von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.     

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.     

Non‐self‐adjoint singular second‐order dynamic operators on time scale

In this study, maximal dissipative second‐order dynamic operators on semi‐infinite time scale are studied in the Hilbert space , that the extensions of a minimal symmetric operator in limit‐point case. We construct a self‐adjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations so that we can determine the scattering function of the dilation as stated in the scheme of Lax‐Phillips. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl‐Titchmarsh function of a self‐adjoint second‐order dynamic operator. Finally, we prove the theorems on completeness of the system of root functions of the dissipative and accumulative dynamic operators.     

The double deflating technique for irreducible singular M-matrix algebraic Riccati equations in the critical case

As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective.     

三维位势问题的梯度边界积分方程的新解法

三维位势问题的边界元分析中,关于坐标变量的边界位势梯度的计算是一个困难的问题. 已有一些方法着手解决这个问题,然而,这些方法需要复杂的理论推导和大量的数值计算. 本文提出求解一般边界位势梯度边界积分方程的辅助边值问题法. 该方法构造了与原边界值问题具有相同解域的辅助边值问题,该辅助边值问题具有已知解,因此通过求解此辅助边值问题,可获得梯度边界积分方程对应的系统矩阵,然后将此系统矩阵应用于求解原边值问题,求解过程非常简单,只需求解一个线性系统即可获得原边值问题的解. 值得注意的是,在求解原边值问题时,不再需要重新计算系统矩阵,因此辅助边值问题法的效率并不很差. 辅助边值问题法避免了强奇异积分的计算,具有数学理论简单、程序设计容易、计算精度高等优点,为坐标变量梯度边界积分方程的求解提供了一个新的途径. 3个标准的数值算例验证了方法的有效性.      

Screening gas-phase chemical kinetic models: Collision limit compliance and ultrafast timescales

Detailed gas-phase chemical kinetic models are widely used in combustion research, and many new mechanisms for different fuels and reacting conditions are developed each year. Recent works have highlighted the need for error checking when preparing such models, but a useful community tool to perform such analysis is missing. In this work, we present a simple online tool to screen chemical kinetic mechanisms for bimolecular reactions exceeding collision limits. The tool is implemented on a user-friendly website, cloudflame.kaust.edu.sa, and checks three different classes of bimolecular reactions; (ie, pressure independent, pressure-dependent falloff, and pressure-dependent PLOG). In addition, two other online modules are provided to check thermodynamic properties and transport parameters to help kinetic model developers determine the sources of errors for reactions that are not collision limit compliant. Furthermore, issues related to unphysically fast timescales can remain an issue even if all bimolecular reactions are within collision limits. Therefore, we also present a procedure to screen ultrafast reaction timescales using computational singular perturbation. For demonstration purposes only, three versions of the rigorously developed AramcoMech are screened for collision limit compliance and ultrafast timescales, and recommendations are made for improving the models. Larger models for biodiesel surrogates, tetrahydropyran, and gasoline surrogates are also analyzed for exemplary purposes. Numerical simulations with updated kinetic parameters are presented to show improvements in wall-clock time when resolving ultrafast timescales.     

计算代数方程组孤立奇异解的符号数值方法

求解代数方程组是计算代数几何的最基本问题之一,孤立奇异解的计算则是其中最具挑战性的课题之一,在科学与工程计算中有着广泛的应用,如机器人、计算机视觉、机器学习、人工智能、运筹学、密码学和控制论等.本文结合作者的研究成果,综述了符号数值方法在计算代数系统孤立奇异解、特别是近似奇异解精化与验证方面的研究进展,并对未来的研究方向提出了展望.