Abstract: Aiming at the computational difficulty of partial differential equations (PDEs) in high-dimensional problems, this paper proposes a new numerical solution to the uncertainty quantification problem based on the Feynman-Kac formula and probability statistics to effectively deal with the computational difficulty of traditional methods in high-dimensional problems. The curse of dimensionality in the scene and the accuracy limitations in low-dimensional scenes. Specifically, this paper innovatively designs a sampling strategy, which improves the representativeness and statistical reliability of the sample set by increasing the number of sampling points and reducing the number of single-point simulations. At the same time, the polynomial regression model is used to replace the traditional interpolation method to avoid the overfitting problem, and the neural network is combined to deal with the complexity of high-dimensional nonlinear problems. Numerical experiments show that the improved method significantly reduces the error in both low-dimensional and high-dimensional cases, and the relative error of the neural network in high-dimensional problems is kept below 2%, which is better than the polynomial regression model. This study provides new ideas for the numerical solution and uncertainty quantification of high-dimensional partial differential equations and demonstrates its significant advantages in low-dimensional high precision and high-dimensional high efficiency.

Abstract: The feedback particle filter (FPF) is an innovative, control-oriented and resampling-free adaptation of the traditional particle filter (PF). In the FPF, individual particles are regulated via a feedback gain, and the corresponding gain function serves as the solution to the Poisson’s equation equipped with a probability-weighted Laplacian. Owing to the fact that closed-form expressions can only be computed under specific circumstances, approximate solutions are typically indispensable. This paper is centered around the development of a novel algorithm for approximating the gain function in the FPF. The fundamental concept lies in decomposing the Poisson’s equation into two equations that can be precisely solved, provided that the observation function is a polynomial. A free parameter is astutely incorporated to guarantee exact solvability. The computational complexity of the proposed decomposition method shows a linear correlation with the number of particles and the polynomial degree of the observation function. We perform comprehensive numerical comparisons between our method, the PF, and the FPF using the constant-gain approximation and the kernel-based approach. Our decomposition method outperforms the PF and the FPF with constant-gain approximation in terms of accuracy. Additionally, it has the shortest CPU time among all the compared methods with comparable performance.

Abstract: Abstract This study aims to investigate the effect of porous material properties on the tilt bearings characteristics. The tilt state of the bearing has a significant effect on the pressure distribution in the air film domain. The stiffness reaches its maximum value of 483 Nμm and the rotational stiffness is 9.02×10 5 Nm rad when the gas film thickness is 6 μm , the porous material thickness is 8 mm, the circle width is 25 mm and the permeability is 2×10 —15 m 2 . Changes in the permeability of the porous material have a greater effect on the bearing characteristics than changes in other bearing parameters, with the increased rate of the gas mass flow reaching 146%. For the characteristics of tilt porous mass aerostatic bearings, the porous permeability and the thickness are strong sensitivity parameters, and the circle width is the weak sensitivity parameter.

Abstract: Abstract We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval $$(-1,1)$$ ( - 1 , 1 ) . We assume that the given source term and reaction coefficient are analytic in $$[-1,1]$$ [ - 1 , 1 ] . The expression rate bounds in Sobolev norms in terms of the NN size are robust, i.e. uniform with respect to the singular perturbation parameter $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and $$\tanh $$ tanh - and sigmoid-activated NNs. The latter activations can represent “exponential boundary layer solution features” explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. All DNN architectures allow robust exponential solution expression in so-called ‘energy’ as well as in ‘balanced’ Sobolev norms, for analytic input data.

Abstract: Numerical methods form the core of modern computational science, providing efficient techniques for solving mathematical problems that lack closed-form analytical solutions. This paper explores the development, implementation, and application of key numerical methods including root-finding algorithms, numerical integration, finite difference methods, and iterative solutions of linear and nonlinear systems. Emphasis is placed on their significance in solving differential equations, optimization problems, and real-time simulations in engineering and physical sciences. The paper also discusses error analysis, convergence criteria, and stability aspects that govern the accuracy and reliability of numerical techniques. Through case studies in fluid mechanics, heat transfer, and structural analysis, the utility and adaptability of these methods are demonstrated. This study aims to offer researchers and engineers a comprehensive understanding of how numerical methods enhance problem-solving capabilities across diverse domains.

Abstract: ABSTRACT This article proposes two novel numerical techniques for solving two‐dimensional stochastic Itô–Volterra Fredholm integral equations. Many two‐dimensional stochastic integral equations present significant challenges for analytical solutions. Consequently, possessing an efficient method to get very precise numerical solutions for these stochastic integral equations is of paramount importance. In this article, two novel numerical techniques based on two‐dimensional wavelets have been successfully introduced to acquire the numerical solutions of the two‐dimensional stochastic Itô–Volterra Fredholm integral equations. This efficient and noteworthy method transforms the stochastic integral equation into a system of algebraic equations, which is then solved using an appropriate numerical technique. Moreover, convergence analysis and the error analysis have also been well established successfully. At the end, some test problems have been presented to illustrate the accuracy, efficiency, simplicity, and plausibility of the proposed techniques.

