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Shuxian Xu

Publications and source records attributed to Shuxian Xu.

6 recordsLinked to original sources

Reproducing human biases in route choice using large language models: Toward scalable behavioral modeling

Human choice behavior, including route choice, exhibits systematic behavioral biases that deviate from the assumptions of full rationality. Cumulative prospect theory (CPT) has been widely recognized as an effective framework for characterizing such behavioral patterns. However, its large-scale application, particularly in simulation and agent-based modeling, critically depends on specifying individual-level CPT parameters, which remain a major bottleneck. Conventional approaches typically rely on surveys and controlled experiments to calibrate CPT parameters, yet these methods are difficult to generalize and often fail to capture the full diversity of human decision-making. To address this challenge, this paper investigates whether large language models (LLMs) can reproduce human behavioral biases in choice-making without explicit specification of prospect-theoretic parameters. Using route choice as a representative scenario, we design a behavioral evaluation framework and systematically compare LLM-generated decisions with established human behavioral patterns predicted by CPT. Experimental results demonstrate that LLMs are capable of reproducing non-rational human choice biases and can exhibit decision behaviors consistent with prospect-theoretic effects under uncertainty. These findings suggest that generative AI models may provide a scalable alternative for modeling human decision processes and offer a promising foundation for next-generation large-scale agent-based simulation and AI-driven behavioral research.

cs.AI

New characterizations of weak CMP inverses

In 2025, Mosić defined the weak CMP inverse utilizing a minimal rank weak Drazin inverse instead of the Drazin inverse. The weak CMP inverse is a new wider class of generalized inverses, of which the CMP and MPCEP inverse are particular cases. In this paper, we provide several expressions, along with a number of new characterizations and properties for the weak CMP inverse. Moreover, we investigate the relationships between the weak CMP inverse and some well-known generalized inverses, such as the Moore-Penrose inverse and weak MPD inverse. Finally, we show that the weak CMP inverse, weak MPD inverse and weak DMP inverse are all strong Bott-Duffin $(e,f)$-inverses.

math.RA

Stability and Accuracy analysis of the $θ$ Method and 3-Point Time filter

This paper analyzes a $θ$-method and 3-point time filter. This approach adds one additional line of code to the existing source code of $θ$-method. We prove the method's $0$-stability, accuracy, and $A$-stability for both constant time step and variable time step. Some numerical tests are performed to validate the theoretical results.

math.NA

A Doubly Adaptive Penalty Method for the Navier Stokes Equations

We develop, analyze and test adaptive penalty parameter methods. We prove unconditional stability for velocity when adapting the penalty parameter, $ε,$ and stability of the velocity time derivative under a condition on the change of the penalty parameter, $ε(t_{n+1})-ε(t_n)$. The analysis and tests show that adapting $ε(t_{n+1})$ in response to $\nabla\cdot u(t_n)$ removes the problem of picking $ε$ and yields good approximations for the velocity. We provide error analysis and numerical tests to support these results. We supplement the adaptive-$ε$ method by also adapting the time-step. The penalty parameter $ε$ and time-step are adapted independently. We further compare first, second and variable order time-step algorithms. Accurate recovery of pressure remains an open problem.

math.NA

Conditioning of linear systems arising from penalty methods

Penalizing incompressibility in the Stokes problem leads, under mild assumptions, to matrices with condition numbers $κ=\mathcal{O} (\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $ mesh width $<1$. Although $κ=\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solving these systems. In the SPD case, using the conjugate gradient method, this is usually explained by spectral gaps occurring in the penalized coefficient matrix. Herein we point out a second contributing factor. Since the solution is approximately incompressible, solution components in the eigenspaces associated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.

math.NA

Stability in 3d of a sparse grad-div approximation of the Navier-Stokes equations

Inclusion of a term $-γ\nabla\nabla\cdot u$, forcing $\nabla\cdot u$ to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let $G^{\ast}=-diag(\partial_{x}^{2},\partial_{y}^{2},\partial_{z}^{2})$ denote the diagonal of $G=-\nabla\nabla\cdot$, and $α\geq0$ an adjustable parameter. The 2-step method considered is $$\begin{eqnarray} 1 &:&\frac{\widetilde{u}^{n+1}-u^{n}}{k}+u^{n}\cdot \nabla \widetilde{u}^{n+1}+\nabla p^{n+1}-νΔ\widetilde{u}^{n+1}=f\text{ & }\nabla \cdot \widetilde{u}^{n+1}=0,\\ 2 &:&\left[ \frac{1}{k}I+(γ+α)G^{\ast }\right] u^{n+1}=\frac{1}{k }\widetilde{u}^{n+1}+\left[ (γ+α)G^{\ast }-γG\right] u^{n}. \end{eqnarray}$$ We prove its unconditional, nonlinear, long time stability in $3d$ for $α\geq0.5γ$. The analysis also establishes that the method controls the persistent size of $||\nabla\cdot u||$ in general and controls the transients in $||\nabla\cdot u||$ when $u(x,0)=0$ and $f(x,t)\neq0$ provided $α>0.5γ$. Consistent numerical tests are presented.

math.NA