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## Über dieses Buch

The two-volume set LNCS 11973 and 11974 constitute revised selected papers from the Third International Conference on Numerical Computations: Theory and Algorithms, NUMTA 2019, held in Crotone, Italy, in June 2019.

This volume, LNCS 11974, consists of 19 full and 32 short papers chosen among regular papers presented at the the Conference including also the paper of the winner (Lorenzo Fiaschi, Pisa, Italy) of The Springer Young Researcher Prize for the best NUMTA 2019 presentation made by a young scientist.

The papers in part II explore the advanced research developments in such interconnected fields as local and global optimization, machine learning, approximation, and differential equations. A special focus is given to advanced ideas related to methods and applications using emerging computational paradigms.

## Inhaltsverzeichnis

### Numerical Algorithms for the Parametric Continuation of Stiff ODEs Deriving from the Modeling of Combustion with Detailed Chemical Mechanisms

The use of detailed chemical mechanisms is becoming increasingly necessary during the actual transition of energy production from fossil to renewable fuels. Indeed, the modern renewable fuels are characterized by a composition more complex than traditional fossil fuels due to the variability of the properties of the primary source, i.e. biomass. The parametric continuation can be a formidable tool to study the behavior of these new fuels allowing to promptly assess equilibrium conditions varying the main operative parameters. However, parametric continuation is a very computationally demanding procedure, both for the number of elementary operations needed and for the memory requirements. Actually, only very recently some approaches that allow affording this computation with chemical mechanisms composed of hundreds of chemical species and thousands of reactions have been proposed [1, 2, 37]. Starting from the procedure presented in [1], this paper illustrates further improvements of key steps that usually represents a bottleneck for the effective computation of parametric continuations and for the identification of bifurcation points.

Luigi Acampora, Francesco S. Marra

### Stability Analysis of DESA Optimization Algorithm

This paper investigates the dynamics of the hybrid evolutionary optimization algorithm, Differential Evolution-Simulated Annealing (DESA) algorithm with the binomial crossover and SA-like selection operators. A detailed mathematical framework of the operators of the DESA/rand/1/bin algorithm is provided to characterize the behavior of the DESA-population system. In DESA, the SA-like selection operation provides a nonzero probability of accepting a deteriorated solution that decreases with a sufficient number of generations. This paper shows that the system defined by the DESA-population is stable. Moreover, the DESA-population system time constant, learning and momentum rates are dependent on the value of the crossover constant and the probability of accepting deterioration in the quality of the objective function.

### Are Humans Bayesian in the Optimization of Black-Box Functions?

Many real-world problems have complicated objective functions whose optimization requires sophisticated sequential decision-making strategies. Modelling human function learning has been the subject of intense research in cognitive sciences. The topic is relevant in black-box optimization where information about the objective and/or constraints is not available and must be learned through function evaluations. The Gaussian Process based Bayesian learning paradigm is central in the development of active learning approaches balancing exploration/exploitation in uncertain conditions towards effective generalization in large decision spaces. In this paper we focus on Bayesian Optimization and analyse experimentally how it compares to humans while searching for the maximum of an unknown 2D function. A set of controlled experiments with 53 subjects confirm that Gaussian Processes provide a general model to explain different patterns of learning enabled search and optimization in humans.

Antonio Candelieri, Riccardo Perego, Ilaria Giordani, Francesco Archetti

### A New Syntax for Diagrammatic Logic: A Generic Figures Approach

In this paper we propose a new syntactical representation of C.S. Peirce’s diagrammatic systems for propositional and predicate logic. In particular, we use the categorical notion of generic figures to represent the syntax of the diagrammatic language as a category of functors from a suitable, simple category into the category of sets, highlighting the relational nature of Peirce’s diagrammatic logic.

Gianluca Caterina, Rocco Gangle

### Objective and Violation Upper Bounds on a DIRECT-Filter Method for Global Optimization

This paper addresses the problem of solving a constrained global optimization problem using a modification of the DIRECT method that incorporates the filter methodology to simultaneously minimize the objective function and the constraints violation. Thus, in the “Selection” step of the herein proposed DIRECT-filter algorithm, the hyperrectangles are classified in four categories and subsequently handled separately. The new algorithm also imposes upper bounds on the objective function and constraints violation aiming to discard some hyperrectangles from the process of identifying the potentially optimal ones. A heuristic to avoid the exploration of the hyperrectangles that have been mostly divided is also implemented. Preliminary numerical experiments are carried out to show the effectiveness of the imposed upper bounds on the objective and violation as well as the goodness of the heuristic.

M. Fernanda P. Costa, Ana Maria A. C. Rocha, Edite M. G. P. Fernandes

### The Approximate Synthesis of Optimal Control for Heterogeneous Discrete Systems with Intermediate Criteria

We consider one of the classes of hybrid systems, heterogeneous discrete systems (HDSs). The mathematical model of an HDS is a two-level model, where the lower level represents descriptions of homogeneous discrete processes at separate stages and the upper (discrete) level connects these descriptions into a single process and controls the functioning of the entire system to ensure a minimum of functionality. In addition, each homogeneous subsystem has its own goal. A method of the approximate synthesis of optimal control is constructed on the basis of Krotov-type sufficient optimality conditions obtained for such a model in two forms. A theorem on the convergence of the method with respect to a function is proved, and an illustrative example is given.