Abstract: Indicator functions of taking values of zero or one are essential to numerous applications in machine learning and statistics. The corresponding primal optimization model has been researched in several recent works. However, its dual problem is a more challenging topic that has not been well addressed. One possible reason is that the Fenchel conjugate of any indicator function is finite only at the origin. This work aims to explore the dual optimization for the sum of a strongly convex function and a composite term with indicator functions on positive intervals. For the first time, a dual problem is constructed by extending the classic conjugate subgradient property to the indicator function. This extension further helps us establish the equivalence between the primal and dual solutions. The dual problem turns out to be a sparse optimization with a $$\ell _0$$ ℓ 0 regularizer and a nonnegative constraint. The proximal operator of the sparse regularizer is used to identify a dual subspace to implement gradient and/or semismooth Newton iteration with low computational complexity. This gives rise to a dual Newton-type method with both global convergence and local superlinear (or quadratic) convergence rate under mild conditions. Finally, when applied to AUC maximization and sparse multi-label classification, our dual Newton method demonstrates satisfactory performance on computational speed and accuracy.

Abstract: <p>Population in statistics means the whole of the information which comes under the purview of statistical <br>investigation. A part of the population selected for study is called a sample. When the sample is drawn <br>properly, it is identical with its population almost in all respect.</p>

Abstract: Sandwich panels are important because they offer a lightweight and economical structure that can be used in various fields and has several geometric shapes. For this study, we chose three honeycomb shapes (Hexagonal, RE-entrant and Star fish) with dimensions of 57mm×120mm and a thickness of 10mm. Aluminum type (Al 2024-T3) was used for the material. The design of the three honeycomb structures was carried out with CATIA V5R20 software, while the numerical analysis of their load capacity (tensile and compression) was carried out using the ABAQUS-CAE calculation code. A numerical study was also carried out to compare the results of the samples according to the different types of honeycomb used. The results showed that the RE-shaped structure had the highest load capacity in terms of tensile and compression. The maximum stress of 422.27 MPa was observed for the honeycomb-shaped structure RE-entering in traction. In contrast, the maximum tensile stress was lower for the starfish-shaped honeycomb, reaching a value of 145.5 MPa. For compression, the hexagonal honeycomb structure recorded the highest stress value of 590.5 MPa. The lowest stress value was measured for the starfish-shaped honeycomb structure, reaching 199.5 MPa.

Abstract: In this work, we consider a class of non-linear, singular, non-self-adjoint, fourth-order, and singular BVP arising in the semiconductor industry. We investigate the theoretical and numerical results of solutions for the governing model corresponding to different boundary conditions by considering the domain as a disk of any radius T > 0. Problems are difficult to analyze the solutions due to their characteristics. Moreover, it has no exact solutions. Additionally, there are two solutions to these problems. So, it becomes difficult to apply any discrete approach to obtain both approximate solutions. We develop an iterative method based on boundary conditions that gives numerical approximations. The convergence analysis of the technique is discussed. Some numerical examples are given to show the efficiency and accuracy of the method.

Abstract: Poster presented at the Marie Curie Alumni Association (MCAA) Annual Conference and General Assembly, held on March 21–22, 2025, in Krakow, Poland.

Abstract: Detailed mathematical derivations for the components of matrix Aim in equation (5), provided as a Microsoft Word file (DOCX).

Abstract: Numerical methods form the core of modern computational science, providing efficient techniques for solving mathematical problems that lack closed-form analytical solutions. This paper explores the development, implementation, and application of key numerical methods including root-finding algorithms, numerical integration, finite difference methods, and iterative solutions of linear and nonlinear systems. Emphasis is placed on their significance in solving differential equations, optimization problems, and real-time simulations in engineering and physical sciences. The paper also discusses error analysis, convergence criteria, and stability aspects that govern the accuracy and reliability of numerical techniques. Through case studies in fluid mechanics, heat transfer, and structural analysis, the utility and adaptability of these methods are demonstrated. This study aims to offer researchers and engineers a comprehensive understanding of how numerical methods enhance problem-solving capabilities across diverse domains.

Abstract: <h3>This paper presents a detailed mathematical and numerical analysis of the stability of three-dimensional solitons (Q-balls) in a scalar field dark matter (SFDM) model with a cubic-quartic self-interaction potential V (|Φ|) = m²|Φ|²−<u> </u>^<u>2</u>^<u>α</u>_<u>3</u>|Φ|³+<u> </u>^<u>β</u>_<u>2</u>|Φ|⁴. Using a combination of variational methods and spectral theory, we prove the existence of a critical frequency ω_cat which the solitons lose their orbital stability. Numerical simulations confirm the theoretical predictions with high precision. For model parameters m = 1, α = 2.5, and β = 1.8, the critical frequency was determined to be ω_c= 0.70153 ± 0.00002, which is in full agreement with the Vakhitov-Kolokolov stability criterion. These results provide a quantitative refinement of our previously observed numerical stability threshold and establish a rigorous theoretical foundation for its interpretation.</h3>