Performance optimization of Gaussian mixture algorithms in mathematical analysis

In this paper, we consider the application of brute force computational techniques (BFCTs) for solving computational problems in mathematical analysis and matrix algebra in a floating-point computing environment. These techniques include, among others, simple matrix computations and the analysis of graphs of functions. Since BFCTs are based on matrix calculations, the program system MATLAB® is suitable for their computer realization. The computations in this paper are completed in double precision floating-point arithmetic, obeying the 2019 IEEE Standard for binary floating-point calculations. One of the aims of this paper is to analyze cases where popular algorithms and software fail to produce correct answers, failing to alert the user. In real-time control applications, this may have catastrophic consequences with heavy material damage and human casualties. It is known, or suspected, that a number of man-made catastrophes such as the Dharhan accident (1991), Ariane 5 launch failure (1996), Boeing 737 Max tragedies (2018, 2019) and others are due to errors in the computer software and hardware. Another application of BFCTs is finding good initial guesses for known computational algorithms. Sometimes, simple and relatively fast BFCTs are useful tools in solving computational problems correctly and in real time. Among particular problems considered are the genuine addition of machine numbers, numerically stable computations, finding minimums of arrays, the minimization of functions, solving finite equations, integration and differentiation, computing condensed and canonical forms of matrices and clarifying the concepts of the least squares method in the light of the conflict remainders vs. errors. Usually, BFCTs are applied under the user’s supervision, which is not possible in the automatic implementation of computational methods. To implement BFCTs automatically is a challenging problem in the area of artificial intelligence and of mathematical artificial intelligence in particular. BFCTs allow to reveal the underlying arithmetic in the performance of computational algorithms. Last but not least, this paper has tutorial value, as computational algorithms and mathematical software are often taught without considering the properties of computational algorithms and machine arithmetic.

Solving Problems in Mathematical Analysis, Part II

Advanced optimization methods are studied systematically in a great level. The three main elements of optimization, that is model parameterization, optimization algorithm and merit function, base on this, the relationship among them and their reciprocity is defined and analyzed in detail. Some local and global optimization strategies, especial for zoom systems effect to the integrated capabilities, are discussed. And the results on simulation and evaluation support the view, which demonstrate their validity and potential.

The article presents a detailed analysis of the computing capabilities of the GPU (Graphics Processing Unit) using NVIDIA Compute Unified Device Architecture (NVIDIA CUDA) compared to traditional sequential computing methods. For this purpose, an application implementing the Gaussian blur algorithm was developed. Then, an implementation of the problem was created in the form of a program. The next step presented the methodology of conducting a study comparing the efficiency of solving the problem with several test configurations. Then, research was carried out during which the data obtained in the form of program implementation times were collected. This paper aims to evaluate the computational capabilities of the GPU using NVIDIA CUDA compared to traditional sequential computing methods. The comparison was made through a developed appli-cation that implements the Gaussian fuzzy algorithm. The article can serve as a valuable educational resource for teaching parallel programming and algorithm optimization using GPU and CUDA tech-nologies. The conducted analysis also provides a strong example of an educational project that com-bines algorithm theory with practical application in the context of improving computational perfor-mance.

The paper reviews and introduces the state-of-the-art nature-inspired metaheuristic algorithms in optimization, including Firefly algorithm, PSO algorithms and ABC algorithm. By implementing these algorithms in Matlab, we will use worked examples to show how each algorithm works. Firefly algorithm is one of the evolutionary optimization algorithms, and is inspired by the flashing behaviour of fireflies in nature. There are many noisy non-linear mathematical optimization problems that can be effectively solved by Metaheuristic Algorithms. Mathematical optimization or programming is the study of such planning and design problems using mathematical tools. Nowadays, computer simulations become an indispensable tool for solving such optimization problems with various efficient search algorithms. Nature-inspired algorithms are among the most powerful algorithms for optimization. Firefly algorithm is one of the new metaheuristic algorithms for optimization problems. The algorithm is inspired by the flashing behaviour of fireflies. A Firefly Algorithm (FA) is a recent nature inspired optimization algorithm, which simulates the flash pattern and characteristics of fireflies. It is a powerful swarm intelligence algorithm inspired by the flash phenomenon of the fireflies. In this context, three types of meta-heuristics called Artificial bee Colony algorithm, Particle Swarm Optimization (PSO) and Firefly algorithms were devised to find optimal solutions of noisy non-linear continuous mathematical models. A series of computational experiments using each algorithm were conducted. The stimulation result of this experiment were analyzed and compared to the best solutions found so The Firefly algorithm in each noisy non linear optimization function seems to perform better and efficient.

