Complexity analysis of accelerated MCMC methods for Bayesian inversion

Fostered by advances in computing, simulation and data science have emerged as the third and fourth paradigms enabling scientific discovery. Many mathematical challenges arise when integrating these paradigms to produce new knowledge. One of the most important challenges is the inverse or calibration problem: find the inputs of a simulation such that its outputs agree with a given set of measurements. As both the cost of the complex simulations and the volume of measurement data sets increase, intuition-driven trial-and-error methods for calibration quickly become untenable; modern scientists need more computationally efficient and mathematically rigorous methods. When measurements and simulated predictions contain uncertainty, the Bayesian perspective on statistical inference provides a popular mathematical framework for developing computational methods to solve inverse problems. Markov chain Monte Carlo (MCMC) methods are a mainstay of this framework; they characterize a probability distribution on the space of model inputs that is conditioned on the given measurements. However, despite recent advances, MCMC methods remain impractical for expensive simulations with more than a handful of inputs. At its heart, MCMC searches the space of inputs for corresponding outputs that match measurements, and so it is afflicted with the curse of dimensionality. Additionally, MCMC methods become less efficient as the volume of the measurement data increases, particularly if the measurements are redundant. To make MCMC—and Bayesian methods, in general—amenable to today’s computer models with millions of inputs and terabytes of measurement data, one must devise methods to reduce the dimension of both the simulation inputs and the measurements. For many models used in practice, the outputs vary primarily on a low-dimensional manifold of the nominally high-dimensional input space. Recently developed active subspace methods discover this low-dimensional manifold and exploit it to enable analyses typically suited for low-dimensional settings (e.g., statistical inversion of complex systems). We hypothesize that active subspace methods can provide the necessary dimension reduction, enabling existing Bayesian methods to find solutions for a large class of otherwise intractable statistical inverse problems. We will test the hypothesis with a series of mathematical analyses and computational experiments that will proceed along three research veins: (i) integrating active subspace methods into methods for statistical inversion, (ii) addressing current challenges in active subspace methods, and (iii) incorporating methods for nonlinear dimensionality reduction into active subspace methods. We will apply the methods we develop to real inverse problems in chemical kinetics and turbulent flame modeling. Our team combines expertise in methods for inverse problems, dimension reduction, and reduced order modeling. With a history of active collaboration and thoughtful engagement, we bring enthusiasm and fresh perspectives to a well studied subject. We are excited by the potential opportunity to work together to extend and apply active subspace methods to data-intensive inverse problems.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the diagonal approximate Hessian of the local log posterior. Our algorithm is called the gradient-based Markov chain Monte Carlo (GMCMC) method. The GMCMC FWI method can quantify inversion uncertainties with estimated posterior distribution given sufficiently long Markov chains. By directly sampling the posterior distribution, we obtain a global view of the model space. Theoretically speaking, statistical assessments do not depend on starting models. Our method is applied to the 2D Marmousi model with the frequency-domain FWI setting. Numerical results suggest that our method can be readily applied to 2D cases with affordable computational efforts.

The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems.

Uncertainty quantification of groundwater (GW) aquifer parameters is critical for efficient management and sustainable extraction of GW resources. These uncertainties are introduced by the data, model, and prior information on the parameters. Here, we develop a Bayesian inversion framework that uses Interferometric Synthetic Aperture Radar (InSAR) surface deformation data to infer the laterally heterogeneous permeability of a transient linear poroelastic model of a confined GW aquifer. The Bayesian solution of this inverse problem takes the form of a posterior probability density of the permeability. Exploring this posterior using classical Markov chain Monte Carlo (MCMC) methods is computationally prohibitive due to the large dimension of the discretized permeability field and the expense of solving the poroelastic forward problem. However, in many partial differential equation (PDE)‐based Bayesian inversion problems, the data are only informative in a few directions in parameter space. For the poroelasticity problem, we prove this property theoretically for a one‐dimensional problem and demonstrate it numerically for a three‐dimensional aquifer model. We design a generalized preconditioned Crank‐Nicolson (gpCN) MCMC method that exploits this intrinsic low dimensionality by using a low‐rank‐based Laplace approximation of the posterior as a proposal, which we build scalably. The feasibility of our approach is demonstrated through a real GW aquifer test in Nevada. The inherently two‐dimensional nature of InSAR surface deformation data informs a sufficient number of modes of the permeability field to allow detection of major structures within the aquifer, significantly reducing the uncertainty in the pressure and the displacement quantities of interest.

