Volume 10, Issue 1
Distributional Sensitivity for Uncertainty Quantification

Akil Narayan & Dongbin Xiu

Commun. Comput. Phys., 10 (2011), pp. 140-160.

Published online: 2011-10

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  • Abstract

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numerical model leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

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@Article{CiCP-10-140, author = {}, title = {Distributional Sensitivity for Uncertainty Quantification}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {1}, pages = {140--160}, abstract = {

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numerical model leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.160210.300710a}, url = {http://global-sci.org/intro/article_detail/cicp/7438.html} }
TY - JOUR T1 - Distributional Sensitivity for Uncertainty Quantification JO - Communications in Computational Physics VL - 1 SP - 140 EP - 160 PY - 2011 DA - 2011/10 SN - 10 DO - http://doi.org/10.4208/cicp.160210.300710a UR - https://global-sci.org/intro/article_detail/cicp/7438.html KW - AB -

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numerical model leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

Akil Narayan & Dongbin Xiu. (2020). Distributional Sensitivity for Uncertainty Quantification. Communications in Computational Physics. 10 (1). 140-160. doi:10.4208/cicp.160210.300710a
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