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We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings algorithm that adds an accept–reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein–Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an L2-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require O(105) model runs and are sensitive to internal variability noise, rendering them infeasible for climate models. Here we demonstrate an approach to model calibration and uncertainty quantification that requires only O(102) model runs and can accommodate internal climate variability. The approach consists of three stages: (a) a calibration stage uses variants of ensemble Kalman inversion to calibrate a model by minimizing mismatches between model and data statistics; (b) an emulation stage emulates the parameter‐to‐data map with Gaussian processes (GP), using the model runs in the calibration stage for training; (c) a sampling stage approximates the Bayesian posterior distributions by sampling the GP emulator with MCMC. We demonstrate the feasibility and computational efficiency of this calibrate‐emulate‐sample (CES) approach in a perfect‐model setting. Using an idealized general circulation model, we estimate parameters in a simple convection scheme from synthetic data generated with the model. The CES approach generates probability distributions of the parameters that are good approximations of the Bayesian posteriors, at a fraction of the computational cost usually required to obtain them. Sampling from this approximate posterior allows the generation of climate predictions with quantified parametric uncertainties. Plain Language Summary Calibrating climate models with available data and quantifying their uncertainties are essential to make climate predictions accurate and actionable. A primary source of uncertainties in climate models comes from representation of small‐scale processes such as moist convection. Parameters in convection schemes and other parameterizations are usually calibrated by hand, using only a small fraction of data that are available. As a result, the calibration process may miss information about the small‐scale processes in question. This paper presents a proof‐of‐concept, in an idealized setting, of how parameters in climate models can be calibrated using a substantial fraction of the available data, and how uncertainties in the parameters can be quantified. We employ a new algorithm, called calibrate‐emulate‐sample (CES), which makes such calibration and uncertainty quantification feasible for computationally expensive climate models. CES reduces the hundreds of thousands of model runs usually required to quantify uncertainties in computer models to hundreds, thereby achieving about a factor 1,000 speedup. It leads to more robust calibration and uncertainty quantification in the presence of noise arising from chaotic variability of the climate system. We show how uncertainties in climate model parameters can be translated into quantified uncertainties of climate predictions through ensemble integrations. Key Points We use time averaged climate statistics to calibrate convective parameters and quantify their uncertainties We demonstrate use of the calibrate‐emulate‐sample algorithm to provide efficient calibration and uncertainty quantification Parametric uncertainty in climate predictions is quantified by sampling from the learnt convective parameter distribution

The ensemble Kalman filter (EnKF) is a widely used methodology for state estimation in partially, noisily observed dynamical systems and for parameter estimation in inverse problems. Despite its widespread use in the geophysical sciences, and its gradual adoption in many other areas of application, analysis of the method is in its infancy. Furthermore, much of the existing analysis deals with the large ensemble limit, far from the regime in which the method is typically used. The goal of this paper is to analyze the method when applied to inverse problems with fixed ensemble size. A continuous time limit is derived and the long-time behavior of the resulting dynamical system is studied. Most of the rigorous analysis is confined to the linear forward problem, where we demonstrate that the continuous time limit of the EnKF corresponds to a set of gradient flows for the data misfit in each ensemble member, coupled through a common preconditioner which is the empirical covariance matrix of the ensemble. Numerical results demonstrate that the conclusions of the analysis extend beyond the linear inverse problem setting. Numerical experiments are also given which demonstrate the benefits of various extensions of the basic methodology.

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying computational complexity. In particular, they lead directly to precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have mainly been proved for target measures with a product structure, severely limiting their applicability. The purpose of this paper is to study diffusion limits for a class of naturally occurring high-dimensional measures found from the approximation of measures on a Hubert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit of a random walk Metropolis algorithm to an infinite-dimensional Hubert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm.

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.

We describe a new MCMC method optimized for the sampling of probability measures on Hubert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on ℝN (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hubert space; as a consequence the algorithm explores the approximate target measures on ℝN in a number of steps which is independent of N. We also present the underlying theory for the limiting nonreversible diffusion on Hubert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.