Identification of contaminant sources in rivers is crucial for river protection and emergency response. This study presents an innovative approach for identifying river pollution sources by using Bayesian inference and cellular automata (CA) modeling. A general Bayesian framework is proposed that combines the CA model with observed data to identify unknown sources of river pollution. To reduce the computational burden of the Bayesian inference, a CA contaminant transport model is developed to efficiently simulate pollutant concentration values in the river. These simulated concentration values are then used to calculate the likelihood function of available measurements. The Markov chain Monte Carlo (MCMC) method is used to produce the posterior distribution of contaminant source parameters, which is a sampling-based method that enables the estimation of complex posterior distributions. The suggested methodology is applied to a real case study of the Fen River in Yuncheng City, Shanxi Province, Northern China, and it estimates the release time, release mass, and source location with relative errors below 19%. The research indicates that the proposed methodology is an effective and flexible way to identify the location and concentrations of river contaminant sources.

The Guide to the Expression of Uncertainty in Measurement (GUM) has led to a harmonization of uncertainty evaluation throughout metrology. The simplicity of its employed methodology has fostered the broad acceptance of the GUM among metrologists. However, this simplicity also compromises best practice and does not provide state-of-the-art data analysis. Specifically, metrologists often possess useful prior knowledge about the measurand which cannot be accounted for by the GUM. Bayesian uncertainty evaluation, on the other hand, takes prior knowledge about the measurand as its starting point. While this technique has been successfully applied in several instances in metrology, its broad access for metrologists is still lacking. One reason for this is that application of Bayesian methods and their computation, e.g. by means of Markov chain Monte Carlo (MCMC) methods, usually requires statistical skills.In this work, we propose a simple method for Bayesian uncertainty evaluation applicable to measurement models that depend linearly on a single input quantity for which Type A information is available, and several input quantities for which Type B information is given. This category of measurement models covers many uncertainty evaluations in metrology. The approach utilizes a specific class of prior distributions to encode the prior information about the measurand. Explicit guidance is provided on how to choose the parameters of the prior in the light of one’s prior knowledge. MCMC methods are not required for the calculation of results; instead, a simple Monte Carlo procedure is developed that allows to draw uncorrelated samples from the posterior distribution, which greatly simplifies convergence assessments. The proposed Bayesian approach allows the treatment of small numbers of observations for a Type A uncertainty evaluation, including the case of only a single observation. Examples are provided that illustrate the proposed approach and corresponding open source Python software is made available.

For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources.

Context.The characterization of the interior of an exoplanet is an inverse problem. The solution requires statistical methods such as Bayesian inference. Current methods employ Markov chain Monte Carlo (MCMC) sampling to infer the posterior probability of the planetary structure parameters for a given exoplanet. These methods are time-consuming because they require the evaluation of a planetary structure model ~105times.Aims.To speed up the inference process when characterizing an exoplanet, we propose to use conditional invertible neural networks to calculate the posterior probability of the planetary structure parameters.Methods.Conditional invertible neural networks (cINNs) are a special type of neural network that excels at solving inverse problems. We constructed a cINN following the framework for easily invertible architectures (FreIA). This neural network was then trained on a database of 5.6 × 106internal structure models to recover the inverse mapping between internal structure parameters and observable features (i.e., planetary mass, planetary radius, and elemental composition of the host star). We also show how observational uncertainties can be accounted for.Results.The cINN method was compared to a commonly used Metropolis-Hastings MCMC. To do this, we repeated the characterization of the exoplanet K2-111 b, using both the MCMC method and the trained cINN. We show that the inferred posterior probability distributions of the internal structure parameters from both methods are very similar; the largest differences are seen in the exoplanet water content. Thus, cINNs are a possible alternative to the standard time-consuming sampling methods. cINNs allow infering the composition of an exoplanet that is orders of magnitude faster than what is possible using an MCMC method. The computation of a large database of internal structures to train the neural network is still required, however. Because this database is only computed once, we found that using an invertible neural network is more efficient than an MCMC when more than ten exoplanets are characterized using the same neural network.