Olga Danilenko, Irina Rasina

### Modelling Climate Changes with Stationary Models: Is It Possible or Is It a Paradox?

Climate is changing; many studies of time series confirm this sentence, but this does not imply that the past is no more representative of the future, and then that ‘‘stationarity is dead’’.In fact, “stationarity” and “change” are not mutually exclusive. As examples: (1) according to Newton’s first law, without an external force, the position of a body in motion changes in time but the velocity is unchanged; (2) according to Newton’s second law, a constant force implies a constant acceleration and a changing velocity.Consequently, “non-stationarity” is not synonymous with change; change is a general notion applicable everywhere, including the real (material) world, while stationarity and non-stationarity only regard the adopted models. Thus, stationary models can be also adopted for environmental changes.With this aim, in this work Authors show some numerical experiments concerning rainfall processes. In detail, a Neymann Scott Rectangular Pulse model (NRSP), with some changing temporal scenarios for its parameters, is adopted, and the derived Annual Maximum Rainfall (AMR) time series are investigated for several temporal resolutions (sub-hourly and hourly scales). The goal is to analyze if there are some particular scales in which the assumed temporal changes in parameters could be “hidden” when AMR series (which are nowadays more available and longer than high-resolution continuous time series for many sites in the world) are studied, and then stationary models for Extreme Value distributions could be adopted.The results confirm what is obtained from analysis of AMR series in some parts of Italy, for which it is not essential to remove the hypothesis of stationary parameters: significant trends could not appear only from the observed AMR data, as a relevant rate of outlier events also occurred in the central part of the last century.

Davide Luciano De Luca, Andrea Petroselli, Luciano Galasso

### Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields

Properties of operators of generalized attenuated ray transforms (ART) are investigated. Starting with Radon transform in the mathematical model of computer tomography, attenuated ray transform in emission tomography and longitudinal ray transform in tensor tomography, we come to the operators of ART of order k over symmetric m-tensor fields, depending on spatial and temporal variables. The operators of ART of order k over tensor fields contain complex-valued absorption, different weights, and depend on time. Connections between ART of various orders are established by means of application of linear part of transport equation. This connections lead to the inhomogeneous k-th order differential equations for the ART of order k over symmetric m-tensor field. The right hand parts of such equations are m-homogeneous polynomials containing the components of the tensor field as the coefficients. The polynomial variables are the components $$\xi ^j$$ of direction vector $$\xi$$ participating in differential part of transport equation. Uniqueness theorems of boundary-value and initial boundary-value problems for the obtained equations are proved, with significant application of Gauss-Ostrogradsky theorem. The connections of specified operators with integral geometry of tensor fields, emission tomography, photometry and wave optics allow to treat the problem of inversion of the ART of order k as the inverse problem of determining the right hand part of certain differential equation.

Evgeny Yu. Derevtsov, Yuriy S. Volkov, Thomas Schuster

### Multiextremal Optimization in Feasible Regions with Computable Boundaries on the Base of the Adaptive Nested Scheme

The paper is devoted to consideration of multidimensional optimization problems with multiextremal objective functions over search domains determined by constraints, which form a special type of domain boundaries called computable ones, which, in general case, are non-linear and multiextremal. The regions of this class can be very complicated, in particular, non-convex, non-simply connected, and even disconnected. For solving such problems, a new global optimization technique based on the adaptive nested scheme developed recently for unconstrained optimization is proposed. The novelty consists in combination of the adaptive scheme with a technique for reducing the constraints to an explicit form of feasible subregions in internal subproblems of the nested scheme that allows one to evaluate the objective function at the feasible points only. For efficiency estimation of the proposed adaptive nested algorithm in comparison with the classical nested optimization and the penalty function method, a representative numerical experiment on the test classes of multidimensional multiextremal functions has been carried out. The results of the experiment demonstrate a significant advantage of the adaptive scheme over its competitors.

Victor Gergel, Vladimir Grishagin, Ruslan Israfilov

### On Polyhedral Estimates of Reachable Sets of Discrete-Time Systems with Uncertain Matrices and Integral Bounds on Additive Terms

We consider discrete-time systems of bilinear type for the case when interval bounds on the coefficients of the system are imposed, additive input terms are restricted by integral non-quadratic constraints, and initial states belong to given sets, which are assumed to be parallelepipeds. An approach for estimating the reachable sets is presented. It is based on considering reachable sets in the “extended” space and constructing external and internal estimates of them in the form of polytopes of some special shape. The specific cross-sections of these polytopes provide the parallelepiped-valued or parallelotope-valued estimates of the reachable sets in the “initial” space. Evolution of the estimates in the “extended” space is determined by recurrence relations. All the estimates can be calculated by explicit formulas. The main attention is paid to internal estimates. Illustrative examples are presented.

