On the Convergence of Adaptive Stochastic Collocation for Elliptic Partial Differential Equations with Affine Diffusion

DOI10.1137/20M1364722WikidataQ114074156 ScholiaQ114074156MaRDI QIDQ5070555

Martin Eigel, Björn Sprungk, Oliver G. Ernst, Lorenzo Tamellini

Publication date: 13 April 2022

Published in: SIAM Journal on Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2008.07186

zbMATH Keywords

sparse gridsadaptive algorithmshigh-dimensional interpolationhigh-dimensional approximationparametric PDEsstochastic collocationrandom PDEs

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Random operators and equations (aspects of stochastic analysis) (60H25) Numerical interpolation (65D05) Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs (65N50) PDEs with randomness, stochastic partial differential equations (35R60) Numerical solutions to stochastic differential and integral equations (65C30) Algorithms for approximation of functions (65D15)

Related Items

Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part II: Multilevel Approximation ⋮ Error Estimation and Adaptivity for Stochastic Collocation Finite Elements Part I: Single-Level Approximation ⋮ Adaptive quasi-Monte Carlo finite element methods for parametric elliptic PDEs ⋮ Adaptive experimental design for multi‐fidelity surrogate modeling of multi‐disciplinary systems ⋮ Efficient adaptive stochastic collocation strategies for advection-diffusion problems with uncertain inputs

Uses Software

• ALEA

Cites Work

• Unnamed Item
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• High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs
• Adaptive stochastic Galerkin FEM
• Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
• Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations
• Sensitivity-driven adaptive sparse stochastic approximations in plasma microinstability analysis
• Sparse high order FEM for elliptic sPDEs
• An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations
• Dimension-adaptive tensor-product quadrature
• Smolyak cubature of given polynomial degree with few nodes for increasing dimension
• On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection
• Goal-oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain inputs
• Convergence of adaptive stochastic collocation with finite elements
• On expansions and nodes for sparse grid collocation of lognormal elliptic PDEs
• A fully adaptive multilevel stochastic collocation strategy for solving elliptic PDEs with random data
• Adaptive stochastic Galerkin FEM with hierarchical tensor representations
• An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient
• Comparison of Clenshaw–Curtis and Leja Quasi-Optimal Sparse Grids for the Approximation of Random PDEs
• Sparse, adaptive Smolyak quadratures for Bayesian inverse problems
• Energy Norm A Posteriori Error Estimation for Parametric Operator Equations
• New Bounds on the Lebesgue Constants of Leja Sequences on the Unit Disc and on $$\mathfrak {R}$$-Leja Sequences
• Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
• Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
• Optimized general sparse grid approximation spaces for operator equations
• ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S
• A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.
• ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS
• Local Equilibration Error Estimators for Guaranteed Error Control in Adaptive Stochastic Higher-Order Galerkin Finite Element Methods
• A convergent adaptive stochastic Galerkin finite element method with quasi-optimal spatial meshes
• A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
• An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
• On the Lebesgue Function for Polynomial Interpolation
• A Posteriori Error Estimation for the Stochastic Collocation Finite Element Method
• Multivariate Approximation in Downward Closed Polynomial Spaces
• Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)
• Uncertainty Quantification in CFD Simulations: A Stochastic Spectral Approach
• A Convergent Adaptive Algorithm for Poisson’s Equation
• Sparse quadrature for high-dimensional integration with Gaussian measure
• Convergence rates of high dimensional Smolyak quadrature
• On a certain class of Lebesgue constants
• Efficient Adaptive Multilevel Stochastic Galerkin Approximation Using Implicit A Posteriori Error Estimation
• Convergence of Adaptive Stochastic Galerkin FEM
• Adjoint Error Estimation for Stochastic Collocation Methods
• N-TERM WIENER CHAOS APPROXIMATION RATES FOR ELLIPTIC PDEs WITH LOGNORMAL GAUSSIAN RANDOM INPUTS
• High-Order Collocation Methods for Differential Equations with Random Inputs
• Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations
• A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data