2014/10/20 by Akshay Mittal, Gianluca Iaccarino, Mittal, Akshay +1
Computer Science · Decision Sciences · Physics and Astronomy · #FOS: Mathematics #Image and Signal Denoising Methods #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1410.5308
openalex publication_date 2014/10/20 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Coupled partial differential equation (PDE) systems, which often represent\nmulti-physics models, are naturally suited for modular numerical solution\nmethods. However, several challenges yet remain in extending the benefits of\nmodularization practices to the task of uncertainty propagation. Since the cost\nof each deterministic PDE solve can be usually expected to be quite\nsignificant, statistical sampling based methods like Monte-Carlo (MC) are\ninefficient because they do not take advantage of the mathematical structure of\nthe problem, and suffer for poor convergence properties. On the other hand,\neven if each module contains a moderate number of uncertain parameters,\nimplementing spectral methods on the combined high-dimensional parameter space\ncan be prohibitively expensive due to the curse of dimensionality. In this\nwork, we present a module-based and efficient intrusive spectral projection\n(ISP) method for uncertainty propagation. In our proposed method, each\nsubproblem is separated and modularized via block Gauss-Seidel (BGS)\ntechniques, such that each module only needs to tackle the local stochastic\nparameter space. Moreover, the computational costs are significantly mitigated\nby constructing reduced chaos approximations of the input data that enter each\nmodule. We demonstrate implementations of our proposed method and its\ncomputational gains over the standard ISP method using numerical examples.\n