Elena K. Kostousova

### Numerical Simulation of Hyperbolic Conservation Laws Using High Resolution Schemes with the Indulgence of Fuzzy Logic

The aim of this paper is to solve numerically a class of problems on conservation laws, modelled by hyperbolic partial differential equations. In this paper, primary focus is over the idea of fuzzy logic-based operators for the simulation of problems related to hyperbolic conservation laws. Present approach considers a novel computational procedure which relies on using some operators from fuzzy logic to reconstruct several higher-order numerical methods known as the flux-limited methods. Further optimization of the flux limiters is discussed. The approach ensures better convergence of the approximation and preserves the basic properties of the solution of the problem under consideration. The new limiters are further applied to several real-life problems like the advection problem to demonstrate that the optimized schemes ensure better results. Simulation results are included wherever required.

Ruchika Lochab, Vivek Kumar

### Methodology for Interval-Valued Matrix Games with 2-Tuple Fuzzy Linguistic Information

In this paper, we consider a non-cooperative 2-player zero-sum interval-valued 2-tuple fuzzy linguistic (IVTFL) matrix game and develop a methodology to evaluate its saddle point and optimal interval-valued linguistic value of the game. In this direction, we have constructed an auxiliary pair of interval-valued linguistic linear programming (IVLLP) problem that is further transformed into conventional interval linear programming (ILP) problem to obtain optimal strategy sets of both players as the region that is not only completely feasible but also totally optimal. The proposed method is illustrated via a hypothetical example to show its applicability in the real world. To validate the suggested solution scheme, the transformed ILP problems are solved using best-worst case (BWC) approach, enhanced-interval linear programming (EILP) method and linguistic linear programming (LLP) technique of solving interval linguistic matrix game problems and lastly the obtained results are compared.

Tanya Malhotra, Anjana Gupta, Anjali Singh

### Set-Membership Computation of Integrals with Uncertain Endpoints

An efficient guaranteed method for the computation of the integral of a nonlinear continuous function between two interval endpoints is proposed. This computation can be of interest for the computation of global optimization problems where such integrals occur like in robotics. The method results in the computation of the minimum and maximum of these integrals and provides the endpoints at stake. The complexity of the resulting algorithms is discussed, it depends on the number of roots of the function to be integrated. The computation is illustrated on several examples.

Olivier Mullier, Julien Alexandre dit Sandretto

### Epidemic Spreading Curing Strategy Over Directed Networks

Epidemic processes on networks have been thoroughly investigated in different research fields including physics, biology, computer science and medicine. Within this research area, a challenge is the definition of curing strategies able to suppress the epidemic spreading while exploiting a minimal quantity of curing resources. In this paper, we model the network under analysis as a directed graph where a virus spreads from node to node with different spreading and curing rates. Specifically, we adopt an approximation of the Susceptible-Infected-Susceptible (SIS) epidemic model, the N-Intertwined Mean Field Approximation (NIMFA). In order to control the diffusion of the virus while limiting the total cost needed for curing the whole network, we formalize the problem of finding an Optimal Curing Policy (OCP) as a constrained optimization problem and propose a genetic algorithm (GA) to solve it. Differently from a previous work where we proposed a GA for solving the OCP problem on undirected networks, here we consider the formulation of the optimization problem for directed weighted networks and extend the GA method to deal with not symmetric adjacency matrices that are not diagonally symmetrizable.

Clara Pizzuti, Annalisa Socievole

### Learning Aerial Image Similarity Using Triplet Networks

Unmanned aerial vehicles (UAV) faces localization challenges in satellite navigation systems denied environments. Images taken from on-board cameras can be used to compare against orthophotographical map to support visual localization algorithms. Image similarity estimation can be achieved calculating various similarity metrics. Pearson correlation was found to be the best choice for evaluating areal images similarity in our experiments. Still is not robust against image displacement caused by aircraft frame movement. We propose a new architecture of triplet neural network to learn image similarity measure. The proposed architecture incorporates VGG16 network base layers. Top layer structure, loss function and performance metrics being suggested by authors. Images were matched to the maps from satellite photo. The matching results from proposed neural network architecture were compared and evaluated against Pearson correlation.

Vytautas Valaitis, Virginijus Marcinkevicius, Rokas Jurevicius

### In the Quest for Invariant Structures Through Graph Theory, Groups and Mechanics: Methodological Aspects in the History of Applied Mathematics

The purpose of this paper is to analyze a geometrical case study as a sample of an intended methodology based on invariant theory’s strategies, which have been developed particularly throughout the nineteenth century as one of the cornerstones of mathematics [15, p. 41], and whose resolution was reached by means of a combination of different disciplines: graph theory, mechanics and group theory, among others.This case study presents the “perfect squared rectangle problem”, that is an exhaustive classification of the dissection of a rectangle into a finite number of unequal squares. Despite its simplicity, in both description and mathematical resolution, it provides plausible elements of generalization from “the ‘applied field’ of mathematics” [8, p. 658], as a special case of applied mathematical toolkit [1, p. 715], related to the practice of invariant strategies that remain fixed through changes.

