A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

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Nonlinear Model Reduction via Discrete Empirical Interpolation

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

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Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

Generalized multiscale finite element methods (GMsFEM)

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

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Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We propose a new reduced-basis output bound method for the symmetric eigenvalue problem. The numerical procedure consists of two stages: the pre-processing stage, in which the reduced basis and associated functions are computed—“off-line”—at a prescribed set of points in parameter space; and the real-time stage, in which the approximate output of interest and corresponding rigorous error bounds are computed—“on-line”—for any new parameter value of interest. The real time calculation is very inexpensive as it requires only the solution or evaluation of very small systems. We introduce the procedure; prove the asymptotic bounding properties and optimal convergence rate of the error estimator; discuss computational considerations; and, finally, present corroborating numerical results.

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

Detailed calculation of heating, evaporation, and reaction processes of a thin liquid layer of hydrocarbon fuel

A one-dimensional model for crevice HC post-flame oxidation is used to calculate and understand the effect of operating parameters and fuel type (propane and isooctane) on the extent of crevice hydrocarbon and the product distribution in the post flame environment. The calculations show that the main parameters controlling oxidation are: bulk burned gas temperatures, wall temperatures, turbulent diffusivity, and fuel oxidation rates. Calculated extents of oxidation agree well with experimental values, and the sensitivities to operating conditions (wall temperatures, equivalence ratio, fuel type) are reasonably well captured. Whereas the bulk gas temperatures largely determine the extent of oxidation, the hydrocarbon product distribution is not very much affected by the burned gas temperatures, but mostly by diffusion rates. Uncertainties in both turbulent diffusion rates as well as in mechanisms are an important factor limiting the predictive capabilities of the model. However, it seems well suited to sensitivity calculations about a baseline. Copyright © 1999 Society of Automotive Engineers, Inc.

Effect of operating conditions and fuel type on crevice hc emissions: Model results and comparison with experiments

The solution of a single optimization problem often requires computationally-demanding evaluations; this is especially true in optimal design of engineering components and systems described by partial differential equations. We present a technique for the rapid and reliable optimization of systems characterized by linear-functional outputs of partial differential equations with affine parameter dependence. The critical ingredients of the method are: (i) reduced-basis techniques for dimension reduction in computational requirements; (ii) an “off-line/on-line” computational decomposition for the rapid calculation of outputs of interest and respective sensitivities in the limit of many queries; (iii) a posteriori error bounds for rigorous uncertainty and feasibility control; (iv) Interior Point Methods (IPMs) for efficient solution of the optimization problem; and (v) a trust-region Sequential Quadratic Programming (SQP) interpretation of IPMs for treatment of possibly non-convex costs and constraints.

Ivan B. Oliveira

Papers

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

Detailed calculation of heating, evaporation, and reaction processes of a thin liquid layer of hydrocarbon fuel

Effect of operating conditions and fuel type on crevice hc emissions: Model results and comparison with experiments

Reliable Real-Time Optimization of Nonconvex Systems Described by Parametrized Partial Differential Equations