The class of SDD _1 matrices is an important subclass of H-matrices. In this paper, we develop new two-sided determinant bounds for SDD _1 matrices by analyzing the structure of their Schur complements. Compared with existing bounds, our results are shown to be significantly tighter in most cases while remaining computable with similar effort.
Here, we consider the regular N-dimensional fractional Sturm-Liouville problems (N-D FSLPs) of order ν =(ν _1,ν _2,⋯ ,ν _N)∈ (0,1] . The FSLP is defined using a fractional version of the gradient operator in terms of left and right Caputo fractional derivatives (CFDs). We establish the essential properties of the eigenvalues (EVs) and eigenfunctions (EFs) of the FSLP using the principle of fractional variational calculus. To validate our theoretical study, we take the fractional Laplace equation as an example and demonstrate that the EVs exist for fractional order ν∈ (0,1] .
This paper re-explores the coupling between shape-from-shading (SfS) and multi-view stereo (MVS), i.e., reconstructing the geometry of a shaded surface from a set of views. We first provide intuitive evidence for the uniqueness of a solution based on epipolar geometry and Lambertian assumption arguments. Then, we explore several alternatives for approximating this solution, from the semi-Lagrangian approximation of the underlying Hamilton-Jacobi equation to the local optimisation of a neural implicit surface. We qualitatively and quantitatively evaluate the latter approach, which we denote SfS-NeuS, on several synthetic and real-world examples and show that it outperforms comparable state-of-the-art 3D reconstruction methods.
This study introduces an enhanced implicit exponentially fitted Runge-Kutta (EFRK) method for the efficient solution of first-order initial value problems (IVPs) with exponential characteristics. The proposed approach minimizes the propagation of internal-stage errors to the final-stage error by deriving new coefficients without relying on frequency-evaluation algorithms. Unlike previous works that mainly addressed specific schemes, this research focuses on the general formulation of implicit EFRK methods. To ensure the conclusion is general and independent of the reference set, several cases of internal and external reference sets are considered. Numerical experiments are conducted to compare the proposed methods with existing ones through linear, nonlinear, and stiff test problems. Additionally, the stability analysis of both the proposed and the standard methods is conducted. The results reveal a larger stability region for the proposed methods, a more accurate solution when applied to stiff problems, and comparable errors for non-stiff problems compared to the standard methods.
In this research work, we investigate a residual-based approach for a posteriori error estimation within the framework of the weak Galerkin finite element method (WGFEM), focusing on singularly perturbed reaction-diffusion problems in the energy norm. Capturing the layer behaviour in the reaction-dominated regime is more complicated compared to the convection-dominated regime, a weighted robust estimator is employed for accurately capturing the layer behaviour for this model problem, which has a reaction-dominated regime. The global reliability and efficiency are verified by deriving both the upper and lower bounds of the proposed estimator. An adaptive WGFEM is developed using this estimator, employing a mesh refinement strategy to effectively resolve the layers for the complicated reaction-dominated regime. Theoretically, the proposed estimator is both reliable and efficient, while numerical results on the unit square and the unit circular domains confirm its effectiveness.
