Volume 36, Issue 2
Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

Commun. Math. Res., 36 (2020), pp. 128-153.

Published online: 2020-05

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• Abstract

One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via $ℓ$1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned $ℓ$1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures (such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.

• Keywords

Uncertainty quantification, data-driven polynomial chaos expansions, sparse approximation, equilibrium measure, induced measure.

41A10, 65D15, 62E17

11749318@mail.sustech.edu.cn (Yongle Liu)

tzhou@lsec.cc.ac.cn (Tao Zhou)

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@Article{CMR-36-128, author = {Ling and Guo and and 7670 and and Ling Guo and Akil and Narayan and and 7671 and and Akil Narayan and Yongle and Liu and 11749318@mail.sustech.edu.cn and 5846 and Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China and Yongle Liu and Tao and Zhou and tzhou@lsec.cc.ac.cn and 7042 and CMIS and LSEC, Institute of Computational Mathematics and Scientific/ Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China and Tao Zhou}, title = {Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {2}, pages = {128--153}, abstract = {

One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via $ℓ$1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned $ℓ$1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures (such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0010}, url = {http://global-sci.org/intro/article_detail/cmr/16926.html} }
TY - JOUR T1 - Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach AU - Guo , Ling AU - Narayan , Akil AU - Liu , Yongle AU - Zhou , Tao JO - Communications in Mathematical Research VL - 2 SP - 128 EP - 153 PY - 2020 DA - 2020/05 SN - 36 DO - http://doi.org/10.4208/cmr.2020-0010 UR - https://global-sci.org/intro/article_detail/cmr/16926.html KW - Uncertainty quantification, data-driven polynomial chaos expansions, sparse approximation, equilibrium measure, induced measure. AB -

One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs. Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems, particularly with high dimensional random parameters. We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approximation via $ℓ$1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned $ℓ$1 minimization problem, we sample from the so-called induced distribution, instead of using Monte Carlo (MC) sampling from the original, unknown probability measure. We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures (such as the equilibrium measure) when we have incomplete information about the distribution. We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus.

Ling Guo, Akil Narayan, Yongle Liu & Tao Zhou. (2020). Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach. Communications in Mathematical Research . 36 (2). 128-153. doi:10.4208/cmr.2020-0010
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