Sandra Visokolskis, Carla Trillini

### Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions

Generalizations of the traditional intermediate value theorem are presented. The obtained generalized theorems are particular useful for the existence of solutions of systems of nonlinear equations in several variables as well as for the existence of fixed points of continuous functions. Based on the corresponding criteria for the existence of a solution emanated by the intermediate value theorems, generalized bisection methods for approximating fixed points and zeros of continuous functions are given. These bisection methods require only algebraic signs of the function values and are of major importance for tackling problems with imprecise (not exactly known) information.

Michael N. Vrahatis

### Modelling Population Size Using Horvitz-Thompson Approach Based on the Zero-Truncated Poisson Lindley Distribution

Capture-recapture analysis is applied to estimate population size in ecology, biology, social science, medicine, linguistics and software engineering. The Poisson distribution is one of the simplest models for count data and appropriate for homogeneous populations. On the other hand, it is found to underestimate the counts for overdispersed data. In this study, population size estimation using the mixture of Poisson and Lindley distribution is proposed. It can exhibit overdispersed, equidispersed and underdispersed data. Additionally, it is able to present count data with long tail. As a result of the problem of unobserved individuals, the zero-truncated Poisson Lindley distribution is considered. The parameter of distribution can be estimated using the maximum likelihood estimation. The Horvitz-Thompson estimator based on the zero-truncated Poisson Lindley distribution for modelling the population size is investigated in this study. Point and interval estimation of the target population are presented. The technique of conditioning is used for variance estimation of the population size. Relative bias, relative variance and relative mean square error are used for measuring the accuracy of the estimator. The simulation results show that the Horvitz-Thompson estimator under the zero-truncated Poisson Lindley distribution provides a good fit when compared to the zero-truncated Poisson distribution.

Ratchaneewan Wongprachan

### Numerical Analysis of the Model of Optimal Consumption and Borrowing with Random Time Scale

This work is dedicated to modelling economic dynamics with random time scale. We propose a solution in the form a continuous time model where interactions of agents are random exchanges of finite portions of products and money at random points in time. In this framework, the economic agent determines the volume, but not the moments of the transactions and their order. The paper presents a correct formal description of optimal consumption and borrowing as a stochastic optimal control problem, which we study using the optimality conditions in the Lagrange’s form. The solution appears to have a boundary layer near the end of planning horizon where the optimal control satisfies the specific functional equation. This equation was studied numerically using the functional Newton method adapted to a two-dimensional case.

Aleksandra Zhukova, Igor Pospelov

### Conditions of the Stability Preservation Under Discretization of a Class of Nonlinear Time-Delay Systems

Nonlinear differential systems with nonlinearities satisfying sector constraints and with constant delays are studied. Such systems belong to well-known class of Persidskii-type systems, and they are widely used for modeling automatic control systems and neural networks. With the aid of the Lyapunov direct method and original constructions of Lyapunov–Krasovskii functionals, we derive conditions of the stability preservation under discretization of the considered differential systems. The fulfilment of these conditions guarantees that the zero solutions of the corresponding difference systems are asymptotically stable for arbitrary values of delays. Moreover, estimates of the convergence rate of solutions are obtained. The proposed approaches are used for the stability analysis of a discrete-time model of population dynamics. An example is given to demonstrate the effectiveness of our results.

Alexander Yu. Aleksandrov

### Numerical Investigation of Natural Rough-Bed Flow

The turbulent flow in natural rough-bed watercourses is a rather complex phenomenon, still poorly investigated. The majority of the existing works on this subject is of experimental nature, while the numerical ones are mostly related to artificially and regularly-roughened beds. In the present work a numerical investigation is carried out, in which the fully turbulent flow in an open channel is simulated, where the channel bottom is constituted by natural-pebble layers. In the numerical simulations, the Large Eddy Simulation (LES) approach is used, in conjunction with the Wall-Adapting Local Eddy viscosity (WALE) Sub-Grid Scale (SGS) closure model at Reynolds number 46,500 and Froude number 0.186. The Finite-Volume discretized governing equations are solved numerically by means of the InterFOAM solver, embedded in the OpenFOAM C++ digital library. In order to take into account the free-surface dynamics, the Volume of Fluid (VoF) method has been used. The results of the simulations are compared with those obtained in a companion experiment, mainly in terms of turbulence statistics of different order, obtaining a rather good agreement.

Giancarlo Alfonsi, Domenico Ferraro, Agostino Lauria, Roberto Gaudio

### A Dynamic Precision Floating-Point Arithmetic Based on the Infinity Computer Framework

We introduce a dynamic precision floating-point arithmetic based on the Infinity Computer. This latter is a computational platform which can handle both infinite and infinitesimal quantities epitomized by the positive and negative finite powers of the symbol , which acts as a radix in a corresponding positional numeral system. The idea is to use the positional numeral system from the Infinity Computer to devise a variable precision representation of finite floating-point numbers and to execute arithmetical operations between them using the Infinity Computer Arithmetics. Here, numerals with negative finite powers of will act as infinitesimal-like quantities whose aim is to dynamically improve the accuracy of representation only when needed during the execution of a computation. An application to the iterative refinement technique to solve linear systems is also presented.

Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro, Francesca Mazzia

### Numerical Strategies for Solving Multiparameter Spectral Problems

We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter $$\sigma \in [0,1]$$, that give back the original problem only when $$\sigma =1$$. We compare the strategies by using well known test problems with two and three parameters.

Pierluigi Amodio, Giuseppina Settanni

### Algorithms of 3D Wind Field Reconstructing by Lidar Remote Sensing Data

In this paper, we analyzed the performance of wind vector field recovery from the wind lidar measurements. Wind lidar (LIDAR – Light Identification Detection And Ranging) remotely measures the wind radial speed by using the Doppler principle. Algorithms of the wind vector reconstruction using different versions of the least squares method are considered. In particular, the versions of weighted least squares (WLS) are considered, as well as the use of data spikes filtering procedures in the source data. The weights were calculated inversely with the local approximation error. As the initial data, the data of real measurements obtained in various wind conditions were used. The situations of a stationary wind field, a wind field with speed gusts, a wind field with fluctuations in direction, a wind field of variable speed and direction are considered. Lidar data were obtained for a region with a low-hilly terrain; therefore, even in the case of a stationary in time, the wind field was characterized by spatial heterogeneity. The questions of the use of regularization methods are considered. The analysis of the influence of the size of the averaging region on the quality of the recovery process was carried out.

Nikolay Baranov

### Stationarity Condition for Nonsmooth MPVCs with Constraint Set

We consider a nonsmooth optimization problem with vanishing constraints and constraint set. A new constraint qualification and a necessary condition for M-stationary of the problem are presented. Our results are formulated in terms of Mordukhoivich subdifferential.

David Barilla, Giuseppe Caristi, Nader Kanzi

### Molecular Dynamics Performance Evaluation with Modern Computer Architecture

An important task of chemical biology is to discover the mechanism of recognition and binding between proteins. Despite the simplicity of the ligand-based model, fundamental mechanisms that regulate these interactions are poorly understood. An adequate equipment is mandatory to unravel this scientific challenge, not only through cost savings but also with high-quality results. With this in mind, we performed Molecular Dynamics simulations using the Gromacs package on two promising platforms: Cavium ThunderX2 ARM based cluster setup and shared-memory Intel based single-node machine. Aforementioned tests were also performed on common Intel based servers as a reference. Acquired results shown that shared-memory machine features the higest performance, although ARM and Intel clutsers are only slightly slower when more than four sockets are employed. During measurements, idle and job-execution consumptions were sampled in order to evaluate the energy required by a single simulation step. Results show that ARM and Intel servers are much less power-hungry with respect to shared-memory machine. The latter, on the other hand, features a decrement in power consumption when more resources are employed. Said unexpected behaviour is later discussed.

Emanuele Breuza, Giorgio Colombo, Daniele Gregori, Filippo Marchetti

### Artificial Neural Networks Training Acceleration Through Network Science Strategies

Deep Learning opened artificial intelligence to an unprecedented number of new applications. A critical success factor is the ability to train deeper neural networks, striving for stable and accurate models. This translates into Artificial Neural Networks (ANN) that become unmanageable as the number of features increases. The novelty of our approach is to employ Network Science strategies to tackle the complexity of the actual ANNs at each epoch of the training process. The work presented herein originates in our earlier publications, where we explored the acceleration effects obtained by enforcing, in turn, scale freeness, small worldness, and sparsity during the ANN training process. The efficiency of our approach has also been recently confirmed by independent researchers, who managed to train a million-node ANN on non-specialized laptops. Encouraged by these results, we have now moved into having a closer look at some tunable parameters of our previous approach to pursue a further acceleration effect. We now investigate on the revise fraction parameter, to verify the necessity of the role of its double-check. Our method is independent of specific machine learning algorithms or datasets, since we operate merely on the topology of the ANNs. We demonstrate that the revise phase can be avoided in order to half the overall execution time with an almost negligible loss of quality.

Lucia Cavallaro, Ovidiu Bagdasar, Pasquale De Meo, Giacomo Fiumara, Antonio Liotta

### Grossone Methodology for Lexicographic Mixed-Integer Linear Programming Problems

In this work we have addressed lexicographic multi-objective linear programming problems where some of the variables are constrained to be integer. We have called this class of problems LMILP, which stands for Lexicographic Mixed Integer Linear Programming. Following one of the approach used to solve mixed integer linear programming problems, the branch and bound technique, we have extended it to work with infinitesimal/infinite numbers, exploiting the Grossone Methodology. The new algorithm, called GrossBB, is able to solve this new class of problems, by using internally the GrossSimplex algorithm (a recently introduced Grossone extension of the well-known simplex algorithm, to solve lexicographic LP problems without integer constraints). Finally we have illustrated the working principles of the GrossBB on a test problem.

Marco Cococcioni, Alessandro Cudazzo, Massimo Pappalardo, Yaroslav D. Sergeyev

### Infinite Games on Finite Graphs Using Grossone

In his seminal work, Robert McNaughton (see [1] and [7]) developed a model of infinite games played on finite graphs. This paper presents a new model of infinite games played on finite graphs using the Grossone paradigm. The new Grossone model provides certain advantages such as allowing for draws, which are common in board games, and a more accurate and decisive method for determining the winner when a game is played to infinite duration.

