Sparse Pseudospectral Approximation Method

Abstract

Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients of basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.

Cited by in corpus (44)

• Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial
• A weighted L1-minimization approach for sparse polynomial chaos expansions
• Enhancing ${\ell }_1$-minimization estimates of polynomial chaos expansions using basis selection
• Least Squares Polynomial Chaos Expansion: A Review of Sampling Strategies
• Spectral tensor-train decomposition
• Adaptive construction of surrogates for the Bayesian solution of inverse problems
• Gradient-based stochastic optimization methods in Bayesian experimental design
• Polynomial chaos expansions for dependent random variables
• Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations
• On Polynomial Chaos Expansion via Gradient-enhanced ${\ell }_1$-minimization
• A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions
• Numerical Integration in Multiple Dimensions with Designed Quadrature
• On the influence of over-parameterization in manifold based surrogates and deep neural operators
• Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models
• Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function
• To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case
• Gradient-based Optimization for Regression in the Functional Tensor-Train Format
• Efficient uncertainty propagation for network multiphysics systems
• Uncertainty Quantification of geochemical and mechanical compaction in layered sedimentary basins
• Divide and Conquer: An Incremental Sparsity Promoting Compressive Sampling Approach for Polynomial Chaos Expansions
• Kalman Filtering with Probabilistic Uncertainty in System Parameters
• Data-driven low-fidelity models for multi-fidelity Monte Carlo sampling in plasma micro-turbulence analysis
• Sensitivity-enhanced generalized polynomial chaos for efficient uncertainty quantification
• Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods
• Multivariate sensitivity-adaptive polynomial chaos expansion for high-dimensional surrogate modeling and uncertainty quantification
• Quantifying Uncertainties in Fault Slip Distribution during the Tōhoku Tsunami using Polynomial Chaos
• Polynomial Chaos-based Bayesian Inference of K-Profile Parametrization in a General Circulation Model of the Tropical Pacific
• A preconditioning approach for improved estimation of sparse polynomial chaos expansions
• A data-driven framework for sparsity-enhanced surrogates with arbitrary mutually dependent randomness
• Uncertainty Quantification of Oscillation Suppression during DBS in a Coupled Finite Element and Network Model
• Power Module Heat Sink Design Optimization with Ensembles of Data-Driven Polynomial Chaos Surrogate Models
• Design Analysis for Optimal Calibration of Diffusivity in Reactive Multilayers
• Adaptive Sparse Polynomial Chaos Expansions via Leja Interpolation
• Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification
• Global Sensitivity Analysis via Multi-Fidelity Polynomial Chaos Expansion
• A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration
• Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions using Weighted Leja Interpolation
• Stochastic trajectory optimization for mechanical systems with parametric uncertainties
• Parametric/Stochastic Model Reduction: Low-Rank Representation, Non-Intrusive Bi-Fidelity Approximation, and Convergence Analysis
• Thermal tomography with unknown boundary
• A Flexible Uncertainty Quantification Framework for General Multi-Physics Systems
• Noise-robust multi-fidelity surrogate modelling for parametric partial differential equations
• Multilevel Surrogate-based Control Variates
• Hierarchical surrogate-based Approximate Bayesian Computation for an electric motor test bench