arXiv subjects

Xueli Bai

Publications and source records attributed to Xueli Bai.

14 recordsLinked to original sources

arXiv2025

A Variational Characterization and A Line Search Newton-Noda Method for the unifying spectral problem of nonnegative tensors

We study the general $(\boldsymbolσ,\mathbf{p})$-eigenvalue problem of nonnegative tensors introduced by A. Gautier, F. Tudisco, and M. Hein [SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206--1231], which unifies several well-studied tensor eigenvalue and singular value problems. First, we propose an alternative min-max Collatz--Wielandt formula for the $(\boldsymbolσ,\mathbf{p})$-spectral radius, which bypasses the auxiliary multihomogeneous mapping employed in that work. This variational characterization both recovers several classical results and admits a natural convex reformulation. It arises from an alternative approach that directly connects the $(\boldsymbolσ,\mathbf{p})$-spectral problem to a class of convex programs. We then develop and analyze a line search Newton-Noda method (LS-NNM) for computing the positive $(\boldsymbolσ,\mathbf{p})$-eigenpair of nonnegative tensors. The proposed method integrates Newton method with Noda iteration. The Newton equation is derived from an equivalent nonlinear system, while the eigenvalue sequence is updated by the strategy of the Noda iteration and its variants. To ensure global convergence, we introduce a positivity-preserving line search procedure based on an equivalent constrained optimization problem. The global and quadratic convergence of LS-NNM are established for the class of $(\boldsymbolσ,\mathbf{p})$-spectral problem that admits a unique positive $(\boldsymbolσ,\mathbf{p})$-eigenpair, as guaranteed by the Perron-Frobenius theorem. Finally, numerical experiments are conducted to illustrate the performance of LS-NNM.

math.OC↗

arXiv2025

Non-convergence of the principal eigenvalue of elliptic operators for large advection

This paper investigates the limit of the principal eigenvalue $λ(s)$ as $s\to+\infty$ for the following elliptic equation \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x), \quad x\in Ω \end{align*} in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $λ(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }λ(s)=\pm\infty$). This work constructs an example such that $λ(s)$ is divergent as $s\to+\infty$. This seems to be the first rigorous result demonstrating the non-convergence of the principal eigenvalue for second-order linear elliptic operators with some strong advection. As an application, we demonstrate that for the classical advection-reaction-diffusion model with advective velocity field $\mathbf{v}=\nabla m$, where $m$ is a potential function with infinite oscillations, the principal eigenvalue changes sign infinitely often along a subsequence of $s\to\infty$. This leads to solution behaviors that differ significantly from those observed when $m$ is non-oscillatory.

math.AP↗

arXiv2025

Tumor-associated CD19$^+$ macrophages induce immunosuppressive microenvironment in hepatocellular carcinoma

Tumor-associated macrophages are a key component that contributes to the immunosuppressive microenvironment in human cancers. However, therapeutic targeting of macrophages has been a challenge in clinic due to the limited understanding of their heterogeneous subpopulations and distinct functions. Here, we identify a unique and clinically relevant CD19$^+$ subpopulation of macrophages that is enriched in many types of cancer, particularly in hepatocellular carcinoma (HCC). The CD19$^+$ macrophages exhibit increased levels of PD-L1 and CD73, enhanced mitochondrial oxidation, and compromised phagocytosis, indicating their immunosuppressive functions. Targeting CD19$^+$ macrophages with anti-CD19 chimeric antigen receptor T (CAR-T) cells inhibited HCC tumor growth. We identify PAX5 as a primary driver of up-regulated mitochondrial biogenesis in CD19$^+$ macrophages, which depletes cytoplasmic Ca$^{2+}$, leading to lysosomal deficiency and consequent accumulation of CD73 and PD-L1. Inhibiting CD73 or mitochondrial oxidation enhanced the efficacy of immune checkpoint blockade therapy in treating HCC, suggesting great promise for CD19$^+$ macrophage-targeting therapeutics.

q-bio.CB↗

arXiv2024

Exact blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $N\ge 3$

In this paper, we obtain the exact blow-up profiles of solutions of the Keller-Segel-Patlak system in the space with dimensions $N\ge 3$, which solves an open problem proposed by P. Souplet and M. Winkler in 2019. To establish this achievement, we develop the zero number argument for nonlinear equations with unbounded coefficients and construct a family of auxiliary backward self-similar solutions through nontrivial ODE analysis.

math.AP↗

arXiv2023

A Counterexample for the Principal Eigenvalue of An Elliptic Operator with Large Advection

There are numerous studies focusing on the convergence of the principal eigenvalue $λ(s)$ as $s\to+\infty$ corresponding to the elliptic eigenvalue problem \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x),\quad x\in Ω, \end{align*} where $Ω$ is a bounded domain and the advection term $\mathbf{v}$ under some certain restrictions. In this paper, we construct an infinitely oscillating gradient advection term $\mathbf{v}=\nabla m(x)\in C^1(Ω)$ such that the principal eigenvalue $λ(s)$ does not converge as $s\to+\infty$. As far as we know, this is the first result that guarantee the non-convergence of the principal eigenvalue.