A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

In this paper, especially in Sections 1–3, we give an account of the author’s results concerning J-inner matrix-functions and related continuation problems stated in [1] and whose proofs can be found in [2]. They are supplemented, in Section 4, 5, with some results about the solutions of the continuation problem with maximal jump of the spectral function, which are obtained using a generalization of Caratheodory Theorem on holomorphic contractive scalar functions in the case of matrix-valued functions. For the significance of J-inner matrix-functions in various problems of Mathematical Analysis and their applications, see [3–6], [25, 26].

Hybrid membrane filtration processes involve complex physical, chemical and biological phenomena, thus their mechanistic modelling is challenging. The chief advantages of statistical and artificial neural networks (ANN) models (data-driven models) are that they do not require assumptions and simplifications to establish relationships from data. This paper investigates the characteristics and performance of several data-driven methods to model a hybrid membrane system. The focus is on the application of regression analysis and artificial intelligence based methods to a steady-state system. Among empirically based approaches, ANN neural networks methods were found to be very useful to predict permeate quality and membrane fouling. In the past multivariate nonlinear regression had barely been investigated for process modelling in water and waste water treatment. In this study polynomial multivariate nonlinear regression showed a superior performance. Multivariate parametric nonlinear models could match the performance of the nonparametric ANN models in the empirical modelling of complex systems, especially when combined with advanced optimization methods. This paper gives the methodology of how one could optimize a membrane hybrid system using ANN, validating it with one set of data. The same procedure/methodology can be applied to similar systems.

Contributors

The rapid growth of smart grid systems has resulted in an exponential increase in data volume. This has led to significant challenges for efficient storage and transmission, thereby necessitating the development of advanced data compression techniques. Traditional techniques often struggle to maintain a balance between compression efficiency and data integrity, particularly when dealing with diverse and large datasets. To address this issue, this paper presents an adaptive data compression algorithm based on discrete wavelet transform (DWT) and Horse Herd Optimization (HHO). The proposed technique significantly improves storage by dynamically optimizing key performance metrics, such as signal-to-noise ratio (SNR), mean square error (MSE), and compression ratio (CR). By employing a robust optimization algorithm, it effectively addresses the computational challenges of processing large-scale and real-time data. This adaptability ensures the algorithm is well-suited for dynamic smart grid environments, providing a scalable and reliable solution to modern data management demands. This technique uses HHO to find the optimal thresholding for smart grid data compression. In general, DWT-based data compression is carried out using a universal threshold for ignoring particular wavelet coefficients. But, the performance of data compression varies for different threshold values. Hence, selecting an optimal threshold is a challenging task for data compression. Therefore, to solve this issue, an effective optimization algorithm is needed. In this work, a Multi-Objective Horse Herd Optimization (MO-HHO) algorithm is proposed to find the optimal threshold. The suggested MO-HHO algorithm accurately determines the global optimum threshold. Hence, it maintains a good compromise between SNR, MSE and CR. The effectiveness of the proposed algorithm is examined using three different data sets. Various datasets from the IEEE Power quality wave data, Household Electric Power Consumption data and Real-time dataset from an experimental set-up were used to test the proposed algorithm. The outcome demonstrates that the suggested MO-HHO algorithm performs better than the other conventional methods.

Support Vector Regression (SVR) is one of the frequently used methods for regression estimation. Kernel selection in SVR is of great importance for the training and estimation of the model. Determining the parameters of the selected kernel is at least as important as the selected kernel. The two parameters of the Radial Basis Function (RBF) kernel are determined stochastically with 5 different current and popular meta-heuristic optimization algorithms and their performances are evaluated on 7 different data sets. The performances of Particle Swarm Optimization (PSO), Salp Swarm Algorithm (SSA), Kepler Optimization Algorithm (KOA), Weighted Average Algorithm (WAA) and Hippopotamus Optimization (HO) algorithms in hyper-parameter optimization are compared. Data sets are separated into training and test with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{k}$</tex>-10 cross-validation. WAA was the model with the highest <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{R}^{\mathbf{2}}$</tex> value among the models, with values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$0.8521,0.9493$</tex>, and 0.8392 in 3 of the 7 data sets, respectively. HO was the model with the highest <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{R}^{\mathbf{2}}$</tex> value among the models, with values of 0.9374, and 0.9375 in 2 of the 7 datasets, respectively. The SVR model, which did not undergo hyper-parameter optimization with the optimization algorithm, was the model with the highest <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{R}^{\mathbf{2}}$</tex> value among the models, with values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{0. 8 6 6 9}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{0. 5 9 2 8}$</tex> in 2 of the 7 datasets. Among the algorithms performing hyperparameter optimization, KOA was found to be the fastest algorithm in terms of run time, while HO was found to be the slowest.

