Mathematics and Computers in Simulation最新文献

Quasi-Monte Carlo (QMC) methods have been gaining popularity in computational finance as they are competitive alternatives to Monte Carlo methods that can accelerate numerical accuracy. This paper develops a new approach for reducing the effective dimension combined with a randomized QMC method. A distinctive feature of the proposed approach is its sample-based transformation that enables us to choose a flexible manipulation via regression. In the proposed approach, the first step is to perform a regression using the samples to estimate the parameters of the regression model. An optimal transformation is proposed based on the regression result to minimize the effective dimension. An advantage of this approach is that adopting a statistical approach allows greater flexibility in selecting the regression model. In addition to a linear model, this paper proposes a dimension reduction method based on a linear-quadratic model for regression. In numerical experiments, we focus on pricing different types of exotic options to test the effectiveness of the proposed approach. The numerical results show that different regression models are chosen depending on the underlying risk process and the type of derivative securities. In particular, we show several examples where the proposed method works while existing dimension reductions are ineffective.

Within the context of a two-dimensional framework encompassing interacting species, an examination is conducted in this study on the double Allee effect and prey refuge, considering both species in the interaction. The stability of the feasible equilibrium of the system and diverse bifurcation patterns including codimension-one and codimension-two bifurcations are scrutinized through theoretical and numerical investigations, which reveals the complex dynamics induced by saturated functional response and double Allee effects. Additionally, one-parameter bifurcation diagrams and two-parameter bifurcation diagrams are constructed to intricately evaluate the system’s dynamics indicative of the presence of multiple attractors like bi-stability and tri-stability. Lastly, the sensitivity analysis is performed to delve into the effect of system parameters on species density, which indicates that the parameter $\eta$ proportional to the conversion rate is the most sensitive parameter. A brief discussion further reveals that the model without double Allee effect reduces dynamic complexity.

In this article, we design and analyze a hybrid high-order method for a semilinear Sobolev model on polygonal meshes. The method offers distinct advantages over traditional approaches, demonstrating its capability to achieve higher-order accuracy while reducing the number of unknown coefficients. We derive error estimates for the semi-discrete formulation of the method. Subsequently, these convergence rates are employed in full discretization with the Crank–Nicolson scheme. The method is demonstrated to converge optimally with orders of $O\left({\tau }^2+h^{k+1}\right)$ in the energy-type norm and $O\left({\tau }^2+h^{k+2}\right)$ in the $L^2$ norm. The reported method is supported by a series of computational tests encompassing linear, semilinear and Allen–Cahn models.

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Input-to-state practical stability (ISpS) of a kind of nonlinear systems suffering from unknown exogenous disturbances is explored in this article, where a disturbance observer is established to estimate the information of the exogenous disturbances. To achieve ISpS of the system, the impulsive controller as well as state-feedback controller are both considered to regulate the discrete and continuous dynamics of the system, respectively. Especially, a novel disturbance observer-based event-triggered mechanism is devised to decide the release of impulsive control signal. Furthermore, several adequate conditions are given for excluding the occurrence of Zeno phenomenon. To confirm the feasibility of the proposed results, two numerical instances and their corresponding simulation results are presented.

Surveys are going through massive changes, and the most important innovation is the use of non-probability samples. Non-probability samples are increasingly used for their low research costs and the speed of the attainment of results, but these surveys are expected to have strong selection bias caused by several mechanisms that can eventually lead to unreliable estimates of the population parameters of interest. Thus, the classical methods of statistical inference do not apply because the probabilities of inclusion in the sample for individual members of the population are not known. Therefore, in the last few decades, new possibilities of inference from non-probability sources have appeared.

Statistical theory offers different methods for addressing selection bias based on the availability of auxiliary information about other variables related to the main variable, which must have been measured in the non-probability sample. Two important approaches are inverse probability weighting and mass imputation. Other methods can be regarded as combinations of these two approaches.