math.AP↗

arXiv2023

Finding the spectral radius of a nonnegative irreducible symmetric tensor via DC programming

The Perron-Frobenius theorem says that the spectral radius of an irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of an irreducible nonnegative symmetric tensor. By transferring the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, which is typically a DC (difference of convex functions) programming, we derive a simpler and cheaper iterative method. The proposed method is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the method $Q$-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Preliminary numerical results show that the improved method performs quite well.

math.OC↗

arXiv2022

Optimisation of total population in logistic model with nonlocal dispersals and heterogeneous environments

In this paper, we investigate the issue of maximizing the total equilibrium population with respect to resources distribution m(x) and diffusion rates d under the prescribed total amount of resources in a logistic model with nonlocal dispersals. Among other things, we show that for $d\ge1$, there exist $C_0, C_1>0$, depending on the $\|m\|_{L^1}$ only, such that $$C_0\sqrt{d}<\mbox{supremum~ of~ total~ population}<C_1\sqrt{d}.$$ However, when replaced by random diffusion, a conjecture, proposed by Ni and justified in [3], indicates that in the one-dimensional case, supremum of total population$=3\|m\|_{L^1}$. This reflects serious discrepancies between models with local and nonlocal dispersal strategies.

math.AP↗

arXiv2022

Feasible Newton's methods for symmetric tensor Z-eigenvalue problems

Finding a Z-eigenpair of a symmetric tensor is equivalent to finding a KKT point of a sphere constrained minimization problem. Based on this equivalency, in this paper, we first propose a class of iterative methods to get a Z-eigenpair of a symmetric tensor. Each method can generate a sequence of feasible points such that the sequence of function evaluations is decreasing. These methods can be regarded as extensions of the descent methods for unconstrained optimization problems. We pay particular attention to the Newton's method. We show that under appropriate conditions, the Newton's method is globally and quadratically convergent. Moreover, after finitely many iterations, the unit steplength will always be accepted. We also propose a nonlinear equations based Newton's method and establish its global and quadratic convergence. In the end, we do several numerical experiments to test the proposed Newton's methods. The results show that both Newton's methods are very efficient.

math.OC↗

arXiv2021

Effects of nonlocal dispersal strategies and heterogeneous environment on total population

In this paper, we consider the following single species model with nonlocal dispersal strategy $$ d\mathcal {L} [θ] (x,t) + θ(x,t) [m(x)- θ(x,t)]=0 , $$ where $\mathcal {L}$ denotes the nonlocal diffusion operator, and investigate how the dispersal rate of the species and the distribution of resources affect the total population. First, we show that the upper bound for the ratio between total population and total resource is $C\sqrt{d}$. Moreover, examples are constructed to indicate that this upper bound is optimal. Secondly, for a type of simplified nonlocal diffusion operator, we prove that if $ \frac{\sup m}{\inf m}<\frac{\sqrt{5}+1}{2}$, the total population as a function of dispersal rate $d$ admits exactly one local maximum point in $\displaystyle (\inf m, \sup m)$. These results reveal essential discrepancies between local and nonlocal dispersal strategies.

math.AP↗

arXiv2021

Global dynamics of a parabolic type equation arising from the curvature flow

This paper studies a type of degenerate parabolic problem with nonlocal term \begin{equation*} \begin{cases} u_t=u^p(u_{xx}+u-\bar{u}) & 0 1$, $a>0$. In this paper, the classification of the finite-time blowup/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. Moreover, we combine the applications of Lojasiewicz-Simon inequality and energy estimates to derive that any bounded solution with positive initial data converges to some steady state as $t\rightarrow +\infty$.

math.AP↗

arXiv2020

Unique solvability of weakly homogeneous generalized variational inequalities

An interesting observation is that most pairs of weakly homogeneous mappings have no strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper focuses on studying the unique solvability of the generalized variational inequality with a pair of weakly homogeneous mappings. By using a weaker condition than the strong monotonicity and some additional conditions, we achieve several results on the unique solvability of the underlying problem. These results are exported by making use of the exceptional family of elements or derived from new obtained Karamardian-type theorems or established under the exceptional regularity condition. They are new even when the problem comes down to its important subclasses studied in recent years.

math.OC↗

arXiv2019

A nonnegativity preserving algorithm for multilinear systems with nonsingular M-tensors

This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular $\mathcal{M}$-tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative decreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method. Particularly, when some elements of the right-hand side vector are zeros, the proposed algorithm works well while existing state-of-the-art solvers may not produce a nonnegative solution.

math.OC↗

arXiv2017

Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels

In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.

math.AP↗

arXiv2015

Global dynamics of competition models with nonlocal dispersals I: Symmetric kernels

In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.

math.AP↗