Sparse Approximation of Data-Driven Polynomial Chaos Expansions:An Induced Sampling Approach

Ling Guo; Akil Narayan; Yongle Liu; Tao Zhou

2020

10.4208/cmr.2020-0010

Communications in Mathematical Research   Impact Factor & Quartile

1674-5647

One of the open problems in the field of forward uncertainty quan-tification (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 polyno-mial chaos expansions with a recently developed preconditioned sparse ap-proximation approach for UQ problems. The first task in this two-step process is to employ the procedure developed in [1] to construct an"arbitrary"polyno-mial chaos expansion basis using a finite number of statistical moments of the random inputs. The second step is a novel procedure to effect sparse approx-imation via?1 minimization in order to quantify the forward uncertainty. To enhance the performance of the preconditioned?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 demon-strate 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.

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WanFang

dbsx-e202002002

1.Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2.Department of Mathematics, and Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, USA
3.Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China
4.LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Ling Guo,Akil Narayan,Yongle Liu,et al. Sparse Approximation of Data-Driven Polynomial Chaos Expansions:An Induced Sampling Approach[J]. Communications in Mathematical Research,2020,36(2):128-153.

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.

Ling Guo,et al."Sparse Approximation of Data-Driven Polynomial Chaos Expansions:An Induced Sampling Approach".Communications in Mathematical Research 36.2(2020):128-153.