arXiv subjects

Jingjing Cai

Publications and source records attributed to Jingjing Cai.

7 recordsLinked to original sources

arXiv2026

Uniform-in-Time Weak and Ergodic Error Estimates of a Nonlinearity-Explicit Full Discretization for Superlinear SPDEs Driven by Multiplicative Noise

For a class of superlinear SPDEs driven by multiplicative noise, we prove an (essentially) sharp uniform-in-time (UIT) weak convergence rate for the nonlinearity-explicit Galerkin tamed Euler method (GTEM). Under standard monotonicity assumptions, the proof combines Malliavin calculus with regularity theory for the associated backward Kolmogorov equation (BKE), leading to UIT moment, H\"older, and Malliavin estimates, along with regularity estimates for the BKE solution. These estimates, together with a weak error decomposition and Malliavin integration by parts (IBP) formula, then yield a UIT weak convergence rate $\tau^\rho+\lambda_N^{-(\rho+\gamma/2)}$ for any $\rho \in (0,1)$, where $\gamma\in[0,1)$ quantifies the assumed spatial Sobolev regularity. Consequently, we obtain a sharp ergodic error estimate between the exact and numerical invariant measures. Numerical experiments support the theory.

math.NA↗

arXiv2026

Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise

We establish sharp, uniform-in-time strong error estimates for a nonlinearity-explicit tamed finite element method (FEM) applied to a class of superlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise, including the stochastic Allen--Cahn equation with a moderately thick interface. This tamed-FEM was first introduced in [Z. Liu and J. Shen, arXiv:2502.19117] to ensure long-time unconditional stability and to preserve the Lyapunov structure of this class of SPDEs. We further prove that the scheme is exponentially ergodic and derive the convergence rate between the exact invariant measure and its numerical counterpart in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity as well as the sharpness and time-independence of the strong convergence rates for this tamed-FEM.

math.NA↗

arXiv2025

Kunlun Anomaly Troubleshooter: Enabling Kernel-Level Anomaly Detection and Causal Reasoning for Large Model Distributed Inference

Anomaly troubleshooting for large model distributed inference (LMDI) remains a critical challenge. Resolving anomalies such as inference performance degradation or latency jitter in distributed system demands significant manual efforts from domain experts, resulting in extremely time-consuming diagnosis processes with relatively low accuracy. In this paper, we introduce Kunlun Anomaly Troubleshooter (KAT), the first anomaly troubleshooting framework tailored for LMDI. KAT addresses this problem through two core innovations. First, KAT exploits the synchronicity and consistency of GPU workers, innovatively leverages function trace data to precisely detect kernel-level anomalies and associated hardware components at nanosecond resolution. Second, KAT integrates these detection results into a domain-adapted LLM, delivering systematic causal reasoning and natural language interpretation of complex anomaly symptoms. Evaluations conducted in Alibaba Cloud Service production environment indicate that KAT achieves over 0.884 precision and 0.936 recall in anomaly detection, providing detail anomaly insights that significantly narrow down the diagnostic scope and improve both the efficiency and success rate of troubleshooting.

cs.LG↗

arXiv2024

Convergence rate and exponential stability of backward Euler method for neutral stochastic delay differential equations under generalized monotonicity conditions

This work focuses on the numerical approximations of neutral stochastic delay differential equations with their drift and diffusion coefficients growing super-linearly with respect to both delay variables and state variables. Under generalized monotonicity conditions, we prove that the backward Euler method not only converges strongly in the mean square sense with order $1/2$, but also inherit the mean square exponential stability of the original equations. As a byproduct, we obtain the same results on convergence rate and exponential stability of the backward Euler method for stochastic delay differential equations with generalized monotonicity conditions. These theoretical results are finally supported by several numerical experiments.

math.NA↗

arXiv2020

Off-grid Multi-Source Passive Localization Using a Moving Array

A novel direct passive localization technique through a single moving array is proposed in this paper using the sparse representation of the array covariance matrix in spatial domain. The measurement is constructed by stacking the vectorized version of all the array covariance matrices at different observing positions. First, an on-grid compressive sensing (CS) based method is developed, where the dictionary is composed of the steering vectors from the searching grids to the observing positions. Convex optimization is applied to solve the `1-norm minimization problem. Second, to get much finer target positions, we develop an on-grid CS based method, where the majorization-minimization technique replaces the atan-sum objective function in each iteration by a quadratic convex function which can be easily minimized. The objective function,atan-sum, is more similar to `0-norm, and more sparsity encouraging than the log-sum function.This method also works more robustly at conditions of low SNR, and fewer observing positions are needed than in the traditional ones. The simulation experiments verify the promises of the proposed algorithm.

eess.SP↗

arXiv2020

An Improved DOA Estimation Method for a Mixture of Circular and Non-Circular Signals Based on Sparse Arrays

Sparse arrays have attracted a lot of interests recently for their capability of providing more degrees of freedom than traditional uniform linear arrays. For a mixture of circular and noncircular signals, most of the existing direction of arrival (DOA) estimation methods are based on various uniform arrays. Recently, a class of DOA estimation algorithms based on sparse arrays was developed for a mixture of circular and noncircular signals. To further improve its performance, in this work, a modified algorithm is presented, which can resolve the same number of signals, and simulation results are provided to verified its performance.

eess.SP↗

arXiv2014

Asymptotic behavior of solutions of a reaction diffusion equation with free boundary conditions

We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+α$ and $h'(t)=-u_x(t,g(t))-α$ for some $α>0$. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundaries representing the expanding fronts. When $α=0$, the problem was recently investigated by \cite{DuLin, DuLou}. In this paper we consider the case $α>0$. In this case shrinking (i.e. $h(t)-g(t)\to 0$) may happen, which is quite different from the case $α=0$. Moreover, we show that, under certain conditions on $f$, shrinking is equivalent to vanishing (i.e. $u\to 0$), both of them happen as $t$ tends to some finite time. On the other hand, every bounded and positive time-global solution converges to a nonzero stationary solution as $t\to \infty$. As applications, we consider monostable and bistable types of nonlinearities, and obtain a complete description on the asymptotic behavior of the solutions.

math.AP↗