Stability Analysis and Performance Evaluation of Additive Mixed-Precision Runge-Kutta Methods

Additive Runge-Kutta methods designed for preserving highly accurate solutions in mixed-precision computation were previously proposed and analyzed.These specially designed methods use reduced precision for the implicit computations and full precision for the explicit computations.In this work,we analyze the stability properties of these methods and their sensitivity to the low-precision rounding errors,and demonstrate their performance in terms of accuracy and efficiency.We develop codes in FORTRAN and Julia to solve nonlinear systems of ODEs and PDEs using the mixed-precision additive Runge-Kutta(MP-ARK)methods.The convergence,accuracy,and runtime of these methods are explored.We show that for a given level of accuracy,suitably chosen MP-ARK methods may provide significant reductions in runtime.

Thermal transport in composition graded silicene/germanene heterostructures

Through equilibrium and non-equilibrium molecular dynamics simulations,we have demonstrated the inhibitory effect of composition graded interface on thermal transport behavior in lateral heterostructures.Specifically,we investigated the influence of composition gradient length and heterogeneous particles at the silicene/germanene(SIL/GER)heterostructure interface on heat conduction.Our results indicate that composition graded interface at the interface diminishes the thermal conductivity of the heterostructure,with a further reduction observed as the length increases,while the effect of the heterogeneous particles can be considered negligible.To unveil the influence of composition graded interface on thermal transport,we conducted phonon analysis and identified the presence of phonon localization within the interface composition graded region.Through these analyses,we have determined that the decrease in thermal conductivity is correlated with phonon localization within the heterostructure,where a stronger degree of phonon localization signifies poorer thermal conductivity in the material.Our research findings not only contribute to understanding the impact of interface gradient-induced phonon localization on thermal transport but also offer insights into the modulation of thermal conductivity in heterostructures.

Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algorithm.We show numerical simulations of these algorithms on various benchmarks.

Dimension by Dimension Finite Volume HWENO Method for Hyperbolic Conservation Laws

In this paper,we propose a finite volume Hermite weighted essentially non-oscillatory(HWENO)method based on the dimension by dimension framework to solve hyperbolic conservation laws.It can maintain the high accuracy in the smooth region and obtain the high resolution solution when the discontinuity appears,and it is compact which will be good for giving the numerical boundary conditions.Furthermore,it avoids complicated least square procedure when we implement the genuine two dimensional(2D)finite volume HWENO reconstruction,and it can be regarded as a generalization of the one dimensional(1D)HWENO method.Extensive numerical tests are performed to verify the high resolution and high accuracy of the scheme.

RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L^(2) projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steady-state simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

Investigation of an electrode-driven hydrogen plasma method for in situ cleaning of tin-based contamination

To prolong the service life of optics,the feasibility of in situ cleaning of the multilayer mirror(MLM)of tin and its oxidized contamination was investigated using hydrogen plasma at different power levels.Granular tin-based contamination consisting of micro-and macroparticles was deposited on silicon via physical vapor deposition(PVD).The electrodedriven hydrogen plasma at different power levels was systematically diagnosed using a Langmuir probe and a retarding field ion energy analyzer(RFEA).Moreover,the magnitude of the self-biasing voltage was measured at different power levels,and the peak ion energy was corrected for the difference between the RFEA measurements and the self-biasing voltage(E_(RFEA)-eV_(self)).XPS analysis of O 1s and Sn 3d peaks demonstrated the chemical reduction process after 1 W cleaning.Analysis of surface and cross-section morphology revealed that holes emerged on the upper part of the macroparticles while its bottom remained smooth.Hills and folds appeared on the upper part of the microparticles,confirming the top-down cleaning mode with hydrogen plasma.This study provides an in situ electrode-driven hydrogen plasma etching process for tin-based contamination and will provide meaningful guidance for understanding the chemical mechanism of reduction and etching.

