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Mathematics
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Numerical Analysis with Applications in Machine Learning
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Numerical Analysis with Applications in Machine Learning

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (20 January 2023) | Viewed by 16381

Special Issue Editor

Prof. Dr. Panagiota Tsompanopoulou

E-Mail Website
Guest Editor
Department of Electrical and Computer Engineering, University of Thessaly, 38221 Volos, Greece
Interests: numerical analysis; scientific computing; numerical linear algebra; machine learning; parallel computing; agent computing; problem solving environments

Special Issue Information

Dear Colleagues,

The collection of large amounts of data produced by an enormous variety of users has been a fact for many years now. Therefore, there is a tremendous need to study, analyze, and process these data in order to clear out the possible noise and derive the substantial information. The methods used for such problems constitute the scientific area of machine learning. Looking closer to the theory that supports these methods, one recognizes many fields of numerical analysis, such as Euclidean spaces with metrics and norms, approximation theory, optimization theory, theory of matrices, etc. The use of existing methods and the tuning of their parameters still gives very interesting results for the treated problems. Wishing to go further in the solution of existing or new more difficult problems, scientists have to go back to the roots of the mathematics used in machine learning, in order to research and create new, stable, and accurate methods.

Through this Special Issue, we invite our colleagues to submit articles that rely on numerical analysis methods to address problems in the field of machine learning, presenting both theoretical and experimental results. The fields of interest originate from mathematics and computer science, including (but not limited to) numerical linear algebra, Euclidean, pseudo-Euclidean and metric spaces, theory of matrices, theory of approximation and optimization, machine learning, computer vision, classification, clustering, and pattern recognition.

Prof. Dr. Panagiota Tsompanopoulou
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Numerical analysis
  • Numerical linear algebra
  • Euclidean space
  • Metric space
  • Theory of matrices
  • Approximation theory
  • Optimization theory
  • Machine learning
  • Computer vision
  • Classification
  • Pattern recognition

Published Papers (6 papers)

