In this paper, we derive a new preconditioner and the corresponding preconditioned Jacobi type methods for the multi-linear systems with ℳ -tensors, which have important applications in many scientific and engineering fields. The new preconditioner is constructed by merging the preconditioner proposed in [10] with some elements of the first column of the majorization matrix associated with the coefficient tensor. Then we construct three new preconditioned Jacobi type iteration methods for solving the multi-linear systems with ℳ -tensors according to three different Jacobi type splittings of the preconditioned system tensor. Theoretically, we prove that the three proposed preconditioned Jacobi type methods are convergent. Then some comparison theorems are established, which show that the convergence factors of the three proposed preconditioned Jacobi type methods are monotonically decreasing, and the third one has the fastest convergence rate and performs better than the original Jacobi method and the ones in [10] and [13]. Also, the monotonic theorem is studied. Finally, several numerical experiments including some practical applications in the higher-order Markov chain and directed graph are given to show the correctness of theoretical results, and the effectiveness, feasibilities and superiorities of the proposed preconditioner and preconditioned Jacobi type methods.
The risk transfer of insurers to reinsurers is one of the most vital operations in insurance markets. Stop-loss contracts are the most widely adopted contract type for such operations. Specifying the main factors, such as retention and maximum (cap) levels in relation to on random loss, is essential to profit from stop-loss contracts. Thus, this study revisits the stop-loss contract modeling explored by [1], where the geometric Brownian motion (GBM) is provided and extends it by using the time-changed Brownian motion, namely, the variance gamma (VG) and normal inverse Gaussian (NIG) processes. The study aims to utilize these processes to model losses and use their advantages of skewness and kurtosis control and calibration. Further, the study illustrates simulations and compares the simulations with the GBM. The numerical experiments show that the VG and NIG processes are promising alternatives to the GBM in modeling the losses.
This paper addresses the problem of detecting an obstacle immersed in a Stokes flow from boundary measurements. The inverse problem is formulated within the framework of shape optimization, as the minimization of a least-squares misfit functional. We first establish a shape identifiability result for non-axisymmetric fluid-obstacle configurations and a perfect slip condition on the obstacle boundary and we discuss the axisymmetric case. Next, under suitable smoothness assumptions, first and second order shape derivatives of the cost functional are computed. As a consequence, by showing that the Riesz operator associated with the shape Hessian is compact, we conclude to the instability of the considered inverse problem.
We are concerned with a generalization of the comparison theorem for solutions of two SDEs that are driven by the same Brownian motion. The purpose of the present note is two folded: first we aim to show that in the framework of the noncausal stochastic calculus (cf. Ogawa,S.,“Noncausal Stochastic Calculus”, 2017 Springer) the subject can be treated in a much simpler way, second we show that this approach permits us to establish similar comparison theorems for more general case where the diffusion coefficients of SDEs are different. Moreover we show as one of the merits of our noncausal approach that we can extend our main results to a case of genuine noncausal SDEs, that is the SDE with noncausal initial data.
This paper investigates the optimal asset-liability strategy for an insurer with uncertain exit time under the mean-variance criterion. We model the risky asset price process and the liability process as general jump-diffusion processes that incorporate random parameters, and the correlation between the risky asset price process and liability process is incorporated into our research framework. The sufficient and necessary conditions for optimal asset-liability strategies are deduced by using the Malliavin calculus. Finally, a closed-form solution for the optimal strategy is also obtained in a special case.
A single-loop automated guided vehicle (AGV) system, in which multiple vehicles run on a single closed loop, is widely used in smart factories for material transport. The advantages of the system include its cost-effectiveness, ease of installation, and expansion capabilities. However, scheduling vehicles in a single-loop system is challenging because of potential interference among vehicles. For example, when a vehicle is picking up or dropping off loads at a station, the following vehicles must wait until the station becomes free. This paper presents a novel mixed-integer programming (MIP) approach to optimize vehicle scheduling while considering such interference. Building upon the core concept of modeling cycles in a looped path as consecutive linear paths, we develop a MIP formulation that incorporates both an objective function minimizing the total completion time for a set of transportation jobs and the constraints addressing interference among vehicles. We validate our approach through experiments using real-world data from a smart factory of a major pharmaceutical company. The results demonstrate that our proposed method reduces the completion time for a set of jobs by 10
Recently, Du proposed regularized randomized iterative (RRI) methods for solving structured, factorized linear systems of the form 𝐀𝐁𝐱 = 𝐛 , where A∈ℝ^m ×ℓ , B∈ℝ^ℓ× n , and b∈ℝ^m . Inspired by the efficiency of surrogate hyperplane projection techniques, we first introduce a novel surrogate hyperplane Gauss-Seidel (SHGS) method tailored for linear least squares problems. Leveraging this foundation, we further develop two new surrogate hyperplane regularized iterative algorithms designed to recover structured solutions, such as sparse or other regularized solutions to the factorized system 𝐀𝐁𝐱 = 𝐛 . We establish a comprehensive convergence theory for both methods, proving their linear convergence under appropriate conditions. Numerical experiments and real-data applications demonstrate that the proposed algorithms not only successfully identify sparse (least squares) solutions but also achieve a substantially accelerated convergence rate compared to existing RRI approaches.