The Origin of the Flat Rotation Curves in Spiral Galaxies: The Hidden Roles of Glitching SMDEOs and Emission of Gravitational Waves

Supermassive DEOs (SMDEOs) are cosmologically evolved objects made of irreducible incompressible supranuclear dense superfluids: The state we consider to govern the matter inside the cores of massive neutron stars. These cores are practically trapped in false vacua, rendering their detection by outside observers impossible. Based on massive parallel computations and theoretical investigations, we show that SMDEOs at the centres of spiral galaxies that are surrounded by massive rotating torii of normal matter may serve as powerful sources for gravitational waves carrying away roughly 1042 erg/s. Due to the extensive cooling by GWs, the SMDEO-Torus systems undergo glitching, through which both rotational and gravitational energies are abruptly ejected into the ambient media, during which the topologies of the embedding spacetimes change from curved into flatter ones, thereby triggering a burst gravitational energy of order 1059 erg. Also, the effects of glitches found to alter the force balance of objects in the Lagrangian-L1 region between the central SMDEO-Torus system and the bulge, enforcing the enclosed objects to develop violent motions, that may explain the origin of the rotational curve irregularities observed in the innermost part of spiral galaxies. Our study shows that the generated GWs at the centres of galaxies, which traverse billions of objects during their outward propagations throughout the entire galaxy, lose energy due to repeatedly squeezing and stretching the objects. Here, we find that these interactions may serve as damping processes that give rise to the formation of collective forces f∝m(r)/r, that point outward, endowing the objects with the observed flat rotation curves. Our approach predicts a correlation between the baryonic mass and the rotation velocities in galaxies, which is in line with the Tully-Fisher relation. The here-presented self-consistent approach explains nicely the observed rotation curves without invoking dark matter or modifying Newtonian gravitation in the low-field approximation.

Fetal MRI Artifacts: Semi-Supervised Generative Adversarial Neural Network for Motion Artifacts Reducing in Fetal Magnetic Resonance Images

This study addresses challenges in fetal magnetic resonance imaging (MRI) related to motion artifacts, maternal respiration, and hardware limitations. To enhance MRI quality, we employ deep learning techniques, specifically utilizing Cycle GAN. Synthetic pairs of images, simulating artifacts in fetal MRI, are generated to train the model. Our primary contribution is the use of Cycle GAN for fetal MRI restoration, augmented by artificially corrupted data. We compare three approaches (supervised Cycle GAN, Pix2Pix, and Mobile Unet) for artifact removal. Experimental results demonstrate that the proposed supervised Cycle GAN effectively removes artifacts while preserving image details, as validated through Structural Similarity Index Measure (SSIM) and normalized Mean Absolute Error (MAE). The method proves comparable to alternatives but avoids the generation of spurious regions, which is crucial for medical accuracy.

微尺度气体润滑的非平衡流效应分析

微尺度气体润滑中,引起微尺度气体润滑压力降低的原因在于润滑流的非平衡流效应。微尺度气体润滑的非平衡流效应不仅会引起壁面气流速度滑移,而且会引起润滑气流热通量的变化。本文首先给出了基于Boltzmann矩方程的微尺度气体润滑模型(矩方程润滑模型);其次,与FK润滑模型和直接模拟Monte Carlo方法的分析结果比较,验证基于矩方程润滑模型的有效性;再次,基于矩方程润滑模型,分析不同轴承数情形下微尺度气体润滑的非平衡流效应。研究表明,随着轴承数的增大,微尺度气体润滑的非平衡流效应增强,表现为壁面速度滑移及热通量均增大。与壁面速度滑移的变化相反,随着轴承数的增大,非平衡流效应引起气膜压力及承载力的变化较小。这一反常现象的原因在于气体的可压缩性。

角质层渗透性质的三维数值模拟

目的从数值模拟的角度研究药物渗透皮肤角质层的过程,为实际的药物工作者提供理论上的指导和帮助。方法用有限元方法离散偏微分方程的差分格式,将角质层的角蛋白细胞近似为物理学上经典的由6个四边形和8个六边形围成的三维十四面体结构,同时将角蛋白细胞间的脂质体视为十四面体结构的一部分,离散其相应的单元网格。结果得到了药物渗透皮肤角质层的三维数值模拟,并给出了模拟过程中浓度和速度变化的可视化效果示意图。结论本文所采用的几何模型与实际的生理结构模型非常接近,数值模拟结果证明这一方法对实际工作有参考意义。

角质层渗透性质的有限元数值模拟计算

角质层是皮肤屏障作用的最主要部分,它决定可外界物质对皮肤的渗透情况。本研究在假设角质层细胞为一种三维的十四面体(物理学经典的Tetrakaidecahedron体)下,进行对角质层渗透性质的数值模拟工作。为此,首先完成了对角质层空间结构的网格拆分,拆分过程分两步进行:首先对角蛋白细胞的网格拆分;然后对角蛋白细胞周围的网状脂质体的网格拆分。在数值模拟过程中,则用有限元法得到方程的离散格式,用多重网格算法降低高频误差,提高计算精度。最后,给出了数值模拟结果的可视化效果图。

