5 research outputs found
A Bayesian Approach to Estimating Background Flows from a Passive Scalar
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.Comment: Streamlined, moved IS & MALA to appendix, and added appendix on
fluids observable
GPU-Accelerated Particle Methods for Evaluation of Sparse Observations for PDE-Constrained Inverse Problems
We consider the inverse problem of estimating parameters of a driven
diffusion (e.g., the underlying fluid flow, diffusion coefficient, or source
terms) from point measurements of a passive scalar (e.g., the concentration of
a pollutant). We present two particle methods that leverage the structure of
the inverse problem to enable efficient computation of the forward map, one for
time evolution problems and one for a Dirichlet boundary-value problem. The
methods scale in a natural fashion to modern computational architectures,
enabling substantial speedup for applications involving sparse observations and
high-dimensional unknowns. Numerical examples of applications to Bayesian
inference and numerical optimization are provided
A Statistical Framework for Domain Shape Estimation in Stokes Flows
We develop and implement a Bayesian approach for the estimation of the shape
of a two dimensional annular domain enclosing a Stokes flow from sparse and
noisy observations of the enclosed fluid. Our setup includes the case of direct
observations of the flow field as well as the measurement of concentrations of
a solute passively advected by and diffusing within the flow. Adopting a
statistical approach provides estimates of uncertainty in the shape due both to
the non-invertibility of the forward map and to error in the measurements. When
the shape represents a design problem of attempting to match desired target
outcomes, this "uncertainty" can be interpreted as identifying remaining
degrees of freedom available to the designer. We demonstrate the viability of
our framework on three concrete test problems. These problems illustrate the
promise of our framework for applications while providing a collection of test
cases for recently developed Markov Chain Monte Carlo (MCMC) algorithms
designed to resolve infinite dimensional statistical quantities
On the accept-reject mechanism for Metropolis-Hastings algorithms
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 random walk or diffusion-type sampling methods with more
complicated "extended phase space" algorithms based around ideas from
Hamiltonian dynamics. Our starting point 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
proposal structure suggests a scope which includes Hamiltonian-type kernels, we
demonstrate that our abstract result is, in an appropriate sense, equivalent to
an earlier general state space setting developed in [Tierney, Annals of Applied
Probability, 1998] where the connection to Hamiltonian methods was more
obscure. Altogether, the theoretical unity and reach of our main result
provides a basis for deriving novel sampling algorithms while laying bare
important relationships between existing methods
Methodological reconstruction of historical seismic events from anecdotal accounts of destructive tsunamis: a case study for the great 1852 Banda arc mega-thrust earthquake and tsunami
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. 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