In this paper, we consider a stochastic tumor-immune system for breast cancer with pulsed comprehensive therapy. By using stochastic Lyapunov analysis and the strong law of large numbers, we first prove the existence, uniqueness and the stochastic ultimate boundedness of the global positive solution. And then we derive the sufficient conditions for the extinction of disease. Finally, we will present numerical simulations to verify the theoretical results obtained in the paper.
In the liquefied petroleum gas industry, service providers must regularly visit customers to replace gas cylinders. Efficiently scheduling these visits gives rise to the Cylinder Replacement Problem (CRP), a variant of the classical capacitated vehicle routing problem that involves additional complexity due to strict, customer-specific time windows determined by remaining gas levels. Recently, neural combinatorial solvers have been actively studied as a means of quickly inferring heuristic solutions. However, existing solvers are unsuitable for directly solving CRPs because (i) they cannot adequately handle strict customer-specific delivery constraints, and (ii) obtaining optimal solutions for training is computationally expensive. To address these issues, we propose a hybrid optimization framework that combines a neural combinatorial solver with traveling salesperson problem (TSP) solvers. We observe that the delivery constraints in the bilevel approach can be reformulated as an optimal transport problem. For end-to-end training, we first train a surrogate network to estimate the optimal TSP cost and then freeze its parameters before training the cost-estimation network. The remaining route optimization consists of a collection of small traveling salesperson problems, which can be solved efficiently. Experimental results show that our framework significantly outperforms the Gurobi Optimizer in both feasibility and computational efficiency. In particular, it achieves more than a 2 × speedup and improves feasibility up to 31
This paper considers a stochastic evolution of epidemiological model designed for insurance and healthcare management in the presence of diffusion processes. The evolution of stochastic SIDRS (s-SIDRS) epidemic model involves susceptibles that face the risk of infection, infectives that face the risk of death, recovered that face the risk of re-infection and deceased that face the risk of loss of asset, as a result of the disease as stipulated in the insurance contract. It then lead to a system of non-linear stochastic differential equations. Consequently, the basic reproduction number, global asymptotic stability and local asymptotic stability of the model are obtained. This paper aims at determining a Discounted Hospitalization Insurance Liability (DHIL) to a Total Discounted Insurance Liability (TDIL) ratio, optimal insurance reserve, optimal premium rate and optimal claim rate for policy holders (PHs) over time. In determining the dynamics of the insurance liabilities, we derive a DHIL-to-TDIL ratio and other fundamental ratios. It is found that DHIL-to-TDIL ratio will help the insurer and the insured to determine insurance pricing for hospitalization, death benefit and other liabilities. From the s-SIDRS model, we constructed the insurance reserve dynamics and solve using dynamic programming approach. As a result, the HJB equation for our model is obtained. Furthermore, the optimal insurance reserve, optimal premiums from susceptible PHs and a fraction of recovered PHs who chooses to continue with insurance policy after recovery are determined. Also determined are the optimal claims by hospitalized and deceased PHs. In this paper, five different utility functions are considered in the determination of the optimal insurance group products. The utility functions consider include quadratic, logarithm, CRRA, exponential and HARA utility functions. Some numerical results are also presented in this paper.
The block Korkine–Zolotarev (BKZ) algorithm is a strong reduction algorithm for solving lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). For a large blocksize β , the BKZ algorithm requires many calls to an exact-SVP algorithm over a local projected block lattice of rank β , and thus in practice it is customary to terminate the reduction process prematurely. In this paper, we propose a new BKZ-type algorithm with provable termination. We name it “PotBKZ" since we use the potential of a lattice basis. Specifically, we develop “PotENUM", an enumeration algorithm to find lattice vectors whose insertion can reduce the potential. In PotBKZ, we call PotENUM instead of an exact-SVP algorithm to reduce the potential monotonically. We prove that PotBKZ terminates in a polynomial number of calls to PotENUM. Furthermore, we develop a self-dual variant of PotBKZ to reduce the potential more effectively with provable termination.