一类非稳态热耦合方程组解的存在性

研究了一类椭圆抛物耦合方程组解的存在性。在假设耦合系数σ(s)、κ(s)∈W1,∞(R),b∈[L∞(Ω)]2,c∈L∞(Ω)且满足c-1/2▽.b≥-(κ1-α)λ1条件下,λ1为-Δ的第一特征值,α>0。运用Faedo-Galerkin方法构造近似解,首先得出近似解在局部时间内存在,然后得出一些近似解的先验估计证明解可以延拓到区间[0,T],利用紧性定理得出解关于时间t和n无关,最后对逼近方程取极限,得出解整体存在。若耦合系数σ(s)、κ(s)退化,构造其截断函数,仍可得出解存在且有界。

一种多层三维几何结构的网格拆分方法

为了采用偏微分方程求解流体的多层复杂几何结构的渗透性质,本文提出了一种多层三维复杂几何结构的网格拆分方法,尤其对多层结构中的核心——1个由6个四边形和8个六边形围成的十四面体(即物理学上经典的Tetrakaidecahedron体)的空间拓扑几何结构进行了详细的分析．在完成单元体向整体结构拼接的过程中,采用一种将单元体视为内外两层的思路,既有效地存贮了网格点的信息,又大大减少计算量．并讨论了影响十四面体几何性质的参数对所生成的网格性质的影响．

面向复杂多流形高维数据的t-SNE降维方法

针对t-SNE方法不能很好地区分相互交叉的多个流形的问题,提出一种可视化降维方法.在t-SNE方法的基础上,在计算高维概率时考虑欧几里得度量和局部主成分分析以区分不同流形.然后可直接使用t-SNE的梯度求解方法得到降维结果.最后分别用3个人工生成的三维数据集和2个通用的机器学习数据集进行实验,并根据不同流形的区分度和流形内的邻域可信度2个指标对降维结果进行量化分析.结果表明,该方法在处理有交叉的多流形数据时的效果要明显优于原来的t-SNE方法,并能够较好地保持每个流形的邻域结构.

Redesign of a conformal boundary recovery algorithm for 3D Delaunay triangulation

Boundary recovery is one of the main obstacles in applying the Delaunay criterion to mesh generation. A stan- dard resolution is to add Steiner points directly at the intersection positions between missing boundaries and triangulations. We redesign the algorithm with the aid of some new concepts, data structures and operations, which make its implementation routine. Furthermore, all possible intersection cases and their solutions are presented, some of which are seldom discussed in the litera- ture. Finally, numerical results are presented to evaluate the performance of the new algorithm.

A Globally Convergent Polak-Ribiere-Polyak Conjugate Gradient Method with Armijo-Type Line Search

In this paper, we propose a globally convergent Polak-Ribiere-Polyak (PRP) conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic modelΦ(t):= f(xk)+tgkTdk+(1/2) t2dkTQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.

RNA in human sperm

We have yet to develop a fundamental understanding of the molecular complexities of human spermatozoa.This encompasses the unique packaging and structure of the sperm genome along with their paternally derived RNAs in preparation for their delivery to the egg.The diversity of these transcripts is vast,including several anti-sense mol- ecules resembling known regulatory micro-RNAs.The field is still grasping with its delivery to the oocyte at fertiliza- tion and possible significance.It remains tempting to analogize them to maternally-derived transcripts active in early embryo patterning.Irrespective of their role in the embryo,their use as a means to assess male factor infertility is promising.

Advances in Studies and Applications of Centroidal Voronoi Tessellations

Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.

人类精子RNA的重要性

我们对人类精子分子组成复杂性的理解仍处于初步阶段。这包括随时准备与卵细胞结合的精子基因组及其RNA具有独特的包装和结构。两者凸现了我们对遗传学因素如何有助于一个健康的孩子的出生理解不足。最近从人类精液中分离出的RNA可被用来显示转录的男性配子的轮廓。完全可以预期,具有正常生育能力的精子基因指纹特征能被鉴定,由此轮廓产生的背离能被检测。

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.