Based on the parametric deterministic formulation of Bayesian inverse problems with unknown input parameter from infinite-dimensional, separable Banach spaces proposed in Schwab and Stuart (2012 Inverse Problems 28 045003), we develop a practical computational algorithm whose convergence rates are provably higher than those of Monte Carlo (MC) and Markov chain Monte Carlo methods, in terms of the number of solutions of the forward problem. In the formulation of Schwab and Stuart, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown diffusion coefficients from noisy observations comprising linear functionals of the system’s response. The sparsity of the generalized polynomial chaos representation of the posterior density being implied by sparsity assumptions on the class of the prior (Schwab and Stuart 2012), we design, analyze and implement a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient approximate numerical evaluation of expectations under the posterior, given data. The proposed, deterministic quadrature algorithm is based on a greedy, iterative identification of finite sets of most significant, ‘active’ chaos polynomials in the posterior density analogous to recently proposed algorithms for adaptive interpolation (Chkifa et al 2012 Report 2012-NN, 2013 Math. Modelling Numer. Anal. 47 253–80). Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown, but are bounded independently of the number of random variables activated by the adaptive algorithm. Numerical results for a model problem of coefficient identification with point measurements in a diffusion problem confirm the theoretical results.

In this paper, we study an inverse transmission scattering problem of a time-harmonic acoustic wave from the viewpoint of Bayesian statistics. In Bayesian inversion, the solution of the inverse problem is the posterior distribution of the unknown parameters conditioned on the observational data. The shape of the scatterer will be reconstructed from full-aperture and limited-aperture far-field measurement data. We first prove a well-posedness result for the posterior distribution in the sense of the Hellinger metric. Then, we employ the Markov chain Monte Carlo method based on the preconditioned Crank-Nicolson algorithm to extract the posterior distribution information. Numerical results are given to demonstrate the effectiveness of the proposed method.

Geoscientists use observed data to estimate properties of the Earth's interior. This often requires non‐linear inverse problems to be solved and uncertainties to be estimated. Bayesian inference solves inverse problems under a probabilistic framework, in which uncertainty is represented by a so‐called posterior probability distribution. Recently, variational inference has emerged as an efficient method to estimate Bayesian solutions. By seeking the closest approximation to the posterior distribution within any chosen family of distributions, variational inference yields a fully probabilistic solution. It is important to define expressive variational families so that the posterior distribution can be represented accurately. We introduce boosting variational inference (BVI) as a computationally efficient means to construct a flexible approximating family comprising all possible finite mixtures of simpler component distributions. We use Gaussian mixture components due to their fully parametric nature and the ease with which they can be optimized. We apply BVI to seismic travel time tomography and full waveform inversion, comparing its performance with other methods of solution. The results demonstrate that BVI achieves reasonable efficiency and accuracy while enabling the construction of a fully analytic expression for the posterior distribution. Samples that represent major components of uncertainty in the solution can be obtained analytically from each mixture component. We demonstrate that these samples can be used to solve an interrogation problem: to assess the size of a subsurface target structure. To the best of our knowledge, this is the first method in geophysics that provides both analytic and reasonably accurate probabilistic solutions to fully non‐linear, high‐dimensional Bayesian full waveform inversion problems.

<p>For strongly non-linear inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF). Particle methods are very well suited for parallelization and offer an alternative approach whereby the posterior PDF is approximated using the states and weights of a population of evolving particles. In addition, it provides reliable estimates of the evidence (marginal likelihood) that is needed for Bayesian model selection at essentially no cost. We consider adaptive sequential Monte Carlo (ASMC), which is an extension of annealed importance sampling (AIS). In these methods, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The main advantages of ASMC with respect to AIS are that it adaptively tunes the tempering between neighboring distributions and it performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field designed such that the posterior facies distribution includes two distinct modes with equal probability. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multi-point statistics (MPS) simulations. We focus here on the ability of ASMC to explore the posterior PDF and compare it with previously published results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget involving 24 particles for ASMC and 24 chains for PT, the ASMC implementation outperforms the results obtained by PT: the models fit the data better and the reference likelihood value is contained in the ASMC sampled likelihood range, while this is not the case for PT range. Moreover, we show that ASMC recovers both reference modes, while none of them is recovered by PT. However, with 24 particles there is one of the modes that has a higher weight than the other while the approximation is improved when moving to a larger number of particles. As a future development, we suggest that including fast surrogate modeling (e.g., polynomial chaos expansion) within ASMC for the MCMC steps used to evolve the particles in-between importance sampling steps would strongly reduce the computational cost while still ensuring results of similar quality as the importance sampling steps could still be performed using the regular more costly forward solver.</p>

