Stephen Russell

Stephen Russell, Niall Madden

Our goal is to present an elementary approach to the analysis and programming of sparse grid finite element methods. This family of schemes can compute accurate solutions to partial differential equations, but using far fewer degrees of freedom than their classical counterparts. After a brief discussion of the classical Galerkin finite element method with bilinear elements, we give a short analysis of what is probably the simplest sparse grid method: the two-scale technique of Lin et al. (2001). We then demonstrate how to extend this to a multiscale sparse grid method which, up to choice of basis, is equivalent to the hierarchical approach, as described by, e.g., Bungartz and Griebel (2004). However, by presenting it as an extension of the two-scale method, we can give an elementary treatment of its analysis and implementation. For each method considered, we provide MATLAB code, and a comparison of accuracy and computational costs.

Satyajit Movidi, Stephen Russell

AIVisor, an agentic retrieval-augmented LLM for student advising, was used to examine how personalization affects system performance across multiple evaluation dimensions. Using twelve authentic advising questions intentionally designed to stress lexical precision, we compared ten personalized and non-personalized system configurations and analyzed outcomes with a Linear Mixed-Effects Model across lexical (BLEU, ROUGE-L), semantic (METEOR, BERTScore), and grounding (RAGAS) metrics. Results showed a consistent trade-off: personalization reliably improved reasoning quality and grounding, yet introduced a significant negative interaction on semantic similarity, driven not by poorer answers but by the limits of current metrics, which penalize meaningful personalized deviations from generic reference texts. This reveals a structural flaw in prevailing LLM evaluation methods, which are ill-suited for assessing user-specific responses. The fully integrated personalized configuration produced the highest overall gains, suggesting that personalization can enhance system effectiveness when evaluated with appropriate multidimensional metrics. Overall, the study demonstrates that personalization produces metric-dependent shifts rather than uniform improvements and provides a methodological foundation for more transparent and robust personalization in agentic AI.

Stephen Russell

Most semantic drift studies report multiple signals e.g., embedding displacement, neighbor changes, distributional divergence, and recursive trajectory instability, without a shared explanatory theory that relates them. This paper proposes a formalization of these signals in one time-indexed substrate, $S_t=(X,d_t,P_t)$, combining embedding geometry with local diffusion. Within this substrate, node-level neighborhood drift measures changes in local conditional distributions, coarse Ricci curvature measures local contractivity of semantic diffusion, and recursive drift probes stability of iterated semantic operators. This manuscript specifies the formal model, assumptions, and tests that can refute the model. Herein, the paper introduces bridge mass, a node-level aggregate of incident negative curvature, as a predictor of future neighborhood rewiring. This paper provides the theory and test contracts; empirical performance is deferred to subsequent studies.