Persistent homology (PH) is a cornerstone of topological data analysis; however, standard filtrations often overlook the anisotropic structure in empirical data. We introduce Ellipse Cloud, a preprocessing pipeline that accentuates anisotropy within PH. The method builds a Vietoris-Rips filtration from ellipse tangency times rather than pairwise Euclidean distances, thereby accommodating anisotropy while preserving compatibility with existing computational tools. The central computational challenge-identifying the first contact times of expanding ellipses-is resolved with an efficient and provably stable numerical algorithm.
This paper first reviews the hypercircle method for a posteriori error estimation of finite element (FE) solutions to Poisson's equation, which originates from the classical Prager- Synge theorem and employs basic P-1 and H(div) elements. With new numerical results, we examine a simplified post-processing method for P-1 solutions that avoids mixed FE systems. While this approach offers practical advantages, it lacks rigorous error analysis. Although the numerical results are promising, the theoretical foundation remains open, and further investigation is needed.
This study aims to derive an integrable discretization of the Bernoulli equation that achieves arbitrary higher-order accuracy while preserving the original solution structure. Using the Fade approximation, we develop discretizations for nonhomogeneous first-order linear differential equations with both constant and variable coefficients. By transforming the independent variable, we obtain the discretization of the Bernoulli equation and its general solution.
We perform some simulations of the semilinear Klein-Gordon equation with a power-law nonlinear term and propose each of the quantitative evaluation methods for the stability and convergence of numerical solutions. We also investigate each of the thresholds in the methods by varying the amplitude of the initial value and the mass, and propose appropriate values.
In this letter, we establish a recurrence relation for the characteristic polynomial of alternatingoriented tetragonal comb graphs. Using this relation, we express the polynomial in terms of Chebyshev polynomials of the second kind. This formulation allows us to explicitly determine the eigenvalues of the graphs. Furthermore, we analyze the geometric structure of the spectra in the complex plane and propose conjectures on generalized comb graphs involving higherorder polygons and various gluing configurations.
Parallel-in-time methods for time-dependent problems have risen to prominence over the past decade as massively parallel computers push core counts into the millions. Among them, the block e-circulant (BEC) preconditioned solver achieves outstanding convergence, yet the preconditioner itself dominates the solver runtime. We propose a mixed-precision strategy that uses single precision for BEC preconditioning and double precision elsewhere, thereby reducing the solver runtime especially on CPU-GPU systems. On an NVIDIA GH200 cluster, the mixed-precision BEC-GMRES solver achieves a 1.42x speedup for two-dimensional advectiondiffusion problems without significant loss of accuracy.
This paper presents a theorem of existence for an optimum point and methods to obtain component-wise verified solutions in constrained convex programming. The proposed method is based on the continuous Newton method and Kantorovich's theorem and slightly modifies Oishi and Tanabe's theorem for linear programming. Moreover, a theorem of existence for an optimum point for convex quadratic programming problems (CQPs) was also formulated. The CQP can be rewritten as linear complementarity problems (LCPs). A method to use component-wise verified solution method for LCPs is also presented. Numerical examples show that the proposed method is effective.
This letter considers the double exponential (DE) formula for computing the matrix function -A log(A), where A is a Hermitian positive semidefinite matrix and tr(A) = 1. The motivation of this work is to utilize the DE formula without selecting parameters. In order to accomplish this, we present a method for truncating the infinite interval transformed by the DE transformation based on an error analysis to achieve the required accuracy. We also discuss techniques to select the number of abscissas. In addition, we report that an appropriate choice of a parameter in the DE transformation will improve the convergence.
This paper establishes perturbation theories for the matrix Mittag-Leffler function E-alpha,E-beta (A), where alpha > 0, beta is an element of Rand A is an element of C-nxn. We present upper bounds on ||||E-alpha,E-beta (A + Delta)- E-alpha,E-beta (A)||, larger the presented bounds are compared to ||E-alpha,E-beta (A + Delta) - E-alpha,E-beta( A)||. When alpha = beta = 1, the function reduces to the matrix exponential. We compare the presented bound when alpha = beta = 1 with an existing perturbation bound for the matrix exponential.
Appropriate material constants are essential for performing finite element analysis of industrial equipment. The material constant identification problem is typically formulated as an optimization problem. This paper presents a gradient-based method to solve this problem in an efficient and accurate manner. The adjoint variable method is applied to derive gradients, and the material constants are subsequently updated using the Newton-Raphson method. Numerical examples based on iron cores, widely used in industrial equipment, demonstrate the effectiveness of the proposed method.
The block conjugate gradient (Bl-CG) method is an effective iterative solver for large sparse symmetric positive definite linear systems with multiple right-hand sides. Variable preconditioning, in which different preconditioners can be applied to each iteration, is a versatile approach for improving the convergence of iterative solvers. However, the convergence properties are obscure when variable preconditioning is used. In this study, we present a simple analysis of the error norm behavior of the Bl-CG method with variable preconditioning. Numerical experiments are conducted to demonstrate the convergence analysis.
In this letter, we provide a complete characterization of Pell numbers that can be expressed as sums or differences of two Lucas numbers, using techniques based on linear forms in logarithms and reduction methods. This characterization distinguishes polynomial families and facilitates further applications, particularly in the classification of graphs via their spectral properties.