We consider estimation of scalar functions that determine the dynamics of diffusion processes. It has been recently shown that nonparametric maximum likelihood estimation is ill-posed in this context. We adopt a probabilistic approach to regularize the problem by the adoption of a prior distribution for the unknown functional. A Gaussian prior measure is chosen in the function space by specifying its precision operator as an appropriate differential operator. We establish that a Bayesian-Gaussian conjugate analysis for the drift of one-dimensional nonlinear diffusions is feasible using high-frequency data, by expressing the loglikelihood as a quadratic function of the drift, with sufficient statistics given by the local time process and the end points of the observed path. Computationally efficient posterior inference is carried out using a finite element method. We embed this technology in partially observed situations and adopt a data augmentation approach whereby we iteratively generate missing data paths and draws from the unknown functional. Our methodology is applied to estimate the drift of models used in molecular dynamics and financial econometrics using high-and low-frequency observations. We discuss extensions to other partially observed schemes and connections to other types of nonparametric inference.

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hubert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional approximation of the target will take O(N¹ / ³) steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require O(N) steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of 0.574. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hubert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler-Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

Data required to calibrate uncertain general circulation model (GCM) parameterizations are often only available in limited regions or time periods, for example, observational data from field campaigns, or data generated in local high‐resolution simulations. This raises the question of where and when to acquire additional data to be maximally informative about parameterizations in a GCM. Here we construct a new ensemble‐based parallel algorithm to automatically target data acquisition to regions and times that maximize the uncertainty reduction, or information gain, about GCM parameters. The algorithm uses a Bayesian framework that exploits a quantified distribution of GCM parameters as a measure of uncertainty. This distribution is informed by time‐averaged climate statistics restricted to local regions and times. The algorithm is embedded in the recently developed calibrate‐emulate‐sample framework, which performs efficient model calibration and uncertainty quantification with only O(102)$\\mathcal{O}({10}^{2})$model evaluations, compared with O(105)$\\mathcal{O}({10}^{5})$evaluations typically needed for traditional approaches to Bayesian calibration. We demonstrate the algorithm with an idealized GCM, with which we generate surrogates of local data. In this perfect‐model setting, we calibrate parameters and quantify uncertainties in a quasi‐equilibrium convection scheme in the GCM. We consider targeted data that are (a) localized in space for statistically stationary simulations, and (b) localized in space and time for seasonally varying simulations. In these proof‐of‐concept applications, the calculated information gain reflects the reduction in parametric uncertainty obtained from Bayesian inference when harnessing a targeted sample of data. The largest information gain typically, but not always, results from regions near the intertropical convergence zone. Plain Language Summary Climate models depend on dynamics across many spatial and temporal scales. It is infeasible to resolve all of these scales. Instead, the physics at the smallest scales is represented by parameterization schemes that link what is unresolvable to variables resolved on the grid scale. A dominant source of uncertainty in climate predictions comes from uncertainty in calibrating empirical parameters in such parameterization schemes, and these uncertainties are generally not quantified. The uncertainties can be reduced and quantified with data that may have limited availability in space and time, for example, data from field campaigns or from targeted high‐resolution simulations in limited areas. But the sensitivity of simulated climate statistics, such as precipitation rates, to parameterizations varies in space and time, raising the question of where and when to acquire additional data so as to optimize the information gain from the data. Here we construct an automated algorithm that finds optimal regions and time periods for such data acquisition, to maximize the information the data provides about uncertain parameters. In proof‐of‐concept simulations with an idealized global atmosphere model, we show that our algorithm successfully identifies the informative regions and times, even in cases where physics‐based intuition may lead to sub‐optimal choices. Key Points Climate models can be calibrated with targeted data, for example, from limited‐area high‐resolution simulations We propose an algorithm for choosing target sites for data acquisition that are maximally informative about climate model parameters The algorithm is benchmarked in an idealized aquaplanet general circulation model

The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil–water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification.