Abstract. This paper introduces Phoenix-OC, a novel optimal control software for the solution of large-scale, multi-phase Optimal Control (OC) problems. Phoenix-OC employs segmented collocation methods for the state discretization and B-Splines for the control parameteriza- tion. Each of the parameterized controls is allowed to have a distinct degree and knot grid. Additionally, control derivatives of arbitrary order can be utilized in the model, as well as constraint and cost functions. User-defined functions can be modeled either through an automatic differentiation framework or via a generic C interface supporting the utilization of virtually arbitrary functions, external models, etc. Among other features, the software inherently supports table data interpolation, the computation of post-optimal sensitivities for parametric OC problems, bi-level optimization, homotopy formulations, parallel batch runs, and job dependencies. Furthermore, jobs can be executed locally or through a job scheduler on computer clusters.
Phoenix-OC operates on the Phoenix-CORE engine - a generic sparse evaluation frame- work for both the evaluation and derivative computation of vector-valued functions. Cen- tral to this computational engine is the notion of an Extended Sparsity Pattern (ESP). This novel type of sparsity pattern extends traditional binary-valued sparsity patterns to a new type of floating-point pattern, allowing for advanced structure exploitation. The ex- ploitation of sparse structures based on the ESP, combined with the multi-level parallelism implemented in Phoenix-OC, yields high performance across a range of representative benchmarks from engineering applications.

Abstract. A mathematical model of cerebral blood flow in the form of a dynamical system is considered. The cerebral blood flow autoregulation modeling problem is treated as an output regulation control problem. The cerebral autoregulation mechanism is described in terms of an output feedback control law based on measurements of the arterial-arteriolar blood flow rate values and intracranial pressure estimates made by an asymptotic state observer. Simulation results confirm good performance of the suggested cerebral blood flow autoregulation model in the form of a dynamic output feedback.

Abstract. For predicting cardiovascular diseases, mathematical modelling of the cardiovascular system has been proven to be a powerful asset. The governing idea is to analyse it through compartments as multiple connected subsystems with inputs and outputs. In this paper, models were identified for four subsystems of input-output sequence (left ventricle - left atrium - ascending aorta - descending aorta - left common carotid artery) by modelling frequency response. The data set used for model identification consisted of blood pressure during four consecutive heart contractions of four circulatory segments from clinical trials performed on a pig. The goal is to discover a linear model with a non-integer order that succinctly represents the process, outperforming high-order autoregressive exogenous input (ARX) integer models. This model identification occurs non-parametrically, aiming to achieve the best smooth fit in the frequency domain by minimizing the difference between real measurements and model predictions using the particle swarm optimization (PSO) algorithm.

Abstract. A derivation of so-called “soft-margin support vector machines with kernel” is presented along with elementary proofs that do not rely on concepts from functional analysis such as Mercer’s theorem or reproducing kernel Hilbert spaces which are frequently cited in this context. The analysis leads to new continuity properties of the kernel functions, in particular a self-concordance condition for the kernel. Practical aspects concerning the im- plementation and the choice of the kernel are addressed and illustrated with some numerical examples. The derivations are intended for a general audience, requiring basic knowledge of calculus and linear algebra, while some more advanced results from optimization theory are being introduced in a self-contained form.

Abstract. In this work, we introduce a novel methodology for studying the low-temperature phase
of frustrated spin glass models using convolutional neural networks (CNNs). Our approach addresses the regression of thermodynamic properties, specifically the average energy ⟨E⟩, as a function of temperature T for spin glasses on a square lattice. By modelling the spin glass as a weighted graph, where exchange interaction values Jk are represented by the edges and mapped to lattice coordinates, we explore the functional relationship between ⟨E⟩ and the spatial distribution of J. We evaluate CNNs for their performance across various spin glass sizes and distributions of exchange integrals, demonstrating the potential of CNNs in capturing complex spin interactions and advancing the understanding of frustrated systems.