This study proposes a new estimation technique for non-probability samples. We call this technique model-assisted kernel weighting, which is combined with some machine learning techniques. The proposed technique is evaluated in a simulation study using data from a population and drawing samples using designs with varying levels of complexity for, a study on the relative bias and mean squared error in this estimator under certain conditions. After analyzing the results, we see that the proposed estimator has the smallest value of both the relative bias and the mean squared error when considering different sample sizes, and in general, the kernel weighting methods reduced more bias compared to based on inverse weighting. We also studied the behavior of the estimators using different techniques such us generalized linear regression versus machine learning algorithms, but we have not been able to find a method that is the best in all cases. Finally, we study the influence of the density function used, triangular or standard normal functions, and conclude that they work similarly.

A case study involving a non-probability sample that took place during the COVID-19 lockdown was conducted to verify the real performance of the proposed methodology, obtain a better estimate, and control the value of the variance.

Deep learning methods have been developed to solve interface problems, benefiting from meshless features and the ability to approximate complex interfaces. However, existing deep neural network (DNN) methods for usual partial differential equations encounter accuracy limitations where after reaching a certain error level, further increases in network width, depth, and iteration steps do not enhance accuracy. This limitation becomes more notable in interface problems where the solution and its gradients may exhibit significant jumps across the interface. To improve accuracy, we propose a piecewise extreme learning machine (PELM) for addressing interface problems. An ELM is a type of shallow neural network where weight/bias coefficients in activation functions are randomly sampled and then fixed instead of being updated during the training process. Considering the solution jumps across the interface, we use a PELM scheme — setting one ELM function for each side of the interface. The two ELM functions are coupled using the interface conditions. Our numerical experiments demonstrate that the proposed PELM for the interface problem significantly improves the accuracy compared to conventional DNN solvers. The advantage of new method is shown for addressing interface problems that feature complex interface curves.

It is well-known that decision-making problems from stochastic control can be formulated by means of a forward–backward stochastic differential equation (FBSDE). Recently, the authors of Ji et al. (2022) proposed an efficient deep learning algorithm based on the stochastic maximum principle (SMP). In this paper, we provide a convergence result for this deep SMP-BSDE algorithm and compare its performance with other existing methods. In particular, by adopting a strategy as in Han and Long (2020), we derive a-posteriori estimate, and show that the total approximation error can be bounded by the value of the loss functional and the discretization error. We present numerical examples for high-dimensional stochastic control problems, both in the cases of drift- and diffusion control, which showcase superior performance compared to existing algorithms.

Metaheuristic approaches commonly disregard the individual strategies of each agent within a population, focusing primarily on the collective best solution discovered so far. While this methodology can yield promising results, it also has several significant drawbacks, such as premature convergence. This study introduces a new metaheuristic approach that emphasizes the balance between individual and social learning in agents. In this approach, each agent employs two strategies: an individual search technique performed by the agent and a social or collective strategy involving the best-known solution. The search strategy is considered a learning problem, and agents must adjust the use of both individual and social strategies accordingly. The equilibrium of this adjustment is determined by a counter randomly set for each agent, which determines the frequency of use invested in each strategy. This mechanism promotes diverse search patterns and fosters a dynamic and adaptive process, potentially improving problem-solving efficiency in intricate spaces. The proposed method was assessed by comparing it with several well-established metaheuristic algorithms using 21 test functions. The results demonstrate that the new method surpasses popular metaheuristic algorithms by offering superior solutions and attaining quicker convergence.

In this paper, our primary objective is to discuss the weak convergence of the split-step backward Euler (SSBE) method, renowned for its exceptional stability when used to solve a class of stochastic delay integro-differential equations (SDIDEs) characterized by global Lipschitz coefficients. Traditional weak convergence analysis techniques are not directly applicable to SDIDEs due to the absence of a Kolmogorov equation. To bridge this gap, we employ modified equations to establish an equivalence between the SSBE method used for solving the original SDIDEs and the Euler–Maruyama method applied to modified equations. By demonstrating first-order strong convergence between the solutions of SDIDEs and the modified equations, we establish the first-order weak convergence of the SSBE method for SDIDEs. Finally, we present numerical simulations to validate our theoretical findings.