Conjugate Gradient (CG) methods are widely recognized for their effectiveness in solving large-scale nonlinear systems of equations, primarily due to their reliance on efficient vector operations. However, the global convergence analysis of CG methods remains a challenging issue. Motivated by the broad range of applications involving symmetric nonlinear equations, this study develops two optimal strategies for the modified Riavie-Mamat-Ismail-Leong (RMIL) CG method. The first strategy is constructed by minimizing an appropriate measure function, while the second combines the modified RMIL direction with the classical quasi-Newton direction. To further enhance robustness, the resulting CG parameters are incorporated with the Li and Fukushima approximate gradient, leading to the formulation of a new CG-type algorithm for large-scale systems of symmetric nonlinear equations. The global convergence of the proposed algorithms is established under standard assumptions. Numerical experiments on benchmark problems demonstrate that the proposed methods exhibit superior efficiency and robustness compared to several existing approaches for symmetric nonlinear equations.
In recent years, Spectral Conjugate Gradient (SCG) algorithms have gained increasing attention for image restoration problem. This study proposed a three efficient SCG method for solving convex-constrained nonlinear equations with image recovery. The proposed algorithms ensure bounded search directions that exhibit sufficient descent. Furthermore, the algorithm’s global and Q-linear convergence is established under a reasonable assumptions. To assess its effectiveness, we conduct numerical tests, comparing the proposed algorithms with existing methods. Finally, we apply the proposed method to image restoration problems.
In this work, we investigate three mathematical models for the dynamics of American Cutaneous Leishmaniasis (ACL) transmission that are based on ordinary differential equations. The subpopulations of each model consists of infected incidental hosts, infected reservoir hosts, and infected vectors. The number of infected individual hosts is thought to influence the incidental host’s infection rate in the first model and the vectors’ death rate in the second whereas the third model integrates both of the impacts. There are two equilibrium points in each of the three models: the endemic equilibrium point and the disease-free equilibrium point. We noticed that the basic reproduction number is influenced by the overall populations of reservoir hosts and vectors, as well as the rates of infection and recovery of the reservoir hosts. For each model, the local and global stability of the two equilibrium points were investigated, and numerical simulations were carried out to support the analytical findings.
This study aims to offer mathematical tools constructed using numerical verification methods that are easy for people engaged in the simulation of phenomena to use. We propose a common formula for several ODE systems with conserved quantities that provides bounds for solution trajectories. Computing the formula using verified numerics is simpler than computing the solution trajectories; therefore, the formula can be a useful tool for researchers unfamiliar with numerical verification methods.
The generalized product-type method based on the biconjugate gradient (hereafter referred to as the GPBiCG method) has been recognized as an efficient Krylov subspace method for solving nonsymmetric linear systems. In this paper, tensor form of the GPBiCG method and its preconditioned variant are presented for solving the Stein tensor equations. The effectiveness of the proposed methods is verified by numerical experiments.
The matrix Riemann-Hilbert problem of the Schrödinger-Hirota equation in nonlinear optics is investigated. The derivation of a Riemann-Hilbert problem from the Lax pair associated with the 2 × 2 matrix spectral problem, which is linked to the Schrödinger-Hirota equation, is achieved through the application of spectral analysis to the Lax pair. Furthermore, by solving a special Riemann-Hilbert problem with vanishing scattering coefficients, an explicit representation of the N-soliton solution to the Schrödinger-Hirota equation is derived based on the asymptotic characteristics of the Riemann-Hilbert problem. It is noteworthy that the real and imaginary parts of the N-soliton solution exhibit two perfectly complementary N-breather solutions.
This paper presents a characterization of the Method of Fundamental Solutions for solving 2D Dirichlet problems from a fundamental perspective. By contrasting the inherent characteristics of the conventional and invariant schemes, we show that in particular cases these two schemes are partially equivalent with regard to their linear systems, where a decomposition by using orthogonalization of the problem into two systems highlights the invariant subspaces of the coefficient matrix. Moreover, we explore their inherent reciprocity properties with a particular focus on their values of sources and boundary values. A key contribution of this work is the proofs of reciprocity preservation in both the conventional and invariant schemes, and providing examples and remarks on them. We demonstrate that this property holds between inner and outer problems, regardless of the domain’s shape complexity. As an application of the reciprocity property, we demonstrate that an approximate solution can be obtained by solving underdetermined systems, even in cases where some collocation points are missing. The numerical experiments suggest that, the reciprocity-based method may even surpass the desirable effects of regularization—namely, suppressing error while keeping the residual small.
