Snapshot selection for groundwater model reduction using proper orthogonal decomposition

Siade, Adam; Putti, Mario; Yeh, William W.‐G.

doi:10.1029/2009wr008792

Cited by 58 publications

(70 citation statements)

References 23 publications

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“…POD has been shown to have the ability to reduce the dimension of a groundwater model by several orders of magnitude while maintaining more than 99% accuracy [McPhee and Yeh, 2008]. Siade et al [2010] demonstrated that by applying POD, a groundwater model that originally contained more than 200,000 spatial nodes could be reduced to a model containing only 10 spatial nodes, resulting in an approximately 1000 times increase in speed of solving the model. Note that the temporal dimension remains untouched.…”

Section: Experimental Designmentioning

confidence: 99%

“…at time t. Equation (6) is referred to as the full model. In most cases of interest matrices, A and B are large, sparse, and positive definite [Siade et al, 2010] (note that, throughout the text, a bold face letter indicates a matrix or vector, whereas a non-bold face letter indicates a scalar. For example, s t indicates the vector of all drawdown values at time t, whereas s t i denotes the drawdown in the ith node at time t).…”

Section: Confined Aquifer Groundwater-flow Modelmentioning

confidence: 99%

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Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

Ushijima

Yeh

2013

Water Resour. Res.

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[1] An optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined, anisotropic aquifer. The design uses a maximal information criterion that chooses, among competing designs, the design that maximizes the sum of squared sensitivities while conforming to specified design constraints. The formulated optimization problem is non-convex and contains integer variables necessitating a combinatorial search. Given a realistic large-scale model, the size of the combinatorial search required can make the problem difficult, if not impossible, to solve using traditional mathematical programming techniques. Genetic algorithms (GAs) can be used to perform the global search ; however, because a GA requires a large number of calls to a groundwater model, the formulated optimization problem still may be infeasible to solve. As a result, proper orthogonal decomposition (POD) is applied to the groundwater model to reduce its dimensionality. Then, the information matrix in the full model space can be searched without solving the full model. Results from a small-scale test case show identical optimal solutions among the GA, integer programming, and exhaustive search methods. This demonstrates the GA's ability to determine the optimal solution. In addition, the results show that a GA with POD model reduction is several orders of magnitude faster in finding the optimal solution than a GA using the full model. The proposed experimental design algorithm is applied to a realistic, two-dimensional, large-scale groundwater problem. The GA converged to a solution for this large-scale problem.Citation: Ushijima, T. T., and W. W.-G. Yeh (2013), Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model, Water Resour. Res., 49,[6688][6689][6690][6691][6692][6693][6694][6695][6696][6697][6698][6699]

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Section: Experimental Designmentioning

confidence: 99%

Section: Confined Aquifer Groundwater-flow Modelmentioning

confidence: 99%

Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

Ushijima

Yeh

2013

Water Resour. Res.

Self Cite

View full text Add to dashboard Cite

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“…Let Y 2 R n ´b e the matrix ofś napshots, Y D OEy r 1 ; : : : ; y r´, where r i , i D 1; : : : ;´are the time steps at which the solutions are collected. In the POD approach, the basis vectors are the m eigenvectors corresponding to the m largest eigenvalues of the matrix Y Y T [13,26]. This procedure resembles the Principal Component Analysis and for this reason the basis vectors are also called principal components.…”

Section: Projection-based Reduced Order Modelsmentioning

confidence: 99%

“…To reduce this cost, in many applications, the snapshots are collected only at the beginning of the temporal simulation, for example, as in Vermeulen et al [26], but this approach does not guarantee that the snapshots are representative of the full-model solution for the rest of the simulation. Siade et al [13] proposes an optimal distribution of the snapshots times during the simulation, thus avoiding the computation of the full-system model at each time step. In this work, POD is based on the set of all snapshots evaluated by the full model during the transient simulation over the entire time interval.…”

Section: Snapshot Techniquementioning

confidence: 99%

“…Model order reduction methods, such as Proper Orthogonal Decomposition (POD), are approaches designed to yield low-cost approximations to the solution of linear PDEs [9][10][11][12][13]. The main underlying idea is the Galerkin projection of the PDE onto the space generated by a few spatially distributed and linearly independent basis vectors (forward step).…”

Section: Introductionmentioning

confidence: 99%

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A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

Pasetto

Ferronato

Putti

2016

Numerical Meth Engineering

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SUMMARYThis paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive-definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced-order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced-order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full-model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full-model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade-off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models.

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An efficient framework for hydrologic model calibration on long data periods

Razavi

Tolson

2013

Water Resour. Res.

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.[1] Long periods of hydrologic data records have become available in many watersheds around the globe. Hydrologic model calibration on such long, full-length data periods is typically deemed the most robust approach for calibration but at larger computational costs. Determination of a representative short period as a ''surrogate'' of a long data period that sufficiently embeds its information content is not trivial and is a challenging research question. The representativeness of such a short period is not only a function of data characteristics but also model and calibration error function dependent. Unlike previous studies, this study goes beyond identifying the best surrogate data period to be used in model calibration and proposes an efficient framework that calibrates the hydrologic model to full-length data while running the model only on a short period for the majority of the candidate parameter sets. To this end, a mapping system is developed to approximate the model performance on the full-length data period based on the model performance for the short data period. The basic concepts and the promise of the framework are demonstrated through a computationally expensive hydrologic model case study. Three calibration approaches, namely calibration solely to a surrogate period, calibration to the full period, and calibration through the proposed framework, are evaluated and compared. Results show that within the same computational budget, the proposed framework leads to improved or equal calibration performance compared to the two conventional approaches. Results also indicate that model calibration solely to a short data period may lead to a range of performances from poor to very well depending on the representativeness of the short data period which is typically not known a priori.

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Snapshot selection for groundwater model reduction using proper orthogonal decomposition

Cited by 58 publications

References 23 publications

Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

Experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

An efficient framework for hydrologic model calibration on long data periods

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