2013/06/01 by Maziar Raissi, Raissi, Maziar, Padmanabhan Seshaiyer +1
Decision Sciences · Engineering · Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Nuclear Engineering Thermal-Hydraulics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.1306.0132
openalex publication_date 2013/06/01 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
Over the last few years there have been dramatic advances in our\nunderstanding of mathematical and computational models of complex systems in\nthe presence of uncertainty. This has led to a growth in the area of\nuncertainty quantification as well as the need to develop efficient, scalable,\nstable and convergent computational methods for solving differential equations\nwith random inputs. Stochastic Galerkin methods based on polynomial chaos\nexpansions have shown superiority to other non-sampling and many sampling\ntechniques. However, for complicated governing equations numerical\nimplementations of stochastic Galerkin methods can become non-trivial. On the\nother hand, Monte Carlo and other traditional sampling methods, are\nstraightforward to implement. However, they do not offer as fast convergence\nrates as stochastic Galerkin. Other numerical approaches are the stochastic\ncollocation (SC) methods, which inherit both, the ease of implementation of\nMonte Carlo and the robustness of stochastic Galerkin to a great deal. However,\nstochastic collocation and its powerful extensions, e.g. sparse grid stochastic\ncollocation, can simply fail to handle more levels of complication. The\nseemingly innocent Burgers equation driven by Brownian motion is such an\nexample. In this work we propose a novel enhancement to stochastic collocation\nmethods using locally improved deterministic model reduction techniques that\ncan handle this pathological example and hopefully other more complicated\nequations like Stochastic Navier-Stokes. Local improvements to reduced-order\nmodels are achieved using sensitivity analysis of the proper orthogonal\ndecomposition. Our numerical results show that the proposed technique is not\nonly reliable and robust but also very efficient.\n