Knowledge of the Earth's interior relies on indirect information collected at or near the surface. Typically, data do not uniquely constrain the subsurface properties and are contaminated by noise, and therefore the solution of ill-posed inverse problems is required. Geophysicists have traditionally addressed such problems through deterministic approaches, seeking a single 'best-fitting' model defined by optimality criteria that reflect our understanding of the problem at hand. In recent decades, Bayesian approaches have become increasingly common, as they characterise the posterior probability distribution of the model conditioned on the observations, thereby quantifying uncertainty.In this context, Markov chain Monte Carlo (MCMC) methods have emerged as a crucial tool as they allow sampling from posterior distributions of arbitrary complexity. At the core of many MCMC algorithms lies the Metropolis-Hastings scheme. This combines a proposal distribution with a probabilistic acceptance criterion to construct a Markov chain that has the desired target distribution as its stationary distribution. The algorithm is versatile as it rests on mild technical conditions on the proposal, and is thus widely adopted across a broad range of geoscientific inference problems. Yet when the parameter space is large or the forward models are computationally expensive---both common scenarios in geophysical applications---it can become inefficient, resulting in poor chain mixing and slow convergence to the target (posterior) distribution. These challenges underscore the importance of effective proposal mechanisms.In this presentation, we introduce a novel approach to designing Metropolis-Hastings proposals based on adaptive transport maps. The framework is inspired by recent developments from the field of Applied Mathematics linking Bayesian inference with Optimal Transport theory. The idea is to find a monotone, nonlinear transformation to recast a (complex) target probability distribution into a (simpler) reference distribution that is more amenable to standard MCMC steps. Our key contribution is to parameterise these transformations using invertible neural networks, ensuring monotonicity while gaining the flexibility and expressiveness that neural architectures afford.The proposed method progresses iteratively. We begin with a standard sampling strategy (e.g., a random-walk Metropolis) to obtain initial draws from the target distribution. These samples inform the training of an invertible neural network that learns to map from the target to a simpler reference distribution, specifically a standard (multivariate) Gaussian. Subsequent proposals are then generated in the reference space, either as global independence moves or local perturbations, and are accepted or rejected following a suitably modified Metropolis-Hastings criterion. As more samples accumulate, the network's parameters are updated, improving overall sampling efficiency. Ultimately, the approach yields not only an ensemble of samples representative of the desired target distribution---just as in standard MCMC---but also a compact, learned representation of it in the network's weights.We illustrate the proposed paradigm both theoretically and through examples. Preliminary results indicate that transport-map-enhanced MCMC has the potential to significantly accelerate Bayesian sampling across a range of applications.

In the context of Bayesian inversion for scientific and engineering modeling, Markov chain Monte Carlo (MCMC) sampling strategies have become the benchmark due to their flexibility and robustness in dealing with arbitrary posterior probability density functions (PDFs). However, these algorithms have been shown to be inefficient when sampling from high-dimensional posterior distributions or exhibit multimodality and/or strong parameter correlations. In such contexts, transitional MCMC (TMCMC) provides a more efficient alternative. Despite the recent applicability for Bayesian updating and model selection across a variety of disciplines, TMCMC may require a prohibitive number of tempering stages when the prior pdf is significantly different from the target posterior. Furthermore, the need to start with an initial set of samples from the prior distribution may present a challenge when dealing with implicit priors, e.g., based on feasible regions. Finally, TMCMC cannot be used for inverse problems with improper prior PDFs that represent a lack of prior knowledge on all or a subset of parameters. A generalization of TMCMC is proposed that alleviates such challenges and limitations toward providing a more robust and efficient tempering sampling strategy. We present convergence analysis, proving that the distance between the intermediate distributions and the target posterior distribution monotonically decreases as the algorithm proceeds. We also demonstrate the advantages of the proposed generalization through a series of test problems and an engineering application in the oil and gas industry.