Towards optimal allocation of computer resources: Trade-offs between uncertainty quantification, discretization and model reduction

Leube, P.; Barros, Felipe P. J. de; Nowak, Wolfgang; Rajagopal, Ram

doi:10.1016/j.envsoft.2013.08.008

Cited by 20 publications

(18 citation statements)

References 62 publications

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“…These empirical optimal points are denoted by opt, * and N opt, * mc . This is the approach adopted in Leube et al [22] We point out that the optimal results obtained from the full bruteforce numerical investigation (step 1) should be based on an error formulation consistent with the proposed error definition given in Eq. (13).…”

Section: Evaluating the Performance Of The Methodsmentioning

confidence: 96%

“…The derived expression for the overall error explicitly accounts for the contribution stemming from the physical discretization and the stochastic resolution. As opposed to Leube et al [22], the derived error metric does not rely on the assumption of statistical independence between the physical and stochastic errors. Through a series of examples, we show that the derived overall model response error is capable of efficiently handling the following conflicting objectives: computational cost, discretization error and statistical error.…”

Section: Introductionmentioning

confidence: 98%

“…They evaluated the accuracy of the reduced model as a function of the correlation scale and variance of the log-transmissivity without considering the interaction between computational budget, physical and statistical resolutions. In a recent work, Leube et al [22] quantified the optimal computational resource allocation by jointly considering the trade-off between uncertainty quantification and discretization and model reduction. They obtained cost-to-error surfaces which were generated by simulating different combinations of physical and statistical resolutions which are then used to determine the optimal points for a given computational constraint [22].…”

Section: Introductionmentioning

confidence: 99%

“…In a recent work, Leube et al [22] quantified the optimal computational resource allocation by jointly considering the trade-off between uncertainty quantification and discretization and model reduction. They obtained cost-to-error surfaces which were generated by simulating different combinations of physical and statistical resolutions which are then used to determine the optimal points for a given computational constraint [22]. The limitation of the approach described in [22] is the need to run various numerical experiments in order to develop cost-to-error surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…They obtained cost-to-error surfaces which were generated by simulating different combinations of physical and statistical resolutions which are then used to determine the optimal points for a given computational constraint [22]. The limitation of the approach described in [22] is the need to run various numerical experiments in order to develop cost-to-error surfaces. Thus, the process of finding the optimal physical and statistical resolutions (for a given computation budget) is a byproduct of these numerical investigations which may lead to an excessive computational burden.…”

Section: Introductionmentioning

confidence: 99%

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Optimal allocation of computational resources in hydrogeological models under uncertainty

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Section: Evaluating the Performance Of The Methodsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal allocation of computational resources in hydrogeological models under uncertainty

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Large‐scale hydraulic tomography and joint inversion of head and tracer data using the Principal Component Geostatistical Approach (PCGA)

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The stochastic geostatistical inversion approach is widely used in subsurface inverse problems to estimate unknown parameter fields and corresponding uncertainty from noisy observations. However, the approach requires a large number of forward model runs to determine the Jacobian or sensitivity matrix, thus the computational and storage costs become prohibitive when the number of unknowns, m, and the number of observations, n increase. To overcome this challenge in large-scale geostatistical inversion, the Principal Component Geostatistical Approach (PCGA) has recently been developed as a ''matrixfree'' geostatistical inversion strategy that avoids the direct evaluation of the Jacobian matrix through the principal components (low-rank approximation) of the prior covariance and the drift matrix with a finite difference approximation. As a result, the proposed method requires about K runs of the forward problem in each iteration independently of m and n, where K is the number of principal components and can be much less than m and n for large-scale inverse problems. Furthermore, the PCGA is easily adaptable to different forward simulation models and various data types for which the adjoint-state method may not be implemented suitably. In this paper, we apply the PCGA to representative subsurface inverse problems to illustrate its efficiency and scalability. The low-rank approximation of the large-dimensional dense prior covariance matrix is computed through a randomized eigen decomposition. A hydraulic tomography problem in which the number of observations is typically large is investigated first to validate the accuracy of the PCGA compared with the conventional geostatistical approach. Then the method is applied to a largescale hydraulic tomography with 3 million unknowns and it is shown that underlying subsurface structures are characterized successfully through an inversion that involves an affordable number of forward simulation runs. Lastly, we present a joint inversion of head and tracer test data using MODFLOW and MT3DMS as coupled black-box forward simulation solvers. These applications demonstrate the advantages of the PCGA, i.e., the scalability to high-dimensional inverse problems and the ability to utilize multiple forward models as black boxes.

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Minimum Hydraulic Resistance and Least Resistance Path in Heterogeneous Porous Media

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The transport dynamics of a solute plume in porous media are strictly related to the hydrogeological properties. Despite progress in simulation techniques, quantifying transport in strongly heterogeneous geological formations is still a challenge. It is well established that the heterogeneity of the hydraulic conductivity (K) field is one of the main factors controlling solute transport phenomena. Increasing the heterogeneity level of the K‐field will enhance the probability of having preferential paths, that are fundamental in predicting the first time arrivals. In this work, we focus on the relationship between the connectivity structure of the K‐field to transport quantities. We compute connectivity based on the concept of hydraulic resistance and the corresponding least resistance paths. We present a new efficient algorithm based on graph theory that enables us to extract useful information from the K‐field without resorting to the solution of the governing equations for flow and transport. For this reason, an exhaustive and fast analysis can be carried out using a Monte Carlo framework for randomly generated K‐fields which allows the computation of the least resistance path and its uncertainty. We examine the minimum hydraulic resistance for both multi‐Gaussian (MG) and non‐MG log⁡K‐fields. The analysis carried out indicates that the expected value of the minimum hydraulic resistance between two points scales exponentially with the standard deviation of the log⁡K‐field. Given the strong correlation with plume's first time arrival, our results illustrate how hydraulic resistance and least resistance path can be used as a computationally efficient risk metric.

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Towards optimal allocation of computer resources: Trade-offs between uncertainty quantification, discretization and model reduction

Cited by 20 publications

References 62 publications

Optimal allocation of computational resources in hydrogeological models under uncertainty

Optimal allocation of computational resources in hydrogeological models under uncertainty

Large‐scale hydraulic tomography and joint inversion of head and tracer data using the Principal Component Geostatistical Approach (PCGA)

Minimum Hydraulic Resistance and Least Resistance Path in Heterogeneous Porous Media

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