Louis D’Alotto

### A Novel Geometric Approach to the Problem of Multidimensional Scaling

Multidimensional scaling (MDS) is one of the most popular methods for a visual representation of multidimensional data. A novel geometric interpretation of the stress function and multidimensional scaling in general (Geometric MDS) has been proposed. Following this interpretation, the step size and direction forward the minimum of the stress function are found analytically for a separate point without reference to the analytical expression of the stress function, numerical evaluation of its derivatives and the linear search. It is proved theoretically that the direction coincides with the steepest descent direction, and the analytically found step size guarantees the decrease of stress in this direction. A strategy of application of the discovered option to minimize the stress function is presented and examined. It is compared with SMACOF version of MDS. The novel geometric approach will allow developing a new class of algorithms to minimize MDS stress, including global optimization and high-performance computing.

Gintautas Dzemyda, Martynas Sabaliauskas

### A Simulink-Based Infinity Computer Simulator and Some Applications

This paper is dedicated to the Infinity Computer – a new type of a supercomputer allowing one to work numerically with finite, infinite, and infinitesimal numbers in one general framework. The existent software simulators of the Infinity Computer are used already for solving important real-world problems in applied mathematics. However, they are not efficient for solving difficult problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. For this purpose, the main aim of this paper is to introduce a new Simulink-based solution of the Infinity Computer.

Alberto Falcone, Alfredo Garro, Marat S. Mukhametzhanov, Yaroslav D. Sergeyev

### Generalizing Pure and Impure Iterated Prisoner’s Dilemmas to the Case of Infinite and Infinitesimal Quantities

In this work, a generalization of both Pure and Impure iterated Prisoner’s Dilemmas is presented. More precisely, the generalization concerns the use of non-Archimedean quantities, i.e., payoffs that can be infinite, finite or infinitesimal and probabilities that can be finite or infinitesimal. This new approach allows to model situations that cannot be adequately addressed using iterated games with purely finite quantities. This novel class of models contains, as a special case, the classical known ones. This is an important feature of the proposed methodology, which assures that we are proposing a generalization of the already known games. The properties of the generalized models have also been validated numerically, by using a Matlab simulator of Sergeyev’s Infinity Computer.

Lorenzo Fiaschi, Marco Cococcioni

### Multidimensional Global Search Using Numerical Estimations of Minimized Function Derivatives and Adaptive Nested Optimization Scheme

This paper proposes a novel approach to the solution of time-consuming multivariate multiextremal optimization problems. This approach is based on integrating the global search method using derivatives of minimized functions and the nested scheme for dimensionality reduction. In contrast with related works novelty is that derivative values are calculated numerically and the dimensionality reduction scheme is generalized for adaptive use of the search information. The obtained global optimization method demonstrates a good performance, which has been confirmed by numerical experiments.

Victor Gergel, Alexey Goryachikh

### 2-Approximation Polynomial-Time Algorithm for a Cardinality-Weighted 2-Partitioning Problem of a Sequence

We consider a problem of 2-partitioning a finite sequence of points in Euclidean space into clusters of the given sizes with some constraints. The solution criterion is the minimum of the sum of weighted intracluster sums of squared distances between the elements of each cluster and its center. The weight of the intracluster sum is equal to the cluster size. The center of one cluster is given as input (is the origin without loss of generality), while the center of the other one is unknown and is determined as a geometric center. The following constraints hold: the difference between the indices of two subsequent points included in the first cluster is bounded from above and below by some given constants. In this paper, we have shown that the considered problem is the strongly NP-hard one and propose a polynomial-time 2-approximation algorithm for solving the problem.

Alexander Kel’manov, Sergey Khamidullin, Anna Panasenko

### Exact Linear-Time Algorithm for Parameterized K-Means Problem with Optimized Number of Clusters in the 1D Case

We consider a well-known strongly NP-hard K-Means problem. In this problem, one needs to partition a finite set of N points in Euclidean space into K non-empty clusters minimizing the sum over all clusters of the intracluster sums of the squared distances between the elements of each cluster and its centers. The cluster’s center is defined as the centroid (geometrical center). We analyze the polynomial-solvable one-dimensional case of the problem and propose a novel parameterized approach to this case. Within the framework of this approach, we, firstly, introduce a new parameterized formulation of the problem for this case and, secondly, we show that our approach and proposed algorithm allows one to find an optimal input data partition and, contrary to existing approaches and algorithms, simultaneously find an optimal clusters number in $$\mathcal {O}(N)$$ time.

### Polynomial-Time Approximation Scheme for a Problem of Searching for the Largest Subset with the Constraint on Quadratic Variation

The paper is addressed to one strongly NP-hard problem of searching for the largest subset in the finite set of points in Euclidean space. A restriction is imposed on the searched subset: quadratic variation of its points with respect to the unknown centroid of this subset must not exceed a given value. We present the first polynomial-time approximation scheme for this problem.

