Journal of Computational Mathematics and Data Science最新文献

The performance of exponential-based numerical integrators for the time propagation of the equations associated with the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method for the approximation of the multi-particle Schrödinger equation in one space dimension is assessed. Among the most popular integrators such as Runge–Kutta methods, time-splitting, exponential integrators and Lawson methods, exponential Lawson multistep methods with one predictor–corrector step provide the best stability and accuracy at the least effort. This assessment is based on the observation that the evaluation of the nonlocal terms associated with the potential is the computationally most demanding part of such a calculation in our setting. In addition, the predictor step provides an estimator for the local time-stepping error, thus allowing for adaptive time-stepping which reflects the smoothness of the solution and enables to reliably control the accuracy of a computation in a robust way, without the need to guess an optimal stepsize a priori. One-dimensional model examples are studied to compare different time integrators and demonstrate the successful application of our adaptive methods.

In this paper we discuss the potential of using artificial neural networks as smooth priors in classical methods for inverse problems for PDEs. Exploring that neural networks are global and smooth function approximators, the idea is that neural networks could act as attractive priors for the coefficients to be estimated from noisy data. We illustrate the capabilities of neural networks in the context of the Poisson equation and we show that the neural network approach show robustness with respect to noisy, incomplete data and with respect to mesh and geometry.

This work involves the study of elliptic type systems of equations in three independent variables. The Lie point symmetries of the systems are obtained; some of the symmetries of a particular system are used to perform reduction to an invariant system with one less independent variable. The symmetries of the reduced system are also obtained and used for further reduction to a system of ordinary differential equations (ODEs). The invariant solutions of the system of ODEs are constructed.

Detection and examination of pairwise dependence patterns between continuous variables is among the central tasks in the fields of business and economic statistics. To perform this analysis, practitioners frequently resort to Pearson’s (1895) product–moment correlation coefficient and the related significance tests. However, the use of such tests in isolation involves the risk of missing the nonlinear and particularly non-monotonic associations between the variables. This problem is also relevant in the cases where the dependence prevails between higher-order moments, e.g., variances, rather than means. We present a simple, computationally inexpensive heuristic by which this problem can be addressed and demonstrate its usefulness in a small number of example cases.

In many systems, the state of each of their components can itself be a source of risk affecting the other components, and it is not easy to aggregate these individual values together with the interconnecting structural elements of the network. There are simulation models in the literature that establish propagation curves for the population as a whole, especially in the epidemiological case, but these models do not provide a clear analytical expression of the risk borne by each of the network nodes. Moreover, classical models, such as the generalized cascade model, are not necessarily convergent. Neither can the individual values of each node be aggregated on the same scale as they were measured. This paper proposes a mathematical model that makes it possible to analyze the propagation of risk in the face of a given adverse event that may reach all the elements of a network and precisely calculate the risk borne by each node according to its own vulnerability and the relationships with the other nodes, which may be more or less vulnerable and constitute additional sources of risk. It is shown that the new model ensures convergence and that the aggregated results can be interpreted in terms of the risk measurement scale previously given for each node. In addition, the global import–export network is used to illustrate how political or economic instability in one state can generate crises in other states.

The Muth distribution and its derivation have been used to construct numerous statistical models in recent years, with applications in a variety of fields. In this paper, we use the inverse scheme to introduce the inverse power Muth distribution. It thus constitutes a new three-parameter heavy-tailed lifetime distribution belonging to the family of inverse distributions, which does not appear to have received adequate attention in the literature. We naturally call it inverse power Muth distribution. Two complementary parts compose the article. The first part aims to determine the main statistical properties of the inverse power Muth distribution, such as the shape behavior of the probability density and hazard rate functions, the expression of the quantile function and the related quantities, and some moment measures. The second part is devoted to its practical aspects, with a focus on its modeling capabilities. We examine the estimation of the model parameters via several well-established methods, including classical and Bayesian estimation methods. Then, we illustrate the flexibility and potential usefulness of the inverse power Muth model by means of a simulation study and two real datasets. A fair investigation reveals that it can outperform existing and comparable three-parameter models also based on the inverse scheme.