Boltzmann’s lectures on natural philosophy point out how the principles of mathematics are both an improvement on traditional philosophy and also serve as a necessary foundation of physics or what the English call “Natura Philosophy”, a title which he will retain for his own lectures. We start with lecture #3 and the mathematical contents of his lectures plus a few philosophical comments. Because of the length of the lectures as a whole we can only give the main points of each but organized into a coherent study. Behind his mathematics stands his support of Darwinian evolution interpreted in a partly Lamarckian way. He also supported non-Euclidean geometry. Much of Boltzmann’s analysis of mathematics is an attempt to refute Kant’s static a priori categories and his identification of space with “non-sensuous intuition”. Boltzmann’s strong attention toward discreteness in mathematics can be seen throughout the lectures. Part II of this paper will touch on the historical background of atomism and focus on the discrete way of thinking with which Boltzmann approaches problems in mathematics and beyond. Part III briefly points out how Boltzmann related mathematics and discreteness to music.

Several optimization problems from various types of applications have been efficiently resolved using available meta-heuristic algorithms such as Particle Swarm Optimization and Genetic Algorithm. Recently, many meta-heuristic optimization techniques have been extensively reported in the literature. Nevertheless, there is still room for new optimization techniques and strategies since, according to the literature, there is no meta-heuristic optimization algorithm that may be considered as the best choice to cope with all modern optimization problems. This paper introduces a novel meta-heuristic optimization algorithm named Dynamic Group-based Optimization Algorithm (DGCO). The proposed algorithm is inspired by the cooperative behavior adopted by swarm individuals to achieve their global goals. DGCO has been validated and tested against twenty-three mathematical optimization problems, and the results have been verified by a comparative study with respect to state-of-the-art optimization algorithms that are already available. The results have shown the high exploration capabilities of DGCO as well as its ability to avoid local optima. Moreover, the performance of DGCO has also been verified against five constrained engineering design problems. The results demonstrate the competitive performance and capabilities of DGCO with respect to well-known state-of-the-art meta-heuristic optimization algorithms. Finally, a sensitivity analysis is performed to study the effect of different parameters on the performance of the DGCO algorithm.

Artificial intelligence applications within the geosciences are becoming increasingly common, yet there are still many challenges involved in adapting established techniques to geoscience data sets. Applications in the realm of volcanic hazards assessment show great promise for addressing such challenges. Here, we describe a Jupyter Notebook we developed that ingests real-time Global Navigation Satellite System (GNSS) data streams from the EarthCube CHORDS (Cloud-Hosted Real-time Data Services for the geosciences) portal TZVOLCANO, applies unsupervised learning algorithms to perform automated data quality control (“noise reduction”), and explores autonomous detection of unusual volcanic activity using a neural network. The TZVOLCANO CHORDS portal streams real-time GNSS positioning data in 1[Formula: see text]s intervals from the TZVOLCANO network, which monitors the active volcano Ol Doinyo Lengai in Tanzania, through UNAVCO’s real-time GNSS data services. UNAVCO’s real-time data services provide near-real-time positions processed by the Trimble Pivot system. The positioning data (latitude, longitude and height) are imported into the Jupyter Notebook presented in this paper in user-defined time spans. The positioning data are then collected in sets by the Jupyter Notebook and processed to extract a useful calculated variable in preparation for the machine learning algorithms, of which we choose the vector magnitude for further processing. Unsupervised K-means and Gaussian Mixture machine learning algorithms are then utilized to locate and remove data points (“filter”) that are likely caused by noise and unrelated to volcanic signals. We find that both the K-means and Gaussian Mixture machine learning algorithms perform well at identifying regions of high noise within tested GNSS data sets. The filtered data are then used to train an artificial intelligence neural network that predicts volcanic deformation. Our Jupyter Notebook has promise to be used for detecting potentially hazardous volcanic activity in the form of rapid vertical or horizontal displacement of the Earth’s surface.

On the efficiency of chaos optimization algorithms for global optimization