On Polynomial Chaos Expansion via Gradient-enhanced ${\ell }_1$-minimization

Abstract

Gradient-enhanced Uncertainty Quantification (UQ) has received recent attention, in which the derivatives of a Quantity of Interest (QoI) with respect to the uncertain parameters are utilized to improve the surrogate approximation. Polynomial chaos expansions (PCEs) are often employed in UQ, and when the QoI can be represented by a sparse PCE, ${\ell }_1$-minimization can identify the PCE coefficients with a relatively small number of samples. In this work, we investigate a gradient-enhanced ${\ell }_1$-minimization, where derivative information is computed to accelerate the identification of the PCE coefficients. For this approach, stability and convergence analysis are lacking, and thus we address these here with a probabilistic result. In particular, with an appropriate normalization, we show the inclusion of derivative information will almost-surely lead to improved conditions, e.g. related to the null-space and coherence of the measurement matrix, for a successful solution recovery. Further, we demonstrate our analysis empirically via three numerical examples: a manufactured PCE, an elliptic partial differential equation with random inputs, and a plane Poiseuille flow with random boundaries. These examples all suggest that including derivative information admits solution recovery at reduced computational cost.

References in corpus (5)

• Compressive Sampling of Polynomial Chaos Expansions: Convergence Analysis and Sampling Strategies
• A weighted L1-minimization approach for sparse polynomial chaos expansions
• Enhancing ${\ell }_1$-minimization estimates of polynomial chaos expansions using basis selection
• Coherence Motivated Sampling and Convergence Analysis of Least-Squares Polynomial Chaos Regression
• Interpolation via weighted $l_1$ minimization

Cited by in corpus (18)

• Polynomial approximation via compressed sensing of high-dimensional functions on lower sets
• Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design
• Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: application to urban drainage simulation
• Sparse Identification of Nonlinear Dynamical Systems via Reweighted ${\ell }_1$-regularized Least Squares
• Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations
• Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC)
• A gradient enhanced ${\ell }_1$-minimization for sparse approximation of polynomial chaos expansions
• Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
• Learning Dynamical Systems and Bifurcation via Group Sparsity
• A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness
• Extracting structured dynamical systems using sparse optimization with very few samples
• A mixed ${\ell }_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs
• Nonlinear dimension reduction for surrogate modeling using gradient information
• Global Sensitivity Analysis via Multi-Fidelity Polynomial Chaos Expansion
• Identifying the Influential Inputs for Network Output Variance Using Sparse Polynomial Chaos Expansion
• Data-Driven Sensitivity Indices for Models With Dependent Inputs Using the Polynomial Chaos Expansion
• Sliced-Inverse-Regression-Aided Rotated Compressive Sensing Method for Uncertainty Quantification
• Global sensitivity analysis using derivative-based sparse Poincaré chaos expansions