### On a Comparison of Several Numerical Integration Methods for Ordinary Systems of Differential Equations

The paper considers the numerical integration methods for ordinary systems of differential equations in which the end of the integration interval is a priori undefined but is defined during the integration process instead. Moreover, the calculation of right hand sides of such systems is an expensive procedure. The paper describes a new integration strategy based on an implicit fourth order method. The proposed strategy employs the behavior of obtained solution to control the integration process. In addition, the number of integration nodes selected by the mentioned method is minimal at every fixed interval under the limitations defined by the local error which results from the approximation of system derivatives.

Anatoliy G. Korotchenko, Valentina M. Smoryakova

### On Acceleration of Derivative-Free Univariate Lipschitz Global Optimization Methods

Univariate box-constrained global optimization problems are considered, where the objective function is supposed to be Lipschitz continuous and multiextremal. It is assumed that its analytical representation is unknown (the function is given as a “black-box”) and even one its evaluation is a computationally expensive procedure. Geometric and information statistical frameworks for construction of global optimization algorithms are discussed. Several powerful acceleration techniques are described and a number of methods of both classes is constructed by mixing the introduced acceleration ideas. Numerical experiments executed on broad test classes taken from the literature show advantages of the presented techniques with respect to their direct competitors.

Dmitri E. Kvasov, Marat S. Mukhametzhanov, Maria Chiara Nasso, Yaroslav D. Sergeyev

### Numerical Simulation of Ski-Jump Hydraulic Behavior

The hydraulic behavior of ski jumps is investigated numerically using the OpenFOAM digital library. A number of ski-jump cases has been simulated by following the RANS approach (Reynolds Averaged Navier-Stokes equations), using the k-ω SST closure model, and the VoF technique (Volume of Fluid) for the tracking of the free surface. Particular attention is given to the pressure distributions in the zone of impact of the falling jet, and to the length of the jet itself, as defined as the distance along the x-direction between the point of maximum dynamic pressure head, and the origin of the reference frame. A chart is proposed reporting the correlation line (and correspondent formal expression) between the approach Froude numbers and the lengths of the jets, in the limit of other parameters tested. The chart may serve as a useful tool for the determination of the length of the jet taking off from the bucket, starting from the value of the approach Froude number.

Agostino Lauria, Giancarlo Alfonsi

### On Linear Spline Wavelets with Shifted Supports

We examine Faber’s type decompositions for spaces of linear minimal splines constructed on nonuniform grids on a segment. A characteristic feature of the Faber decomposition is that the basis wavelets are centered around the knots that do not belong to the coarse grid. The construction of the lazy wavelets begins with the use of the basis functions in refined spline space centered at the odd knots. We propose to use as wavelets the functions centered at the even knots under some conditions. In contrast to lazy wavelets, in this case the decomposition system of equations has a unique solution, which can be found by the sweep method with the guarantee of well-posedness and stability.

Svetlana Makarova, Anton Makarov

### An Online Learning Approach to a Multi-player N-armed Functional Bandit

Congestion games possess the property of emitting at least one pure Nash equilibrium and have a rich history of practical use in transport modelling. In this paper we approach the problem of modelling equilibrium within congestion games using a decentralised multi-player probabilistic approach via stochastic bandit feedback. Restricting the strategies available to players under the assumption of bounded rationality, we explore an online multiplayer exponential weights algorithm for unweighted atomic routing games and compare this with a $$\epsilon$$-greedy algorithm.

Sam O’Neill, Ovidiu Bagdasar, Antonio Liotta

### The Singular Value Decomposition of the Operators of the Dynamic Ray Transforms Acting on 2D Vector Fields

The problem of dynamic 2D vector tomography is considered. Object motion is a combination of rotation and shifting. Properties of the dynamic ray transform operators are investigated. Singular value decomposition of the operators is constructed with usage of classic orthogonal polynomials.

Anna P. Polyakova, Ivan E. Svetov, Bernadette N. Hahn

### Numerical Investigation into a Combined Ecologically Pure Plasma-Vortex Generator of Hydrogen and Thermal Energy

One of the promising approaches in clean energy is hydrogen production using the hydration reaction of a metal micropowder, stimulated by plasma formations. In present work, we carried out a series of numerical simulations of the turbulent three-dimensional swirling flow with chemical reactions in plasma vortex reactor (PVR) to provide further insights into use of such apparatus. The modelling has demonstrated that desirable behavior with heat transferred mostly downstream can be ensured by use of pipe-like electrode placed downstream from cylindrical one. Then, using exact experimental geometry and flow composition, optimized electrode system and simplified kinetic scheme of plasma-chemical reactions, we tested several operating modes of the PVR and obtained time dependent flow characteristics, which, in turn, will be used for further adjustment of the scheme and overall apparatus.

