Journal of Computational Mathematics and Data Science最新文献

In many situations, uncertainty about the mechanical properties of surrounding soils due to the lack of data and spatial variations requires tools that involve the study of parameters by means of random variables or random functions. Usually only a few measurements of parameters, such as permeability or porosity, are available to build a model, and some measurements of the geomechanical behavior, such as displacements, stresses, and strains are needed to check/calibrate the model. In order to introduce this type of modeling in geomechanical analysis, taking into account the random nature of soil parameters, Bayesian inference techniques are implemented in highly heterogeneous porous media. Within the framework of a coupling algorithm, these are incorporated into the inverse poroelasticity problem, with porosity, permeability and Young modulus treated as stationary random fields obtained by the moving average (MA) method. To this end, the Metropolis–Hasting (MH) algorithm was chosen to seek the geomechanical parameters that yield the lowest misfit. Numerical simulations related to injection problems and fluid withdrawal in a $3D$ domain are performed to compare the performance of this methodology. We conclude with some remarks about numerical experiments.

The discrete Laplace transform (DLT) with $M$ inputs and $N$ outputs has a nominal computational cost of $O\left(MN\right)$. There are approximate DLT algorithms with $O(M+N)$ cost such that the output errors divided by the sum of the inputs are less than a fixed tolerance $\eta$. However, certain important applications of DLTs require a more stringent accuracy criterion, where the output errors divided by the true output values are less than $\eta$. We present a fast DLT algorithm combining two strategies. The bottom-up strategy exploits the Taylor expansion of the Laplace transform kernel. The top-down strategy chooses groups of terms in the DLT to include or neglect, based on the whole summand, and not just on the Laplace transform kernel. The overall effort is $O(M+N)$ when the source and target points are very dense or very sparse, and appears to be $O\left({(M+N)}^{1.5}\right)$ in the intermediate regime. Our algorithm achieves the same accuracy as brute-force evaluation, and is typically 10–100 times faster.

Information theory uses the Kullback–Leibler divergence to compare distributions. In this paper, we apply it to bayesian posterior distributions and we show how it can be used to train a machine learning algorithm as well. The data sample used in this study is an OCTOTelematics set of driving behaviour data.

The Monge–Ampère equation is a full y nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge–Ampère equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.

In this study, we introduce a novel weight function-based eighth order derivative-free method for locating repeated roots of nonlinear equations. It is a three-step Steffensen-type scheme with first order divided differences in place of the first order derivatives. It is noteworthy that so far only few eighth order derivative free multiple root finding scheme exist in literature. Different nonlinear standard and applications based nonlinear functions are used to demonstrate the applicability of the suggested approach and to confirm its strong convergence tendency. Drawing basins of attraction on the graphical regions demonstrates how the offered family of approaches converge.

The problem of interpretability for binary image classification is considered through the lens of kernel two-sample tests and generative modeling. A feature extraction framework coined Deep Interpretable Features is developed, which is used in combination with IntroVAE, a generative model capable of high-resolution image synthesis. Experimental results on a variety of datasets, including COVID-19 chest x-rays demonstrate the benefits of combining deep generative models with the ideas from kernel-based hypothesis testing in moving towards more robust interpretable deep generative models.

Many machine learning methods used for the treatment of sequential data often rely on the construction of vector representations of unitary entities (e.g. words in natural language processing, or $k$-mers in bioinformatics). Traditionally, these representations are constructed with optimization formulations arising from co-occurrence based models. In this work, we propose a new method to embed these entities based on the Distance Geometry Problem: find object positions based on a subset of their pairwise distances or inner products. Considering the empirical Pointwise Mutual Information as a surrogate for the inner product, we discuss two Distance Geometry based algorithms to obtain word vector representations. The main advantage of such algorithms is their significantly lower computational complexity in comparison with state-of-the-art word embedding methods, which allows us to obtain word vector representations much faster. Furthermore, numerical experiments indicate that our word vectors behave quite well on text classification tasks in natural language processing as well as regression tasks in bioinformatics.

In this paper, a two prey–one predator model with different types of growth rate and mixed functional responses is proposed and analysed. Moreover, we considered anti-predation behaviour and constant harvesting effort in both the prey populations. The positivity and boundedness of the system are studied. The criteria for the extinction of the predator–prey populations are discussed. Analytically, we have studied the criteria for existence and stability of different equilibrium points. In addition, we have derived sufficient conditions for local bifurcations such as transcritical and Hopf bifurcation. We discussed that the effect of fear not only reduces prey populations, but also decreases the rate of growth of the predator population. Computer simulations are performed to validate our analytical results. The biological implications of analytical and numerical results are critically discussed.

Analysing the cubic sectors of a real polynomial of degree $n$, a modification of Newton’s Rule of signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the roots of a polynomial is also proposed. Relationship between the quadratic elements of the polynomial is established through its roots and those of its derivatives. Some aspects of polynomial discriminants are also discussed — the relationship between the discriminants of real polynomials, the discriminants of their derivatives, and the quadratic elements, following a “discriminant of the discriminant” approach.

Feature selection (FS) is a common preprocessing step of machine learning that selects informative subset of features which fuels a model to perform better during prediction or classification. It helps in the design of an intelligent and expert system used in computer vision, image processing, gene expression data analysis, intrusion detection and natural language processing. In this paper, we introduce an effective filter method called Joint Mutual Information with Class relevance (JoMIC) using multivariate Joint Mutual Information (JMI) and Mutual Information (MI). Our method considers both JMI and MI of a non selected feature with selected ones w.r.t a given class to select a feature that is highly relevant to the class but non redundant to other selected features. We compare our method with seven other filter-based methods using the machine learning classifiers viz., Logistic Regression, Support Vector Machine, K-nearest Neighbor (KNN), Decision Tree, Random Forest, Naïve Bayes, and Stochastic Gradient Descent on various datasets. Experimental results reveal that our method yields better performance in terms of accuracy, Matthew’s Correlation Coefficient (MCC) and F1-score over 16 benchmark datasets, as compared to other competent methods. The superiority of our proposed method is that it uses an effective objective function that combines both JMI and MI to choose the relevant and non redundant features.