arXiv subjects

Shengbo Ma

Publications and source records attributed to Shengbo Ma.

5 recordsLinked to original sources

arXiv2026

Parametric Sensitivity of POD Reduced-Order Models for Semilinear Evolution Equations with Applications to G-Equations

Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.

math.NA↗

arXiv2026

A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations

Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.

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arXiv2026

TiInsight: A SQL-based Automated Exploratory Data Analysis System through Large Language Models

The SQL-based exploratory data analysis has garnered significant attention within the data analysis community. The emergence of large language models (LLMs) has facilitated the paradigm shift from manual to automated data exploration. However, existing methods generally lack the ability for cross-domain analysis, and the exploration of LLMs capabilities remains insufficient. This paper presents TiInsight, an SQL-based automated cross-domain exploratory data analysis system. First, TiInsight offers a user-friendly GUI enabling users to explore data using natural language queries. Second, TiInsight offers a robust cross-domain exploratory data analysis pipeline: hierarchical data context (i.e., HDC) generation, question clarification and decomposition, text-to-SQL (i.e., TiSQL), and data visualization (i.e., TiChart). Third, we have implemented and deployed TiInsight in the production environment of PingCAP and demonstrated its capabilities using representative datasets. The demo video is available at https://youtu.be/JzYFyYd-emI.

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arXiv2024

Towards Automated Cross-domain Exploratory Data Analysis through Large Language Models

Exploratory data analysis (EDA), coupled with SQL, is essential for data analysts involved in data exploration and analysis. However, data analysts often encounter two primary challenges: (1) the need to craft SQL queries skillfully, and (2) the requirement to generate suitable visualization types that enhance the interpretation of query results. Due to its significance, substantial research efforts have been made to explore different approaches to address these challenges, including leveraging large language models (LLMs). However, existing methods fail to meet real-world data exploration requirements primarily due to (1) complex database schema; (2) unclear user intent; (3) limited cross-domain generalization capability; and (4) insufficient end-to-end text-to-visualization capability. This paper presents TiInsight, an automated SQL-based cross-domain exploratory data analysis system. First, we propose hierarchical data context (i.e., HDC), which leverages LLMs to summarize the contexts related to the database schema, which is crucial for open-world EDA systems to generalize across data domains. Second, the EDA system is divided into four components (i.e., stages): HDC generation, question clarification and decomposition, text-to-SQL generation (i.e., TiSQL), and data visualization (i.e., TiChart). Finally, we implemented an end-to-end EDA system with a user-friendly GUI interface in the production environment at PingCAP. We have also open-sourced all APIs of TiInsight to facilitate research within the EDA community. Through extensive evaluations by a real-world user study, we demonstrate that TiInsight offers remarkable performance compared to human experts. Specifically, TiSQL achieves an execution accuracy of 86.3% on the Spider dataset using GPT-4. It also demonstrates state-of-the-art performance on the Bird dataset.

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arXiv2024

Uncertainty Quantification of Phase Transition Problems with an Injection Boundary

We develop an enthalpy-based modeling and computational framework to quantify uncertainty in Stefan problems with an injection boundary. Inspired by airfoil icing studies, we consider a system featuring an injection boundary inducing domain changes and a free boundary separating phases, resulting in two types of moving boundaries. Our proposed enthalpy-based formulation seamlessly integrates thermal diffusion across the domain with energy fluxes at the boundaries, addressing a modified injection condition for boundary movement. Uncertainty then stems from random variations in the injection boundary. The primary focus of our Uncertainty Quantification (UQ) centers on investigating the effects of uncertainty on free boundary propagation. Through mapping to a reference domain, we derive an enthalpy-based numerical scheme tailored to the transformed coordinate system, facilitating a simple and efficient simulation. Numerical and UQ studies in one and two dimensions validate the proposed model and the extended enthalpy method. They offer intriguing insights into ice accretion and other multiphysics processes involving phase transitions.

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