Abstract. An inverse extremal problem for the Fisher-Kolmogorov model of tumor growth is studied. It is required to minimize the normalized density of tumor cells in a given subdomain, while the drug concentration in the tissue must be limited to specified values. The solvability of the inverse extremal problem is established. An algorithm to find its solution is constructed. The numerical experiments illustrate its efficiency.

Abstract. The paper presents a description of a computational algorithm and applied software designed to find the partial pressure of oxygen in a model domain known as the Krogh cylinder. The algorithm is implemented using the finite element method in the FreeFEM package, the graphical interface is made using Python language. The developed software is adapted for calculating partial pressure of oxygen in the brain tissue of preterm infants.

Abstract. The number of user-to-machine interactions has increased dramatically over the last decade. With the introduction of fully functional generative AI, such as ChatGPT, Copilot, and Gemini, the number of the interactions and the associated amount of information will further increase in the nearest future. This work suggests an algorithm for optimizing the number of question-reply pairs between users and a machine. The goal of the optimization was to find the optimal amount of information for API and its server. As a result of the optimization, the cost of running the services for service providers and for users can be lowered. Furthermore, in this study, the optimization led to 23% increase of customer retention rate, 15% increase of revenue per customer, 17% drop of the customer acquisition cost, and 35% increase of customer engagement with AI.

Abstract. Cellular automata, being an apparatus for the implementation of discrete dynamic models, play a special role in mathematical biology and in silico studies of microorganisms. The study was undertaken to design 3D hybrid cellular automata-based model of bacterial biofilm taking into account the surface spreading mechanism. The model formalization is based on the cellular automaton algorithm of biofilm evolution, a discrete analogy for the diffusion model of nutrient consumption, and an additional inoculation mechanism. The proposed computational procedure allows to conduct simulations under variations of key model parameters: the initial nutrient level, the probability of additional inoculation, and the radius of random inoculation transfer. A series of in silico experiments was conducted to investigate biofilm formation with a focus on ensuring two key factors: maximum space occupation with minimal resource consumption.

Abstract. Microbiological systems have become relevant objects of interdisciplinary research in mathematical biology and bioinformatics and can be analyzed using in silico studies with the implementation of computational experiments. The design of mathematical models of bacterial biomass growth under external inhibition can help to predict states of the microbiological system, reduce antibiotic use, and avoid antimicrobial resistance. The paper is devoted to developing a mathematical model of nutrient-dependent dynamics of bacteria cultured in media, considering external surface growth inhibition. The mathematical problem statement includes governing equations to define spatial-temporal distributions of bacterial biomass concentration, nutrient concentration, and time-dependent dose of antibiotic concentration. We propose a joint numerical scheme based on finite difference methods and specialized program application implemented with Matlab. A series of computational experiments were performed to describe the distributions of the key chemical compounds characterizing bacterial surface growth exposed to antibiotics and predict antibiotic treatment strategies.

Abstract. The paper deals with linear differential games with a fixed terminal instant, convex geometric constraints of the players’ controls, and convex terminal target set. The first player tries to guide the system to the target set at the terminal instant, the second one hinders this. In the 1960’s, L. S. Pontryagin proposed a theoretic geometric procedure for approximate constructing time sections of the maximal stable bridge for games of this type. This procedure is known as the second Pontryagin’s method. At the beginning of the 1980’s in the Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Rus- sia), a computational algorithm for the procedure has been suggested and implemented as a computer program. However, this algorithm is suitable only for games with two- dimensional equivalent phase vector. The authors suggest a procedure suitable for games with a multi-dimensional phase vector. For an implementation of this method, one needs implementations of convex hull construction, Minkowski sum and difference. The authors have taken known algorithms for convex hull construction and Minkowski sum. An al- gorithm for Minkowski difference as well as some procedures for conversion of different representations of multi-dimensional polytopes to each other have been suggested. All these algorithms have been implemented as a computer library in C# by the authors. A series of model differential games has been computed.

