Reduced order modeling for fluid flows based on nonlinear balanced truncation

Sahyoun, Samir; Dong, Jin; Djouadi, Seddik M.

doi:10.1109/acc.2013.6580013

Cited by 7 publications

(7 citation statements)

References 30 publications

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“…Part 2 [34] of this paper will further investigate the nonlinear balancing problem, including a novel balance-and-reduce framework for nonlinear systems and the corresponding manifold approximations. A further benefit of this approach is that the energy functions in the H ∞ and HJB balancing methods automatically (as a byproduct) produce controllers for nonlinear systems, see [51] and [54].…”

Section: Discussionmentioning

confidence: 99%

“…Their subsequent balanced ROMs are illustrated on a four-dimensional ODE. Building on [22], the authors in [51] consider the special class of quadratic models and use Taylor series expansion to solve the HJB equations and balancing transformations. However, no numerical examples are given, and the computational feasibility for large-scale systems remains in question.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Balanced Truncation: Part 1 -- Computing Energy Functions

Kramer¹,

Gugercin²,

Borggaard³

2022

Preprint

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Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open-or closed-loop observability and controllability aspects of the system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the energy functions required for characterization of controllability and observability are solutions of high-dimensional Hamilton-Jacobi-(Bellman) equations, and (b) efficient model reduction and subsequent reduced-order model (ROM) simulation on the resulting nonlinear balanced manifolds. This work proposes a unifying and scalable approach to the challenge (a) by considering a Taylor series-based approach to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of the balancing approach. The value of a formulation parameter provides either openloop balancing or a variety of closed-loop balancing options. To solve for coefficients of the Taylorseries approximation to the energy functions, the presented method derives a linear tensor structure and heavily utilizes this to solve structured linear systems with billions of unknowns. The strength and scalability of the algorithm is demonstrated on two semi-discretized partial differential equations, namely the Burgers equation and the Kuramoto-Sivashinsky equation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear Balanced Truncation: Part 1 -- Computing Energy Functions

Kramer¹,

Gugercin²,

Borggaard³

2022

Preprint

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“…Model reduction in general is an active research area in the last few decades because it is computationally difficult to design control laws for systems described by partial differential equations [19], [25], [8]. A very large number of states are needed to accurately capture the dynamics of such systems and this makes them unsuitable for control design [2], [5].…”

Section: Introductionmentioning

confidence: 99%

Time, space, and space-time hybrid clustering POD with application to the Burgers' equation

Sahyoun

Djouadi

2014

53rd IEEE Conference on Decision and Control

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In this paper, we investigate new methods that make the Proper Orthogonal Decomposition (POD) more accurate in reducing the order of large scale nonlinear systems. The general framework is to apply POD locally to clusters instead of applying it to the global system. Each cluster contains relatively close in distance behavior within itself, and considerably far with respect to other clusters. We introduce three different clustering schemes in time, space and space-time. For time clustering, time snapshots of the solution are grouped into clusters where the solution exhibits significantly different features and a local basis is pre-computed and assigned to each cluster. Space clustering is done in a similar fashion for the space vectors of the solution instead of snapshots, and finally space-time clustering is applied through a hybrid clustering scheme that combines space and time behavior together. We apply our method to reduce a nonlinear convective PDE system governed by the Burgers' equation for fluid flows over 1D and 2D domains and show a significant improvement over conventional POD.

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“…For nonlinear systems, Proper Orthogonal Decomposition (POD) is a model reduction technique that proved efficient performance when used to reduce models that approximate nonlinear infinite dimensional systems by high order finite dimensional systems, especially those who describe the dynamics of fluid flows [8], [4]. POD is a popular model reduction technique used to alleviate the computational expense required for very high dimensional systems.…”

Section: Introductionmentioning

confidence: 99%