Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates

Peherstorfer, Benjamin; Willcox, Karen

doi:10.1137/140989169

Cited by 188 publications

(121 citation statements)

References 39 publications

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“…The surrogate models are constructed with supervised learning techniques. Another body of work combines the high-fidelity model with a surrogate model in the context of the Monte Carlo method with importance sampling [31,30,13,41,38].…”

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confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

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209

233

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Abstract. This work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high-fidelity models. Existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs; however, a general collection of surrogate models, which may include projection-based reduced models, data-fit models, support vector machines, and simplified-physics models, does not necessarily give rise to such a hierarchy. Our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type. Instead of relying on error and cost rates, an optimization problem balances the number of model evaluations across the high-fidelity and surrogate models with respect to error and costs. We show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models. Our multifidelity method makes occasional recourse to the high-fidelity model; in doing so it provides an unbiased estimator of the statistics of the high-fidelity model, even in the absence of error bounds and error estimators for the surrogate models. Numerical experiments with linear and nonlinear examples show that speedups by orders of magnitude are obtained compared to Monte Carlo estimation that invokes a single model only. 1. Introduction. Multilevel techniques have a long and successful history in computational science and engineering, e.g., multigrid for solving systems of equations [8,25,9], multilevel discretizations for representing functions [50,18,10], and multilevel Monte Carlo and multilevel stochastic collocation for estimating mean solutions of partial differential equations (PDEs) with stochastic parameters [27,22,45]. These multilevel techniques typically start with a fine-grid discretization-a high-fidelity model-of the underlying PDE or function. The fine-grid discretization is chosen to guarantee an approximation of the output of interest with the accuracy required by the current problem at hand. Additionally, a hierarchy of coarser discretizations-lowerfidelity surrogate models-is constructed, where a parameter (e.g., mesh width) controls the trade-off between error and computational costs. Changing this parameter gives rise to a multilevel hierarchy of discretizations with known error and cost rates. Multilevel techniques use these error and cost rates to distribute the computational work among the discretizations in the hierarchy, shifting most of the work onto the

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mentioning

confidence: 99%

Optimal Model Management for Multifidelity Monte Carlo Estimation

Peherstorfer¹,

Willcox²,

Gunzburger³

2016

SIAM J. Sci. Comput.

Self Cite

209

233

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“…Based on local error indicators (see [21][22][23][24] for local error indicators for GMsFEM), some local multiscale basis functions are updated and used to compute the online global modes inexpensively. For the adaption of DEIM, we employ a modified low rank updates method investigated in [25], which introduces computational costs that scale linearly in the number of unknowns of the full-order system. We remark that the offline local-global approaches have been studied in the literature [26][27][28][29].…”

Section: Of 21mentioning

confidence: 99%

“…In this section, we summarize the idea of online adaptive DEIM as in [25]. First, we adapt the DEIM space by rank one updates.…”

Section: Online Adaptive Deimmentioning

confidence: 99%

“…It is shown in [25] that the run time cost of the adaption scales linearly with the number of degrees of freedom of the full-order system. Algorithm 2 Online Adaptive DEIM [25] 1: INPUT : (U k−1 , P k−1 ), number of sampling points m s , window size w 2: Select m s points from {1, 2, · · · , n} that are not points in P k−1 . 3: Assemble sampling points matrix S k by combining P k−1 and the selected points.…”

Section: Online Adaptive Deimmentioning

confidence: 99%

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Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

2016

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Abstract:We propose an online adaptive local-global POD-DEIM model reduction method for flows in heterogeneous porous media. The main idea of the proposed method is to use local online indicators to decide on the global update, which is performed via reduced cost local multiscale basis functions. This unique local-global online combination allows (1) developing local indicators that are used for both local and global updates (2) computing global online modes via local multiscale basis functions. The multiscale basis functions consist of offline and some online local basis functions. The approach used for constructing a global reduced system is based on Proper Orthogonal Decomposition (POD) Galerkin projection. The nonlinearities are approximated by the Discrete Empirical Interpolation Method (DEIM). The online adaption is performed by incorporating new data, which become available at the online stage. Once the criterion for updates is satisfied, we adapt the reduced system online by changing the POD subspace and the DEIM approximation of the nonlinear functions. The main contribution of the paper is that the criterion for adaption and the construction of the global online modes are based on local error indicators and local multiscale basis function which can be cheaply computed. Since the adaption is performed infrequently, the new methodology does not add significant computational overhead associated with when and how to adapt the reduced basis. Our approach is particularly useful for situations where it is desired to solve the reduced system for inputs or controls that result in a solution outside the span of the snapshots generated in the offline stage. Our method also offers an alternative of constructing a robust reduced system even if a potential initial poor choice of snapshots is used. Applications to single-phase and two-phase flow problems demonstrate the efficiency of our method.

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“…This means that it is not possible to select more point indices than there are basis vectors spanning U. This is a limiting factor, since there are application scenarios where oversampling is beneficial (see [1]) or even explicitly required (see [13]). …”

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confidence: 99%

An Accelerated Greedy Missing Point Estimation Procedure

Zimmermann¹,

Willcox²

2016

SIAM J. Sci. Comput.

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Abstract. Model reduction via Galerkin projection fails to provide considerable computational savings if applied to general nonlinear systems. This is because the reduced representation of the state vector appears as an argument to the nonlinear function, whose evaluation remains as costly as for the full model. Masked projection approaches, such as the missing point estimation and the (discrete) empirical interpolation method, alleviate this effect by evaluating only a small subset of the components of a given nonlinear term; however, the selection of the evaluated components is a combinatorial problem and is computationally intractable even for systems of small size. This has been addressed through greedy point selection algorithms, which minimize an error indicator by sequentially looping over all components. While doable, this is suboptimal and still costly. This paper introduces an approach to accelerate and improve the greedy search. The method is based on the observation that the greedy algorithm requires solving a sequence of symmetric rank-one modifications to an eigenvalue problem. For doing so, we develop fast approximations that sort the set of candidate vectors that induce the rank-one modifications, without requiring solution of the modified eigenvalue problem. Based on theoretical insights into symmetric rank-one eigenvalue modifications, we derive a variation of the greedy method that is faster than the standard approach and yields better results for the cases studied. The proposed approach is illustrated by numerical experiments, where we observe a speed-up by two orders of magnitude when compared to the standard greedy method while arriving at a better quality reduced model.

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Online Adaptive Model Reduction for Nonlinear Systems via Low-Rank Updates

Cited by 188 publications

References 39 publications

Optimal Model Management for Multifidelity Monte Carlo Estimation

Optimal Model Management for Multifidelity Monte Carlo Estimation

Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

An Accelerated Greedy Missing Point Estimation Procedure

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