An `empirical interpolation' method: Application to efficient reduced-basis discretization of partial differential equations

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DOI10.1016/j.crma.2004.08.006zbMath1061.65118OpenAlexW2047591100MaRDI QIDQ1763517

Anthony T. Patera, Maxime Barrault, Yvon Maday, Ngoc Cuong Nguyen

Publication date: 22 February 2005

Published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.crma.2004.08.006

zbMATH Keywords

numerical results inverse problem finite element method error bound elliptic equation a posteriori estimator interpolation procedure collateral reduced-basis expansion nonaffine parameter dependence reduced-basis discretization

Mathematics Subject Classification ID

Boundary value problems for second-order elliptic equations (35J25) Error bounds for boundary value problems involving PDEs (65N15) Inverse problems for PDEs (35R30) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Numerical methods for inverse problems for boundary value problems involving PDEs (65N21)

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Mapping Automation for Parametric Component Reduced Basis Techniques (RB-COMPONENT), (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov--Galerkin Methods, Optimized Sampling for Multiscale Dynamics, Randomized Residual-Based Error Estimators for Parametrized Equations, Efficient Reduced Basis Methods for Saddle Point Problems with Applications in Groundwater Flow, Reduced Basis Methods for Uncertainty Quantification, Simultaneous Empirical Interpolation and Reduced Basis Method: Application to Non-linear Multi-Physics Problem, Reduced-Order Semi-Implicit Schemes for Fluid-Structure Interaction Problems, Efficient Reduction of PDEs Defined on Domains with Variable Shape, Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry, Leveraging Sparsity and Compressive Sensing for Reduced Order Modeling, Energy Stable Model Order Reduction for the Allen-Cahn Equation, Model Order Reduction of Nonlinear Eddy Current Problems Using Missing Point Estimation, Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis, Estimation of Regularization Parameters in Elliptic Optimal Control Problems by POD Model Reduction, The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena, EVALUATION OF FLUX INTEGRAL OUTPUTS FOR THE REDUCED BASIS METHOD, Space--Time Least-Squares Petrov--Galerkin Projection for Nonlinear Model Reduction, Effective equations governing an active poroelastic medium, Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models, Efficient reduced models anda posteriorierror estimation for parametrized dynamical systems by offline/online decomposition, Fast Bayesian approach for parameter estimation, Numerical solution of parametrized Navier–Stokes equations by reduced basis methods, A New Selection Operator for the Discrete Empirical Interpolation Method---Improved A Priori Error Bound and Extensions, A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement, Functional Regression for State Prediction Using Linear PDE Models and Observations, Energy Stable Model Reduction of Neurons by Nonnegative Discrete Empirical Interpolation, A discontinuous Galerkin reduced basis element method for elliptic problems, Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion, A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems, A ‘best points’ interpolation method for efficient approximation of parametrized functions, A DEIM Induced CUR Factorization, Subspace Acceleration for Large-Scale Parameter-Dependent Hermitian Eigenproblems, Convergence analysis of the Generalized Empirical Interpolation Method, The Random Feature Model for Input-Output Maps between 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wave equation, Nonintrusive Reduced Order Modelling of Convective Boussinesq Flows, Manifold Approximations via Transported Subspaces: Model Reduction for Transport-Dominated Problems, Learning physics-based models from data: perspectives from inverse problems and model reduction, Reduced basis methods for time-dependent problems, A Simplified Newton Method to Generate Snapshots for POD Models of Semilinear Optimal Control Problems, Fast $L^2$ Optimal Mass Transport via Reduced Basis Methods for the Monge--Ampère Equation, An Adaptive Sampling Approach for the Reduced Basis Method, Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control, Online Multiscale Model Reduction for Nonlinear Stochastic PDEs with Multiplicative Noise, Simulation and Control of a Nonsmooth Cahn–Hilliard Navier–Stokes System with Variable Fluid Densities, A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations, Tensor-Based Numerical Method for Stochastic Homogenization, A quadratic bilinear equation arising from the quadratic dynamical system, A Generalized CUR Decomposition for Matrix Pairs, Model reduction for reacting flow applications, Legendre spectral element method (LSEM) to simulate the two-dimensional system of nonlinear stochastic advection–reaction–diffusion models, A posteriorierror bounds for reduced-basis approximations of parametrized parabolic partial differential equations, Reduced Operator Inference for Nonlinear Partial Differential Equations, Piecewise-Global Nonlinear Model Order Reduction for PDE-Constrained Optimization in High-Dimensional Parameter Spaces, Efficient and certified solution of parametrized one-way coupled problems through DEIM-based data projection across non-conforming interfaces, \(\boldsymbol{\mathcal{L}_2}\)-Optimal Reduced-Order Modeling Using Parameter-Separable Forms, A modelling framework for efficient reduced order simulations of parametrised lithium-ion battery 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equations by space-time tensor Galerkin methods, Data-Driven Time Parallelism via Forecasting, Weakly Intrusive Low-Rank Approximation Method for Nonlinear Parameter-Dependent Equations, An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids, Structure-preserving reduced basis methods for Poisson systems, 1 Model reduction in chemical process optimization, 6 Model reduction in computational aerodynamics, 8 Reduced-order modeling for applications to the cardiovascular system, Modeling synchronization in globally coupled oscillatory systems using model order reduction, Projection-based model reduction with dynamically transformed modes, Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, Reduced Basis Methods for Fractional Laplace Equations via Extension, A Least-Squares Finite Element Reduced Basis Method, Stein Variational Reduced Basis Bayesian Inversion, Numerical Model Construction with Closed Observables, Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics, Two-Sided Projection Methods for Nonlinear Model Order Reduction, Reduced basis techniques for nonlinear conservation laws, On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition, Valuation of Structured Financial Products by Adaptive Multiwavelet Methods in High Dimensions, Analysis of a Greedy Reconstruction Algorithm, Reduced Basis Methods for Quasilinear Elliptic PDEs with Applications to Permanent Magnet Synchronous Motors, Structure-Preserving Reduced- Order Modeling of Non-Traditional Shallow Water Equation, Reduced-Order Modeling and ROM-Based Optimization of Batch Chromatography, Model Order Reduction of Differential Algebraic Equations Arising from the Simulation of Gas Transport Networks, MicroROM: An efficient and accurate reduced order method to solve many-query problems in micro-motility, Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier–Stokes equations with model order reduction, Reduced Basis Multiscale Finite Element Methods for Elliptic Problems, Data-driven kinematics-consistent model-order reduction of fluid–structure interaction problems: application to deformable microcapsules in a Stokes flow, A Reduced-Basis Approach to Two-Phase Flow in Porous Media

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