Discretization method

This theme investigates the development and analysis of discretization schemes that guarantee both consistency and convergence when applied to partial differential equations (PDEs)—notably diffusion and flow problems—on complex meshes, including polyhedral grids that may exhibit anisotropic and heterogeneous properties. The importance of this research area lies in resolving challenges arising from grid irregularities, permeability anisotropy, and ensuring that discrete solutions converge to the true solution as mesh refinement proceeds. Addressing these issues affects the fidelity and robustness of numerical simulations in fields such as porous media flow and underground engineering.

This theme addresses the extension of classical discretization approaches through meshless methods such as RBF-FD, emphasizing the numerical stability and conditioning challenges that arise from various parameter choices including stencil size, polynomial augmentation degree, RBF type, shape parameters, and node distributions. The goal is to identify parameter combinations that minimize discretization errors and conditioning issues for PDEs, thereby advancing flexible and efficient numerical methods on structured and unstructured point sets and in higher dimensions.

This research theme focuses on the accuracy, convergence rates, and computational characteristics of Haar wavelet-based discretization methods (HWDM) for solving ordinary differential equations and integro-differential equations. Haar wavelets’ piecewise-constant, orthogonal properties offer computational simplicity but impose challenges due to non-differentiability. Research explores expansions of derivatives in Haar bases, techniques for handling boundary conditions, and enhancements such as Richardson extrapolation, aiming to rigorously establish convergence orders and error bounds that underpin HWDM's reliability in practical applications.

This theme studies the design and implementation of higher-order discretization schemes tailored to nonlinear elliptic PDEs and multiphase flow problems such as fluidized beds. It involves constructing compact and off-step finite difference methods with fourth-order accuracy and mitigating numerical artifacts observed in standard lower-order schemes. In fluidized bed simulations, improving discretization reduces numerical diffusion effects that produce unrealistic bubble shapes, thereby enhancing physical fidelity. The research provides theoretical and computational advances to increase precision and stability of solutions in challenging nonlinear and multiphase contexts.

This theme explores innovative discretization operators that exactly or nearly reproduce the algebraic, analytical, and operational properties of integer and fractional derivatives. It covers infinite-series-based fractional difference operators that embody algebraic correspondence principles and universality, ensuring exact discretization independent of differential equation forms. These developments support solving fractional PDEs with variable orders, minimizing discretization-induced artifacts and facilitating efficient, accurate numerical solutions in complex geometries and materials with nonlocal behaviors.

Population balance equations (PBEs) model particle size distributions evolving under complex mechanisms such as growth, aggregation, breakage, attrition, and nucleation. This theme revolves around numerical methods that accurately and stably discretize PBEs, conserving key moments and capturing interclass fluxes to handle multiple simultaneous mechanisms. Achieving numerical stability and high accuracy on arbitrary discretization grids enables practical solution of PBEs in diverse industrial settings involving particulate systems.

This theme investigates discretization approaches tailored for human activity recognition based on accelerometer data collected via mobile devices. By reducing continuous sensor data into discrete symbols or states, these methods aim to improve classification accuracy for granular activities while minimizing computational expense and energy demands on battery-limited devices. The objective is to balance high recognition granularity with low power consumption and real-time processing capability within resource-constrained systems.

This theme explores supervised discretization methods that partition continuous features to maximize the AUC, a robust, threshold-independent performance metric in classification. By optimizing discretization boundaries to maximize class separation as measured by ROC curves, these methods aim to improve predictive power of classifiers—especially those sensitive to categorical attributes such as Naïve Bayes—while preserving discriminative information better than traditional entropy or chi-square-based discretization.

This theme analyses the propagation and accumulation of local truncation errors through iterative numerical time-stepping methods for initial value ODE problems, leading to global discretization errors. It focuses on providing accurate a priori error bounds and estimates to assess algorithmic quality, guiding adaptive timestep choices aiming to minimize overall global errors. Understanding global error behavior impacts the reliability and efficiency of numerical ODE solvers in scientific computing.

This theme empirically compares common discretization techniques for converting continuous data into categorical features prior to supervised learning tasks. It addresses how these methods impact classifier accuracy, generalization, and training efficiency, aiming to reveal strengths, weaknesses, and best practices in discretization preprocessing across diverse real-world datasets.