A Quasi‐Newton Reformulated Geostatistical Approach on Reduced Dimensions for Large‐Dimensional Inverse Problems

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The main computational costs of gradient-based inverse methods for solving high-dimensional groundwater inverse problems usually include (a) the storage and computation of large-dimensional spatial covariance matrices, which regularize the smoothness of the parameter field to be estimated, (b) a large number of iterative forward model implementations to determine the Jacobian matrix, and (c) forward model simulations on high-resolution parameter fields (Kitanidis, 2015). Many research efforts have been devoted to improving the storage and computation of large matrices and reducing the number of forward model runs. For example, adjoint state methods were developed to evaluate the Jacobian by solving an additional forward model system for each measurement (Sun & Yeh, 1990a, 1990b, 1992, reducing the number of forward model runs to the number of measurements. Geostatistical approach (GA) has been improved to compute large-dimensional covariance matrices and reduce the number of forward model runs (

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

High‐Dimensional Groundwater Flow Inverse Modeling by Upscaled Effective Model on Principal Components

Zhao

Guo

et al. 2022

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“…Options for irregular grids include, for example, the Karhunen‐Loeve decomposition (which relies on a numerical eigendecomposition of the auto‐covariance matrix without the circulant embedding and hence without FFT assistance). While this sounds much slower, the Karhunen‐Loeve decomposition is usually truncated, such as in the dimension‐reduced approach by Zhao and Luo (2021b), and it could be truncated early if desired. Yet another option is to re‐parameterize, on a coarser grid, the random field through pilot points (Doherty et al., 2010), followed by interpolation or conditional simulation in between (Keller et al., 2021).…”

Section: Applicationmentioning

confidence: 99%

Bayesian Inversion of Multi‐Gaussian Log‐Conductivity Fields With Uncertain Hyperparameters: An Extension of Preconditioned Crank‐Nicolson Markov Chain Monte Carlo With Parallel Tempering

Xiao

Reuschen

et al. 2021

The characterization of hydraulic properties of aquifers and soils is essential to better predict water flow in the subsurface and the transport of heat or solutes. Typically, not enough direct data (e.g., hydraulic conductivity) are available to characterize the heterogeneous subsurface. Thus, additional indirect data (e.g., hydraulic heads) are important for improving characterization and, in turn, predictions by subsurface flow and transport models. In the geostatistical context, the resulting inverse problem for subsurface problems is typically underdetermined. Therefore, approaches were developed which limited the number of independent parameters to be estimated, either by defining a limited number of zones with constant parameters (Carrera & Neuman, 1986) or parameterizing the spatially variable parameter field by a geostatistical function with a few unknown parameters (Kitanidis & Vomvoris, 1983). Later, methods were formulated to estimate a series of equally likely solutions to the groundwater inverse problem, either by an ensemble-based variational data assimilation approach (Gómez-Hernández et al., 1997) or by a sequential data assimilation approach for an ensemble of random parameter fields (Chen & Zhang, 2006). In summary, the combination of regularization and casting the problem in a stochastic framework, helped to tackle groundwater inverse problems.Bayesian inversion has been widely used for model parameter inference (

“…The most widely used approach for solving HT inverse problems is geostatistical approach (GA), including quasilinear GA (Fienen et al., 2008; Kitanidis, 1995) and successive linear estimator (SLE) (Yeh et al., 1995) The major bottleneck of GA is that it requires iterative forward model simulations to evaluate the Jacobian matrix, which are computationally expensive for large‐scale, high‐dimensional models. Many efforts have been invested to save the computation of GA through reducing the number of forward simulations or accelerating the computational implementation (Ambikasaran et al., 2013; Broyden, 1965; Kitanidis & Lee, 2014; Klein et al., 2017; Nowak & Cirpka, 2004; Nowak et al., 2003; Saibaba et al., 2012; Zhao & Luo, 2021a; Zhao et al., 2022).…”

Section: Introductionmentioning

confidence: 99%

“…expensive for large-scale, high-dimensional models. Many efforts have been invested to save the computation of GA through reducing the number of forward simulations or accelerating the computational implementation (Ambikasaran et al, 2013;Broyden, 1965;Klein et al, 2017;Nowak & Cirpka, 2004;Nowak et al, 2003;Saibaba et al, 2012;Zhao & Luo, 2021a;Zhao et al, 2022).…”

mentioning

confidence: 99%

Predictive Deep Learning for High‐Dimensional Inverse Modeling of Hydraulic Tomography in Gaussian and Non‐Gaussian Fields

Guo,

Liu,

Luo

2023

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Inverse modeling of hydraulic tomography (HT) is computationally expensive for estimating high‐dimensional hydrogeologic parameter fields. In this work, we develop a novel method called HT‐INV‐NN, which combines dimensionality reduction techniques with a predictive deep learning (DL) model to estimate high‐dimensional Gaussian and non‐Gaussian channel fields. The HT‐INV‐NN model consists of a predictor that directly learns the inverse process from hydraulic head measurements to latent variables of random fields, and a decoder that generates high‐dimensional parameter fields from predicted latent variables. For Gaussian spatially correlated fields, the decoder utilizes principal components derived from spatial covariance, and for non‐Gaussian channel fields, a generative adversarial network (GAN) is trained using generated realizations based on a training image (TI). The predictor is a deep neural network calibrated using the reference data obtained from HT forward simulations, which can be implemented in parallel. HT‐INV‐NN is successfully tested in multiple numerical experiments including steady‐state and transient HT for estimating Gaussian fields in 2D and 3D, as well as binary discontinuous or continuous non‐Gaussian channel fields. The training process is efficient, and the model structure demonstrates robustness for input data with perturbations. The model performance on multiple validation data sets are satisfactory when compared with other numerical and deep learning methods.