scholarly journals On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

2012 ◽  
Vol 46 (6) ◽  
pp. 1555-1576 ◽  
Author(s):  
Toni Lassila ◽  
Andrea Manzoni ◽  
Gianluigi Rozza
Author(s):  
Wei Xing ◽  
Akeel A. Shah ◽  
Prasanth B. Nair

In this paper, Isomap and kernel Isomap are used to dramatically reduce the dimensionality of the output space to efficiently construct a Gaussian process emulator of parametrized partial differential equations. The output space consists of spatial or spatio-temporal fields that are functions of multiple input variables. For such problems, standard multi-output Gaussian process emulation strategies are computationally impractical and/or make restrictive assumptions regarding the correlation structure. The method we develop can be applied without modification to any problem involving vector-valued targets and vector-valued inputs. It also extends a method based on linear dimensionality reduction to response surfaces that cannot be described accurately by a linear subspace of the high dimensional output space. Comparisons to the linear method are made through examples that clearly demonstrate the advantages of nonlinear dimensionality reduction.


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