Dimension Reduction of Large-Scale Systems

Benner, Peter; Mehrmann, Volker; C., Sorensen Danny

doi:10.1007/3-540-27909-1

Cited by 247 publications

(17 citation statements)

References 140 publications

(250 reference statements)

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“…The dimension of the state space is then reduced to say k N through projection onto a low-dimensional subspace. Techniques for dimension reduction include the Proper Orthogonal Decomposition (POD) [6], [5], [22], [32], reduced basis methods, see, e.g., [41], and rational interpolation strategies [1] to mention a few. Furthermore, the complexity of the reduced model can be further decreased through hyper-reduction of the nonlinear term.…”

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confidence: 99%

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Kirsten

2022

JCD

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<p style='text-indent:20px;'>We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.</p>

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confidence: 99%

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Kirsten

2022

JCD

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“…The solution to this minimization problem is given by the SVD of snapshot matrix X(ζ) [4] and selecting the first r columns of the left projection matrix.…”

Section: Proper Orthogonal Decompositionmentioning

confidence: 99%

“…One such technique is Reduced-Order Modeling (ROM), which includes intrusive or non-intrusive projectionbased methods. They do not approximate the physics and rely on reducing the dimensionality of the state-space of fine-scale models [3,4], and thus are usually computationally cheaper than coarsened models [5,6] and more robust than data-driven models [7,8].…”

Section: Introductionmentioning

confidence: 99%

Two-Step Predict and Correct Non-Intrusive Parametric Model Order Reduction for Changing Well Locations Using a Machine Learning Framework

Zalavadia

Gildin

2021

Energies

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The objective of this paper is to develop a two-step predict and correct non-intrusive Parametric Model Order Reduction (PMOR) methodology for the problem of changing well locations in an oil field that can eventually be used for well placement optimization to gain significant computational savings. In this work, we propose a two-step PMOR procedure, where, in the first step, a Proper Orthogonal Decomposition (POD)-based strategy that is non-intrusive to the simulator source code is introduced, as opposed to the convention of using POD as a simulator intrusive procedure. The non-intrusiveness of the proposed technique stems from formulating a novel Machine Learning (ML)-based framework used with POD. The features of the ML model (Random Forest was used here) are designed such that they take into consideration the temporal evolution of the state solutions and thereby avoid simulator access for the time dependency of the solutions. The proposed PMOR method is global, since a single reduced-order model can be used for all the well locations of interest in the reservoir. We address the major challenge of the explicit representation of the well location change as a parameter by introducing geometry-based features and flow diagnostics-inspired physics-based features. In the second step, an error correction model based on reduced model solutions is formulated to correct for discrepancies in the state solutions at well grid blocks expected from POD basis for new well locations. The error correction model proposed uses Artificial Neural Networks (ANNs) that consider the physics-based reduced model solutions as features, and is proved to reduce the error in QoI (Quantities of Interest), such as oil production rates and water cut, significantly. This workflow is applied to a simple homogeneous reservoir and a heterogeneous channelized reservoir using a section of SPE10 model that showed promising results in terms of model accuracy. Speed-ups of about 50×–100× were observed for different cases considered when running the test scenarios. The proposed workflow for Reduced-Order Modeling is “non-intrusive” and hence can increase its applicability to any simulator used. Additionally, the method is formulated such that all the simulation time steps are independent and hence can make use of parallel resources very efficiently and also avoid stability issues that can result from error accumulation over time steps.

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“…In case of controller design, simulation and design optimization, working with these large-scale systems is often problematic due to computational complexity and storage requirements. The idea of model order reduction (MOR) [3,13,28] is to approximate a large scale dynamical system by a substantially lower dimensional system which has nearly the same input-output behavior. This lower dimensional model then, e.g.…”

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confidence: 99%

“…serves as the basis for feedback control design. The system theoretic method balanced truncation (BT) [3,13] is a particular technique which preserves important properties of the system (such as asymptotic stability) and features an a priori bound on the approximation error.…”

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confidence: 99%

Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control

Benner

Saak

Uddin

2016

NACO

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Stabilizing a flow around an unstable equilibrium is a typical problem in flow control. Model-based designed of modern controllers like LQR/LQG or H∞ compensators is often limited by the large-scale of the discretized flow models. Therefore, model reduction is usually needed before designing such a controller. Here we suggest an approach based on applying balanced truncation for unstable systems to the linearized flow equations usually used for compensator design. For this purpose, we modify the ADI iteration for Lyapunov equations to deal with the index-2 structure of the underlying descriptor system efficiently in an implicit way. The resulting algorithm is tested for model reduction and control design of a linearized Navier-Stokes system describing von Kármán vortex shedding.2010 Mathematics Subject Classification. Primary: 93B40, 93B11; Secondary: 65F30, 65F50. Key words and phrases. balanced truncation, unstable system, differential algebraic system, differential index-2, linearized flow model. 1 2 P. BENNER, J. SAAK AND M. M. UDDIN

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Dimension Reduction of Large-Scale Systems

Cited by 247 publications

References 140 publications

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Two-Step Predict and Correct Non-Intrusive Parametric Model Order Reduction for Changing Well Locations Using a Machine Learning Framework

Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control

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