Abstract. In this age where data is growing at an astronomical rate, with unfettered access to digital information, complexities have been introduced to scientific computations, analysis, and inferences. This is because such data could not be easily processed with traditional approaches. However, with innovative designs brought to the fore by NVIDIA and other market players in recent times, there have been productions of state-of-the-art GPUs such as NVIDIA A100 Tensor Core GPU, Tesla V100, and NVIDIA H100 that seamlessly handle complex mathematical simulations and computations, artificial intelligence, machine learning, and high-performance computing, producing highly improved speed and efficiency, with room for scalability. These innovations have made it possible to efficiently deploy many parallel programming models like shared memory, distributed memory, data parallelization, and Partitioned Global Address Space (PGAS) with high-performance metrics. In this work, we analyzed the parquet-formatted New York City yellow taxi dataset on a RAPIDS and DASK supported distributed data-parallel training platform using a high-performance cluster of 7 NVIDIA TITAN RTX GPUs (24GB GDDR6 each) running CUDA 12.4. The dataset was used to train Extreme Gradient Boosting (XGBoost), RandomForest Regressor, and Elastic Net models for trip fare predictions. Our models achieved notable performance metrics. The XGBoost achieved a mean squared error (MSE) of 10.87, R2 of 96.9%, and a training time of 21.1 seconds despite the huge size of the training dataset, showing how computationally efficient the system was. Random- Forest achieved MSE of 27.46, R2 of 92.2% and a training time of 25.9 seconds. In the bid to show the scalability and versatility of our experimental design to different machine learning domains, our multi-GPU accelerated training was extended to image classification tasks by using MobileNet-V3-Large pre-trained architecture on a CIFAR-100 dataset. The following parallelization results were achieved: a low Karp-Flatt metric of 0.013, indicating minimal serialization, 98.7% parallel fraction, demonstrating excellent parallelization, and only 7.1% communication overhead relative to computation time. For the model performance, we achieved a ROC AUC of over 95% for the implementation. This work advances the state-of-the-art in parallel computing through implementation of RAPIDS and DASK frameworks on a distributed data-parallel training platform making use of NVIDIA multi-GPUs. The work is built on a well established theoretical framework using Amdahl and Gustafson’s laws on parallel computation. By integrating RAPIDS and DASK, we contribute to advancing parallel computing capabilities, offering potential applications in smart city development and the field of logistics and transportation management services where rapid fare predictions are very important. The contribution could also be extended to the field of image classification, vision systems, object detection and embedded systems for mobile applications.

Abstract. A problem of safe aircraft flows merging in an airport approach zone is studied. In the situation under consideration, the air routes of aircraft flows have a branched structure, that is, there can be multiple paths leading an aircraft from the entry point of the flow to the final point of the scheme. Also, there may be several merge points, at which an aircraft flow joins other ones, original and/or merged earlier. At each point of the air route scheme, a safe passage of vessels must be provided, that is, the presence of a safe time interval between the instants of aircraft passage must be guaranteed. Regulation of the arrival instants to points of a scheme is carried out by changing aircraft velocities, routes, or usage of special scheme elements: holding areas, point-merge schemes, path alignments, etc. In the problem, some model is considered taking into account directions from an air traffic controller to a pilot connected with changing the aircraft’s velocity and/or route. The resultant schedule of the aircraft arrivals to the scheme points is optimized from the point of view of a criterion minimizing the deviations of aircraft arrivals to the final point of the scheme from the nominal ones and the number of directions from air traffic controllers to pilots. The methodology for constructing the model for such a problem is proposed within the mixed integer linear programming framework. The case of several runways is not included, but the model can be easily extended to cover this case. Results of numerical modeling are given.

Abstract. We consider a zero-sum finite horizon linear-quadratic differential game. Suboptimal state-feedback controls of the players in this game are derived. This derivation is based on the approximate solution (with a novel error’s estimate) of the corresponding Riccati matrix differential equation by the method of artificial parameter. The theoretical results are illustrated by the approximate solution of the problem of pursuit-evasion engagement between two flying vehicles.