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Research

16 pages, 905 KiB  
Article
A Noise-Aware Multiple Imputation Algorithm for Missing Data
by Fangfang Li, Hui Sun, Yu Gu and Ge Yu
Mathematics 2023, 11(1), 73; https://doi.org/10.3390/math11010073 - 25 Dec 2022
Cited by 1 | Viewed by 1027
Abstract
Missing data is a common and inevitable phenomenon. In practical applications, the datasets usually contain noises for various reasons. Most of the existing missing data imputing algorithms are affected by noises which reduce the accuracy of the imputation. This paper proposes a noise-aware [...] Read more.
Missing data is a common and inevitable phenomenon. In practical applications, the datasets usually contain noises for various reasons. Most of the existing missing data imputing algorithms are affected by noises which reduce the accuracy of the imputation. This paper proposes a noise-aware missing data multiple imputation algorithm NPMI in static data. Different multiple imputation models are proposed according to the missing mechanism of data. Secondly, the method to determine the imputation order of multivariablesmissing is given. A random sampling consistency algorithm is proposed to estimate the initial values of the parameters of the multiple imputation model to reduce the influence of noise data and improve the algorithm’s robustness. Experiments on two real datasets and two synthetic datasets verify the accuracy and efficiency of the proposed NPMI algorithm, and the results are analyzed. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
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37 pages, 8202 KiB  
Article
Lie-Group Type Quadcopter Control Design by Dynamics Replacement and the Virtual Attractive-Repulsive Potentials Theory
by Simone Fiori, Luca Bigelli and Federico Polenta
Mathematics 2022, 10(7), 1104; https://doi.org/10.3390/math10071104 - 29 Mar 2022
Cited by 4 | Viewed by 2210
Abstract
The aim of the present research work is to design a control law for a quadcopter drone based on the Virtual Attractive-Repulsive Potentials (VARP) theory. VARP theory, originally designed to enable path following by a small wheeled robot, will be tailored to control [...] Read more.
The aim of the present research work is to design a control law for a quadcopter drone based on the Virtual Attractive-Repulsive Potentials (VARP) theory. VARP theory, originally designed to enable path following by a small wheeled robot, will be tailored to control a quadcopter drone, hence allowing such device to learn flight planning. The proposed strategy combines an instance of VARP method to control a drone’s attitude (SO(3)-VARP) and an instance of VARP method to control a drone’s spatial location (R3-VARP). The resulting control strategy will be referred to as double-VARP method, which aims at making a drone follow a predefined path in space. Since the model of the drone as well as the devised control theory are formulated on a Lie group, their simulation on a computing platform is performed through a numerical analysis method specifically designed for these kinds of numerical simulations. A numerical simulation analysis is used to assess the salient features of the proposed regulation theory. In particular, resilience against shock-type disturbances are assessed numerically. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
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20 pages, 2130 KiB  
Article
Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach
by Kathrin Hellmuth and Christian Klingenberg
Mathematics 2022, 10(3), 489; https://doi.org/10.3390/math10030489 - 03 Feb 2022
Cited by 2 | Viewed by 2393
Abstract
In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model [...] Read more.
In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
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18 pages, 3712 KiB  
Article
Cross-Platform GPU-Based Implementation of Lattice Boltzmann Method Solver Using ArrayFire Library
by Michal Takáč and Ivo Petráš
Mathematics 2021, 9(15), 1793; https://doi.org/10.3390/math9151793 - 28 Jul 2021
Cited by 4 | Viewed by 3371
Abstract
This paper deals with the design and implementation of cross-platform, D2Q9-BGK and D3Q27-MRT, lattice Boltzmann method solver for 2D and 3D flows developed with ArrayFire library for high-performance computing. The solver leverages ArrayFire’s just-in-time compilation engine for compiling high-level code into optimized kernels [...] Read more.
This paper deals with the design and implementation of cross-platform, D2Q9-BGK and D3Q27-MRT, lattice Boltzmann method solver for 2D and 3D flows developed with ArrayFire library for high-performance computing. The solver leverages ArrayFire’s just-in-time compilation engine for compiling high-level code into optimized kernels for both CUDA and OpenCL GPU backends. We also provide C++ and Rust implementations and show that it is possible to produce fast cross-platform lattice Boltzmann method simulations with minimal code, effectively less than 90 lines of code. An illustrative benchmarks (lid-driven cavity and Kármán vortex street) for single and double precision floating-point simulations on 4 different GPUs are provided. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
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16 pages, 301 KiB  
Article
Convergence Analysis of the Straightforward Expansion Perturbation Method for Weakly Nonlinear Vibrations
by Soledad Moreno-Pulido, Francisco Javier García-Pacheco, Alberto Sánchez-Alzola and Alejandro Rincón-Casado
Mathematics 2021, 9(9), 1036; https://doi.org/10.3390/math9091036 - 03 May 2021
Cited by 3 | Viewed by 1429
Abstract
There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters, the Naive Singular Perturbation Method, the Method of Multiple Scales, [...] Read more.
There are typically several perturbation methods for approaching the solution of weakly nonlinear vibrations (where the nonlinear terms are “small” compared to the linear ones): the Method of Strained Parameters, the Naive Singular Perturbation Method, the Method of Multiple Scales, the Method of Harmonic Balance and the Method of Averaging. The Straightforward Expansion Perturbation Method (SEPM) applied to weakly nonlinear vibrations does not usually yield to correct solutions. In this manuscript, we provide mathematical proof of the inaccuracy of the SEPM in general cases. Nevertheless, we also provide a sufficient condition for the SEPM to be successfully applied to weakly nonlinear vibrations. This mathematical formalism is written in the syntax of the first-order formal language of Set Theory under the methodology framework provided by the Category Theory. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
20 pages, 1132 KiB  
Article
Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks
by Beatriz Salvador, Cornelis W. Oosterlee and Remco van der Meer
Mathematics 2021, 9(1), 46; https://doi.org/10.3390/math9010046 - 28 Dec 2020
Cited by 8 | Viewed by 2302
Abstract
Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element [...] Read more.
Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). The classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied here. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black–Scholes equation, whereas for the American option we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements). In the short note, previously published, a brief introduction to this work was given, where some ideas to price vanilla options by ANNs were presented, and only European options were addressed. In the current work, the methodology is introduced in much more detail. Full article
(This article belongs to the Special Issue Numerical Analysis with Applications in Machine Learning)
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