We consider the statistical inverse problem of estimating a background flow field (e.g., of air or water) from the partial and noisy observation of a passive scalar (e.g., the concentration of a solute), a common experimental approach to visualizing complex fluid flows. Here the unknown is a vector field that is specified by a large or infinite number of degrees of freedom. Since the inverse problem is ill-posed, i.e., there may be many or no background flows that match a given set of observations, we adopt a Bayesian approach to regularize it. In doing so, we leverage frameworks developed in recent years for infinite-dimensional Bayesian inference. The contributions in this work are threefold. First, we lay out a functional analytic and Bayesian framework for approaching this problem. Second, we define an adjoint method for efficient computation of the gradient of the log likelihood, a key ingredient in many numerical methods. Finally, we identify interesting example problems that exhibit posterior measures with simple and complex structure. We use these examples to conduct a large-scale benchmark of Markov Chain Monte Carlo methods developed in recent years for infinite-dimensional settings. Our results indicate that these methods are capable of resolving complex multimodal posteriors in high dimensions.

Any acknowledgements must start with my amazing committee. Jeff and Nathan, my coadvisors, thank you for agreeing to work with me; for your patience with me as we navigated various life, logistical, and mathematical hurdles; for always being willing to go the extra mile (or, more accurately, hour) with me; and for all your assistance and advice in building my career. Lizette, thank you for giving me a chance when we started revamping Vector Geometry all those years ago, for encouraging me when I expressed an interest in pursuing a doctorate, and for all of your advice and support along the way. Thank you Tia for your help with various technical points as the work progressed. I am tremendously grateful to have had you as mentors and, more importantly, as friends.I owe a special debt of gratitude to Terry Herdman, my supervisor with Advanced Research Computing (ARC), for supporting my efforts to obtain my Ph.D. From the moment that I proposed the idea to the day of my defense, you have been more patient than I ever could have hoped, and I cannot imagine having been able to complete my degree without your help. Thank you also to my ARC colleagues, especially Bob, Brian, James, and John, who helped me along the way.Thank you to the many members of the Math Department who made this possible: to Peter for your work in building such supportive department and for your advice and friendship over almost two decades, to Nicole for always being there whenever I had a question, and to the many other members of the department who provided support, advice, and friendship over the years.

PACS. 05.70.Ln -Nonequilibrium and irreversible thermodynamics. PACS. 64.60.Cn -Order-disorder transformations; statistical mechanics of model systems. PACS. 89.75.Kd -Patterns.Abstract. -Motivated by an analogy with traffic, we simulate two species of particles ('vehicles'), moving stochastically in opposite directions on a two-lane ring road. Each species prefers one lane over the other, controlled by a parameter 0 ≤ b ≤ 1 such that b = 0 corresponds to random lane choice and b = 1 to perfect 'laning'. We find that the system displays one large cluster ('jam') whose size increases with b, contrary to intuition. Even more remarkably, the lane 'charge' (a measure for the number of particles in their preferred lane) exhibits a region of negative response: even though vehicles experience a stronger preference for the 'right' lane, more of them find themselves in the 'wrong' one! For b very close to 1, a sharp transition restores a homogeneous state. Various characteristics of the system are computed analytically, in good agreement with simulation data.c EDP Sciences

We demonstrate the efficacy of a Bayesian statistical inversion framework for reconstructing the likely characteristics of large pre‐instrumentation earthquakes from historical records of tsunami observations. Our framework is designed and implemented for the estimation of the location and magnitude of seismic events from anecdotal accounts of tsunamis including shoreline wave arrival times, heights, and inundation lengths over a variety of spatially separated observation locations. The primary advantage of this approach is that all of the assumptions made in the inversion process are incorporated explicitly into the mathematical framework. As an initial test case we use our framework to reconstruct the great 1852 earthquake and tsunami of eastern Indonesia. Relying on the assumption that these observations were produced by a subducting thrust event, the posterior distribution indicates that the observables were the result of a massive mega‐thrust event with magnitude near 8.8 Mw and a likely rupture zone in the north‐eastern Banda arc. The distribution of predicted epicentral locations overlaps with the largest major seismic gap in the region as indicated by instrumentally recorded seismic events. These results provide a geologic and seismic context for hazard risk assessment in coastal communities experiencing growing population and urbanization in Indonesia. In addition, the methodology demonstrated here highlights the potential for applying a Bayesian approach to enhance understanding of the seismic history of other subduction zones around the world.

This work develops a powerful and versatile framework for determining acceptance ratios in Metropolis-Hastings type Markov kernels widely used in statistical sampling problems. Our approach allows us to derive new classes of kernels which unify popular random walk or diffusion-type sampling methods with more complicated 'extended phase space' algorithms based on Hamiltonian dynamics. The starting point for our approach is an abstract result developed in the generality of measurable state spaces that addresses proposal kernels that possess a certain involution structure. Note that, while this underlying involution proposal structure suggests a scope which includes Hamiltonian-type kernel structures, we demonstrate that our abstract result also recovers a broad range of kernels encompassed by a previous general state space approach in [Tie98].On the basis of our abstract result we develop two new classes of 'generalized' Hamiltonian Monte Carlo (gHMC) algorithms first in the classical finite-dimensional setting and then in an infinite-dimensional setting as considered in [BRSV08, BPSSS11, CRSW13]. These gHMC algorithms provide a versatile methodology to bypass expensive gradient computations through skillful reduced order modeling and/or data driven approaches as we begin to explore in a forthcoming companion work, [GHKM20]. Moreover, these gHMC algorithms, along with the connection of our abstract result to the framework in [Tie98], provide a unified picture connecting a number of popular existing algorithms which arise as special cases of gHMC under suitable parameter choices. We show that in the finite-dimensional case our gHMC algorithm includes the Metropolis adjusted Langevin algorithm (MALA) and random walk Metropolis (RWM) while in the infinite-dimensional situation, the gHMC algorithm includes the preconditioned Crank-Nicolson (pCN) and ∞MALA algorithms as special cases. In the later infinite-dimensional case we furthermore show that we can rigorously derive the reversibility of the ∞HMC method from [BPSSS11] without resorting to involved finite-dimensionalization arguments.