Galerkin proper orthogonal decomposition methods for parabolic problems
TL;DR: In this article, error bounds for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved and the resulting error bounds depend on the number of POD basis functions and on the time discretization.
Abstract: In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included.
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TL;DR: 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).
Abstract: 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.
1,695 citations
Cites background or methods from "Galerkin proper orthogonal decompos..."
...The separate POD basis used in DEIM to approximate the nonlinearity is very closely related Kunish-Volkwein's inclusion of finite difference snapshots [49]1....
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...Nonlinear problems with Lipschitz continuous nonlinearities are considered in [49] and extended to the Navier-Stokes equations in [50]....
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...1) where the matrix A E Rnxn is constant and the nonlinear function F: [0, T] ~ Y is assumed to be uniformly Lipschitz continuous with respect to the second argument 1 In [49], the finite difference snapshots of the form (YHl- Yi)/h are included into the snapshot set....
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...Similar approaches for deriving the error estimates in the function space setting from [49, 50] were later applied within a finite dimensional Euclidean space setting in [94]....
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...In [49, 50], Kunish and Volkwein derive error estimates for a POD reduced system for a class of nonlinear parabolic PDEs....
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TL;DR: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved and the backward Euler scheme is considered.
Abstract: Error estimates for Galerkin proper orthogonal decomposition (POD) methods for nonlinear parabolic systems arising in fluid dynamics are proved For the time integration the backward Euler scheme is considered The asymptotic estimates involve the singular values of the POD snapshot set and the grid-structure of the time discretization as well as the snapshot locations
752 citations
Cites background or methods or result from "Galerkin proper orthogonal decompos..."
...Third, we focus in this paper on a different class of nonlinearities, including the Navier–Stokes equations in dimension two, which were not included in [16]....
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...It appears that, except for the work in [16], these issues have not been addressed....
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...We extend our earlier results of [16] in three directions....
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TL;DR: In many situations across computational science and engineering, multiple computational models are available that describe a system of interest as discussed by the authors, and these different models have varying evaluation costs, i.e.
Abstract: In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs...
678 citations
TL;DR: A new definition of PGD is introduced, called Minimax PGD, which can be interpreted as a Petrov–Galerkin model reduction technique, where test and trial reduced basis functions are related by an adjoint problem, and improves convergence properties of separated representations with respect to a chosen metric.
298 citations
Cites background from "Galerkin proper orthogonal decompos..."
...Key words: Model reduction, Evolution problems, Proper Orthogonal Decomposition (POD), Proper Generalized Decomposition (PGD), Generalized Spectral Decomposition, Separation of variables Preprint February 10, 2010 ha l-0 04 55 63 5, v er si on 1 - 10 F eb 2...
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01 Jan 2005
TL;DR: In this article, the authors consider the problem of nonlinear evolution in real separable Hilbert spaces, where the inner product in V is given by a symmetric bounded, coercive, bilinear form.
Abstract: Nonlinear Dynamical System Let V and H be real separable Hilbert spaces and suppose that V is dense in H with compact embedding. By 〈· , ·〉H we denote the inner product in H. The inner product in V is given by a symmetric bounded, coercive, bilinear form a : V × V → IR: 〈φ,ψ〉V = a(φ,ψ) for all φ,ψ ∈ V (10.16) with associated norm given by ‖ · ‖V = √ a(· , ·). Since V is continuously injected into H, there exists a constant cV > 0 such that ‖φ‖H ≤ cV ‖φ‖V for all φ ∈ V. (10.17) We associate with a the linear operator A: 〈Aφ,ψ〉V ′,V = a(φ,ψ) for all φ,ψ ∈ V, where 〈· , ·〉V ′,V denotes the duality pairing between V and its dual. Then, by the Lax-Milgram lemma, A is an isomorphism from V onto V ′. Alternatively, A can be considered as a linear unbounded self-adjoint