Denis Porfiriev, Svetlana Kurushina, Nonna Molevich, Igor Zavershinsky

### Computer Modeling of Electrochemical Impedance Spectra for Defected Phospholipid Membranes: Finite Element Analysis

This study deals with application of finite element method to model electrochemical impedance spectra of phospholipid membranes containing defects. Practical issues of choosing mesh and solver parameters are investigated in order to obtain the best combination of solution accuracy and computation times for the given problem. A simple mesh generation strategy suitable for membrane models with various randomly generated defect distributions is presented. Models with varying mesh densities were solved with direct and iterative solvers and solution accuracy was evaluated in terms of EIS spectral features. Computation times of models with various mesh sizes and solver configurations were also measured in two different computing environments.

### A Compact Filter Regularization Method for Solving Sideways Heat Equation

In this paper, the stable approximate solution of the sideways heat equation is numerically investigated. The problem is severely ill-posed because if the solution exists, it does not depend continuously on the data. We introduce the compact filter regularization as a new, simple and convenient regularization method. Furthermore, the numerical implementation of the method is discussed. The numerical example shows that the proposed method is efficient and feasible.

Ankita Shukla, Mani Mehra

### Acceleration of Global Search by Implementing Dual Estimates for Lipschitz Constant

The paper considers global optimization problems with a black-box objective function satisfying the Lipschitz condition. Efficient algorithms for this class of problems require reliable estimates of the Lipschitz constant to be introduced. Various approaches have been proposed to take into account both global and local properties of the objective function. In particular, algorithms using local estimates of the Lipschitz constant have shown their potential. The new approach presented in this paper is based on simultaneous use of two estimates: one is substantially larger than the other. The larger estimate ensures global convergence and the smaller one reduces the total number of trials needed to find the global optimizer. Results of numerical experiments on the random sample of multidimensional functions demonstrate the efficiency of the approach proposed by the authors.

Roman Strongin, Konstantin Barkalov, Semen Bevzuk

### The Method of Approximate Inverse in Slice-by-Slice Vector Tomography Problems

A numerical solution of the problem of recovering the solenoidal part of three-dimensional vector field using the incomplete tomographic data is proposed. Namely, values of the ray transform for all straight lines, which are parallel to one of the coordinate planes, are known. The recovery algorithms are based on the method of approximate inverse.

Ivan E. Svetov, Svetlana V. Maltseva, Alfred K. Louis

### Computational Fluid Dynamics Methods for Wind Resources Assessment

The use of already existing infrastructure for mounting of wind speed sensors could be a promising way of how to assess wind resources instead to install the new meteorological mast. One part of this study is devoted to exploring the impact of the mast on the flow field around it. Computational Fluid Dynamics (CFD) is chosen to predict airflow using Reynolds-Averaged Navier-Stokes equations. In the second part of this research, the typical topology near the Baltic Sea is selected to evaluate numerically the turbulent airflow over coastal terrain. The lidar images are utilized to describe the topology of the interested area. Digital Surface Model is used to generate the ground surface which is applied as the input to develop the high-resolution computational mesh of the terrain. Computational domain parallelization and the computational cluster is applied due to the complexity of the numerical simulations. Obtained results are compared with experimentally measured data from wind speed sensors located on the telecommunication mast.

Sabine Upnere, Valerijs Bezrukovs, Vladislavs Bezrukovs, Normunds Jekabsons, Linda Gulbe

### On Stationary Points of Distance Depending Potentials

We continue investigations started in the previous publications by the authors (LNCS, volumes 8136 (2013) and 9570 (2016)). The structure of stationary point sets is established for the family of functions given as linear combinations of an exponent L of Euclidean distances from a variable point to the fixed points in 2D and 3D spaces. We compare the structure of the stationary point sets for several values of the exponent L, focusing ourselves mainly onto the cases of Coulomb potential and Weber facility location problem. We develop the analytical approach to the problem aiming at finding the exact number of stationary points and their location in relation to the parameters involved.

Alexei Uteshev, Marina Goncharova

### A Partition Based Bayesian Multi-objective Optimization Algorithm

The research is aimed at coping with the inherent computational intensity of Bayesian multi-objective optimization algorithms. We propose the implementation which is based on the rectangular partition of the feasible region and circumvents much of computational burden typical for the traditional implementations of Bayesian algorithms. The included results of the solution of testing and practical problems illustrate the performance of the proposed algorithm.

Antanas Žilinskas, Linas Litvinas

### Problem Statement for Preparing a Single Batch of End Product Under Uncertainty

Oil refining is a key industry of the world economy. Growing hydrocarbon production cost and global competition in the oil market encourage the oil refining industry to optimize the production scheme. The evolution of mathematical tools of automated enterprise control systems is closely connected with the systems development at each level of control. Mathematical models for organizational and economic control of the enterprise and process control models are widely presented in publications and implemented in the enterprise information systems. The management of operational scheduled and dispatching production is one of the most complex problems. The paper deals with the problem of finding an optimal ratio for the components from the tanks to obtain an oil product of the required amount and quality in a commercial tank. The peculiarity of the mathematical models proposed for solving the problem is that only the boundaries for each quality indicator of petroleum product are known. To formalize the emerging uncertainty, models utilizing the interval approach are proposed.

Anna Zykina, Olga Kaneva, Vladimir Savkin, Tatyana Fink

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