Real and Complex Analysis

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.

/pdf/nonlinear-model-reduction-via-discrete-empirical-f3tjqvqhuf.pdf

Nonlinear Model Reduction via Discrete Empirical Interpolation

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

/pdf/a-survey-of-projection-based-model-reduction-methods-for-3yjqbxk305.pdf

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

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

Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

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.

Galerkin proper orthogonal decomposition methods for parabolic problems

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

Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition

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

Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

Proper orthogonal decomposition (POD) is a powerful technique for model reduction of non-linear systems. It is based on a Galerkin type discretization with basis elements created from the dynamical system itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. A method is proposed which avoids this problem of unmodelled dynamics in the proper orthogonal decomposition approach to optimal control. It is referred to as optimality system proper orthogonal decomposition (OS-POD).

/pdf/proper-orthogonal-decomposition-for-optimality-systems-55m4keh5tg.pdf

Stefan Volkwein

Papers

Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

Galerkin proper orthogonal decomposition methods for parabolic problems

Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition

Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

Proper orthogonal decomposition for optimality systems