operator in H with domain D(A) = {φ ∈ V : Aφ ∈ H}. By identifying H and its dual H ′ it follows that 10 POD: Error Estimates and Suboptimal Control 269 D(A) ↪→ V ↪→ H = H ′ ↪→ V ′, each embedding being continuous and dense, when D(A) is endowed with the graph norm of A. Moreover, let F : V × V → V ′ be a bilinear continuous operator mapping D(A) × D(A) into H. To simplify the notation we set F (φ) = F (φ,φ) for φ ∈ V . For given f ∈ C([0, T ];H) and y0 ∈ V we consider the nonlinear evolution problem d dt 〈y(t), φ〉H + a(y(t), φ) + 〈F (y(t)), φ〉V ′,V = 〈f(t), φ〉H (10.18a) for all φ ∈ V and t ∈ (0, T ] a.e. and y(0) = y0 in H. (10.18b) Assumption (A1). For every f ∈ C([0, T ];H) and y0 ∈ V there exists a unique solution of (10.18) satisfying y ∈ C([0, T ];V ) ∩ L(0, T ;D(A)) ∩H(0, T ;H). (10.19) Computation of the POD Basis Throughout we assume that Assumption (A1) holds and we denote by y the unique solution to (10.18) satisfying (10.19). For given n ∈ IN let 0 = t0 < t1 < . . . < tn ≤ T (10.20) denote a grid in the interval [0, T ] and set δtj = tj − tj−1, j = 1, . . . , n. Define ∆t = max (δt1, . . . , δtn) and δt = min (δt1, . . . , δtn). (10.21) Suppose that the snapshots y(tj) of (10.18) at the given time instances tj , j = 0, . . . , n, are known. We set V = span {y0, . . . , y2n}, where yj = y(tj) for j = 0, . . . , n, yj = ∂ty(tj−n) for j = n + 1, . . . , 2n with ∂ty(tj) = (y(tj)−y(tj−1))/δtj , and refer to V as the ensemble consisting of the snapshots {yj} j=0, at least one of which is assumed to be nonzero. Furthermore, we call {tj}j=0 the snapshot grid. Notice that V ⊂ V by construction. Throughout the remainder of this section we let X denote either the space V or H. 270 Michael Hinze and Stefan Volkwein Remark 10.2.1 (compare [KV01, Remark 1]). It may come as a surprise at first that the finite difference quotients ∂ty(tj) are included into the set V of snapshots. To motivate this choice let us point out that while the finite difference quotients are contained in the span of {yj} j=0, the POD bases differ depending on whether {∂ty(tj)}j=1 are included or not. The linear dependence does not constitute a difficulty for the singular value decomposition which is required to compute the POD basis. In fact, the snapshots themselves can be linearly dependent. The resulting POD basis is, in any case, maximally linearly independent in the sense expressed in (P ) and Proposition 10.2.5. Secondly, in anticipation of the rate of convergence results that will be presented in Section 10.3.3 we note that the time derivative of y in (10.18) must be approximated by the Galerkin POD based scheme. In case the terms {∂ty(tj)}j=1 are included in the snapshot ensemble, we are able to utilize the estimate
294 citations
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5,359 citations
Book•
10 Sep 1993
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.
5,038 citations
Book•
01 Jan 1996TL;DR: In this article, the authors present a review of rigor properties of low-dimensional models and their applications in the field of fluid mechanics. But they do not consider the effects of random perturbation on models.
Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.
2,920 citations
Book•
01 Jun 1984
TL;DR: The standard Galerkin method is based on more general approximations of the elliptic problem as discussed by the authors, and is used to solve problems in algebraic systems at the time level.
Abstract: The Standard Galerkin Method.- Methods Based on More General Approximations of the Elliptic Problem.- Nonsmooth Data Error Estimates.- More General Parabolic Equations.- Negative Norm Estimates and Superconvergence.- Maximum-Norm Estimates and Analytic Semigroups.- Single Step Fully Discrete Schemes for the Homogeneous Equation.- Single Step Fully Discrete Schemes for the Inhomogeneous Equation.- Single Step Methods and Rational Approximations of Semigroups.- Multistep Backward Difference Methods.- Incomplete Iterative Solution of the Algebraic Systems at the Time Levels.- The Discontinuous Galerkin Time Stepping Method.- A Nonlinear Problem.- Semilinear Parabolic Equations.- The Method of Lumped Masses.- The H1 and H?1 Methods.- A Mixed Method.- A Singular Problem.- Problems in Polygonal Domains.- Time Discretization by Laplace Transformation and Quadrature.
1,864 citations
TL;DR: POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation to comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system.
Abstract: Proper orthogonal decomposition (POD) is a method to derive reduced-order models for dynamical systems In this paper, POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation The relative simplicity of the equation allows comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system For closed-loop control, suboptimal state feedback strategies are presented
433 citations