The Levenberg‐Marquardt (LM) method is commonly used for inverting models used to describe geothermal, groundwater, or oil and gas reservoirs. In previous studies, LM parameter updates have been made tractable for highly parameterized inverse problems with large data sets by applying matrix factorization methods or iterative linear solvers to approximately solve the update equations. Some studies have shown that basing model updates on the truncated singular value decomposition (TSVD) of a dimensionless sensitivity matrix achieved using Lanczos iteration can speed up the inversion of reservoir models. Lanczos iterations only require the sensitivity matrix times a vector and its transpose times a vector, which are found efficiently using adjoint and direct simulations without the expense of forming a large sensitivity matrix. Nevertheless, Lanczos iteration has the drawback of being a serial process, requiring a separate adjoint solve and direct solve every Lanczos iteration. Randomized methods, developed for low‐rank matrix approximation of large matrices, are more efficient alternatives to the standard Lanczos method. Here we develop LM variants which use randomized methods to find a TSVD of a dimensionless sensitivity matrix when updating parameters. The randomized approach offers improved efficiency by enabling simultaneous solution of all adjoint and direct problems for a parameter update.

We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our main goal is to make standard, "out-of-the-box" Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models by using suitable approximations. To do this, we first show how to pose both the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate so-called posterior-informed model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error approach. This enables the use of a "coarse," approximate model in place of a finer, more expensive model, while accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modeling, a relatively small number of fine model simulations and only modifies the target posterior-any standard MCMC sampling algorithm can be used to sample the new posterior. These corrections can also be used in methods that are not based on MCMC sampling. We show that our approach can achieve significant computational speedups on two geothermal test problems. We also demonstrate the dangers of naively using coarse, approximate models in place of finer models, without accounting for the induced approximation errors. The naive approach tends to give overly confident and biased posteriors while incorporating Bayesian approximation error into our hierarchical framework corrects for this while maintaining computational efficiency and ease of use. Key Points: • We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective • We present a simple method for incorporating posterior-informed approximation errors into a hierarchical Bayesian framework • Our method makes standard out-of-the-box MCMC sampling feasible for more complex models while correcting for bias and overconfidence Supporting Information: • Supporting Information S1 , M. J. (2020). Incorporating posterior-informed approximation errors into a hierarchical framework to facilitate out-of-the-box MCMC sampling for geothermal inverse problems and uncertainty quantification. Water Resources Research, 56, e2018WR024240. https://

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This paper describes practical randomized algorithms for low-rank matrix approximation that accommodate any budget for the number of views of the matrix. The presented algorithms, which are aimed at being as pass efficient as needed, expand and improve on popular randomized algorithms targeting efficient low-rank reconstructions. First, a more flexible subspace iteration algorithm is presented that works for any views v ≥ 2, instead of only allowing an even v. Secondly, we propose more general and more accurate singlepass algorithms. In particular, we propose a more accurate memory efficient single-pass method and a more general single-pass algorithm which, unlike previous methods, does not require prior information to assure near peak performance. Thirdly, combining ideas from subspace and single-pass algorithms, we present a more passefficient randomized block Krylov algorithm, which can achieve a desired accuracy using considerably fewer views than that needed by a subspace or previously studied block Krylov methods. However, the proposed accuracy enhanced block Krylov method is restricted to large matrices that are either accessed a few columns or rows at a time. Recommendations are also given on how to apply the subspace and block Krylov algorithms when estimating either the dominant left or right singular subspace of a matrix, or when estimating a normal matrix, such as those appearing in inverse problems. Computational experiments are carried out that demonstrate the applicability and effectiveness of the presented algorithms. Motivation.The primary motivation for this study was to develop algorithms to speed up inverse methods used to estimate parameters in models describing subsurface flow in geothermal reservoirs [36,3]. Inverting models describing complex geophysical processes, such as fluid flow in the subsurface, frequently involves matching a large data set using highly parameterized computational models. Running the model commonly involves solving an expensive and nonlinear forward problem. Despite the possible nonlinearity of the forward problem, the link between the model parameters and simulated observations is often described in terms of a Jacobian matrix J ∈ R N d ×N m , which locally linearizes the relationship between the parameters and observations. The size of J is therefore determined by the (large) parameter and observation spaces. In this case, explicitly forming J is out of the question since at best it involves solving N m direct problems (linearized forward simulations) or N d adjoint problems (linearized backward simulations) [6,23,35,38]. Nevertheless, the information contained in J can be helpful for the purpose of inverting the model using nonlinear inversion methods such as a Gauss-Newton or Levenberg-Marquardt approach, and for quantifying uncertainty.Using adjoint simulation, direct simulation and randomized algorithms, the necessary information can be extracted from J without ever explicitly forming the large matrix J . Bjarkason et al. [3] showed that inversion of a nonlinea...

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