Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach (Communications in Mathematical Research, 2022)
One of the open problems in the field of forward uncertainty quantification (UQ) is the ability to form accurate assessments of uncertainty havingonly incomplete information about the distribution of random inputs. Anotherchallenge is to efficiently make use of limited training data for UQ predictionsof complex engineering problems, particularly with high dimensional randomparameters. 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 processis to employ the procedure developed in [1] to construct an "arbitrary" polynomial chaos expansion basis using a finite number of statistical moments of therandom inputs. The second step is a novel procedure to effect sparse approximation via $ℓ$1 minimization in order to quantify the forward uncertainty. Toenhance the performance of the preconditioned $ℓ$1 minimization problem, wesample 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 betterchoice compared with sampling from asymptotically optimal measures (suchas the equilibrium measure) when we have incomplete information about thedistribution. We demonstrate the capacity of the proposed induced samplingalgorithm via sparse representation with limited data on test functions, and ona Kirchoff plating bending problem with random Young's modulus.