Local Polynomial Chaos Expansion for Linear Differential Equations with High Dimensional Random Inputs

2015 ◽  
Vol 37 (1) ◽  
pp. A79-A102 ◽  
Author(s):  
Yi Chen ◽  
John Jakeman ◽  
Claude Gittelson ◽  
Dongbin Xiu
Keyword(s):  
Differential Equations ◽  
Polynomial Chaos ◽  
Polynomial Chaos Expansion ◽  
Linear Differential Equations ◽  
High Dimensional ◽  
Chaos Expansion ◽  
Random Inputs ◽  
Local Polynomial ◽  
Linear Differential

A Sparse Data-Driven Polynomial Chaos Expansion Method for Uncertainty Propagation

2016 ◽  
Author(s):  
F. Wang ◽  
F. Xiong ◽  
S. Yang ◽  
Y. Xiong
Keyword(s):  
Polynomial Chaos ◽  
Uncertainty Propagation ◽  
Computational Cost ◽  
Expansion Method ◽  
Polynomial Chaos Expansion ◽  
Data Driven ◽  
High Dimensional ◽  
Chaos Expansion ◽  
Least Angle Regression ◽  
Structure Problem

The data-driven polynomial chaos expansion (DD-PCE) method is claimed to be a more general approach of uncertainty propagation (UP). However, as a common problem of all the full PCE approaches, the size of polynomial terms in the full DD-PCE model is significantly increased with the dimension of random inputs and the order of PCE model, which would greatly increase the computational cost especially for high-dimensional and highly non-linear problems. Therefore, a sparse DD-PCE is developed by employing the least angle regression technique and a stepwise regression strategy to adaptively remove some insignificant terms. Through comparative studies between sparse DD-PCE and the full DD-PCE on three mathematical examples with random input of raw data, common and nontrivial distributions, and a ten-bar structure problem for UP, it is observed that generally both methods yield comparably accurate results, while the computational cost is significantly reduced by sDD-PCE especially for high-dimensional problems, which demonstrates the effectiveness and advantage of the proposed method.

scholarly journals Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation

2018 ◽  
Vol 16 (1) ◽  
pp. 1651-1666 ◽  
Author(s):  
Julia Calatayud ◽  
Juan Carlos Cortés ◽  
Marc Jornet
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Galerkin Projection ◽  
Comparative Case Study ◽  
Random Inputs ◽  
Frobenius Method ◽  
Order Linear ◽  
Generalized Polynomial ◽  
Linear Differential

AbstractThis paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Fröbenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Fröbenius method, in which the random input coefficients may be expressed via a Karhunen-Loève expansion.

scholarly journals Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems

2020 ◽  
Vol 1 (3) ◽  
pp. 153-173
Author(s):  
Jeongeun Son ◽  
Dongping Du ◽  
Yuncheng Du
Keyword(s):  
Uncertainty Quantification ◽  
Polynomial Chaos ◽  
Biological Systems ◽  
Parametric Uncertainty ◽  
Polynomial Chaos Expansion ◽  
High Dimensional ◽  
Chaos Expansion ◽  
Deterministic Models ◽  
Generalized Dimension ◽  
The Impact

Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems.

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