A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems

We address the numerical solution of infinite-dimensional inverse problems in the framework of Bayesian inference. In the Part I companion to this paper (arXiv.org:1308.1313), we considered the linearized infinite-dimensional inverse problem. Here in Part II, we relax the linearization assumption and consider the fully nonlinear infinite-dimensional inverse problem using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling high-dimensional pdfs arising from Bayesian inverse problems governed by PDEs, we build on the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf, whose construction is made tractable by invoking a low-rank approximation of its data misfit component of the Hessian. Here we introduce an approximation of the stochastic Newton proposal in which we compute the low-rank-based Hessian at just the MAP point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method with a MAP-based Hessian converges at least as rapidly as the original stochastic Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian at each step. On the other hand, it is more expensive per sample than the independence sampler; however, its convergence is significantly more rapid, and thus overall it is much cheaper. Finally, we present extensive analysis and interpretation of the posterior distribution, and classify directions in parameter space based on the extent to which they are informed by the prior or the observations.Comment: 31 page

Optimal low-rank approximations of Bayesian linear inverse problems

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space

A computational framework for the solution of infinite-dimensional Bayesian statistical inverse problems with application to global seismic inversion

textQuantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the “best-fit” parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 10⁶ parameters.Computational Science, Engineering, and Mathematic

SIMULATION METAMODELING AND OPTIMIZATION WITH AN ADDITIVE GLOBAL AND LOCAL GAUSSIAN PROCESS MODEL FOR STOCHASTIC SYSTEMS

Ph.DDOCTOR OF PHILOSOPH

A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning

We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utility-based selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments---active user modelling with preferences, and hierarchical reinforcement learning---and a discussion of the pros and cons of Bayesian optimization based on our experiences

Fatigue reliability of ship structures

Today we are sitting on a huge wealth of structural reliability theory but its application in ship design and construction is far behind. Researchers and practitioners face a daunting task of dove-tailing the theoretical achievements into the established processes in the industry. The research is aimed to create a computational framework to facilitate fatigue reliability of ship structures. Modeling, transformation and optimization, the three key elements underlying the success of computational mechanics are adopted as the basic methodology through the research. The whole work is presented in a way that is most suitable for software development. The foundation of the framework is constituted of reliability methods at component level. Looking at the second-moment reliability theory from a minimum distance point of view the author derives a generic set of formulations that incorporate all major first and second order reliability methods (FORM, SORM). Practical ways to treat correlation and non- Gaussian variables are discussed in detail. Monte Carlo simulation (MCS) also accounts for significant part of the research with emphasis on variance reduction techniques in a proposed Markov chain kernel method. Existing response surface methods (RSM) are reviewed and improved with much weight given to sampling techniques and determination of the quadratic form. Time-variant problem is touched upon and methods to convert it to nested reliability problems are discussed. In the upper layer of the framework common fatigue damage models are compared. Random process simulation and rain-flow counting are used to study effect of wide-banded non-Gaussian process. At the center of this layer is spectral fatigue analysis based on SN curve and first-principle stress and hydrodynamic analysis. Pseudo-excitation is introduced to get linear equivalent stress RAO in the non-linear ship-wave system. Finally response surface method is applied to this model to calculate probability of failure and design sensitivity in the case studies of a double hull oil tanker and a bulk carrier

Bayesian Modeling and Estimation Techniques for the Analysis of Neuroimaging Data

Brain function is hallmarked by its adaptivity and robustness, arising from underlying neural activity that admits well-structured representations in the temporal, spatial, or spectral domains. While neuroimaging techniques such as Electroencephalography (EEG) and magnetoencephalography (MEG) can record rapid neural dynamics at high temporal resolutions, they face several signal processing challenges that hinder their full utilization in capturing these characteristics of neural activity. The objective of this dissertation is to devise statistical modeling and estimation methodologies that account for the dynamic and structured representations of neural activity and to demonstrate their utility in application to experimentally-recorded data. The first part of this dissertation concerns spectral analysis of neural data. In order to capture the non-stationarities involved in neural oscillations, we integrate multitaper spectral analysis and state-space modeling in a Bayesian estimation setting. We also present a multitaper spectral analysis method tailored for spike trains that captures the non-linearities involved in neuronal spiking. We apply our proposed algorithms to both EEG and spike recordings, which reveal significant gains in spectral resolution and noise reduction. In the second part, we investigate cortical encoding of speech as manifested in MEG responses. These responses are often modeled via a linear filter, referred to as the temporal response function (TRF). While the TRFs estimated from the sensor-level MEG data have been widely studied, their cortical origins are not fully understood. We define the new notion of Neuro-Current Response Functions (NCRFs) for simultaneously determining the TRFs and their cortical distribution. We develop an efficient algorithm for NCRF estimation and apply it to MEG data, which provides new insights into the cortical dynamics underlying speech processing. Finally, in the third part, we consider the inference of Granger causal (GC) influences in high-dimensional time series models with sparse coupling. We consider a canonical sparse bivariate autoregressive model and define a new statistic for inferring GC influences, which we refer to as the LASSO-based Granger Causal (LGC) statistic. We establish non-asymptotic guarantees for robust identification of GC influences via the LGC statistic. Applications to simulated and real data demonstrate the utility of the LGC statistic in robust GC identification