Advanced Numerical Techniques for Solving High-Dimensional Integral Equations in Environmental Engineering Applications

Abstract

Applications in environmental engineering include pollutant dispersion, groundwater contamination and atmospheric transport which use high-dimensional integral equations. Their solutions are nonlinear, stochastic in variability, and the curse of dimensionality frequently makes their analysis solutions intractable. This paper explores modern numerical methods that are aimed at effectively solving such equations at both accuracy and scalability. Deterministic (such as Galerkin formulations, spectral decomposition, and quadrature-based discretization) are compared with stochastic (such as Monte Carlo (MC), quasi-Monte Carlo (QMC), and sparse grid methods). Hybrid machine learning-assisted solvers are also proposed to further improve performance to surrogate model and accelerate convergence. Case studies are concentrated on two important areas; the transport of groundwater contaminants and prediction of urban air quality. Findings indicate that the sparse grid and QMC techniques are much more efficient and accurate than conventional MC simulations with a potential to reduce the cost of computation by as much as 40 percent. Spectral techniques allow extremely high accuracy with smooth deterministic models at increased computational cost, and hybrid ML-assisted solvers can scale to dimensions of tens of thousands or more (that is, comparable limits) hence are applicable to real-time environmental monitoring. The results endorse the idea that the combination of the deterministic, stochastic, and hybrid strategies produces a powerful computational system to tackle the complicate environmental processes. The stated findings demonstrate the promise of complex numerical integration models in improving predictive performance, resource utilization, and real-time decision-making in environmental engineering problems.