209 research outputs found
Exploiting network topology for large-scale inference of nonlinear reaction models
The development of chemical reaction models aids understanding and prediction
in areas ranging from biology to electrochemistry and combustion. A systematic
approach to building reaction network models uses observational data not only
to estimate unknown parameters, but also to learn model structure. Bayesian
inference provides a natural approach to this data-driven construction of
models. Yet traditional Bayesian model inference methodologies that numerically
evaluate the evidence for each model are often infeasible for nonlinear
reaction network inference, as the number of plausible models can be
combinatorially large. Alternative approaches based on model-space sampling can
enable large-scale network inference, but their realization presents many
challenges. In this paper, we present new computational methods that make
large-scale nonlinear network inference tractable. First, we exploit the
topology of networks describing potential interactions among chemical species
to design improved "between-model" proposals for reversible-jump Markov chain
Monte Carlo. Second, we introduce a sensitivity-based determination of move
types which, when combined with network-aware proposals, yields significant
additional gains in sampling performance. These algorithms are demonstrated on
inference problems drawn from systems biology, with nonlinear differential
equation models of species interactions
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Efficient Localization of Discontinuities in Complex Computational Simulations
Surrogate models for computational simulations are input-output
approximations that allow computationally intensive analyses, such as
uncertainty propagation and inference, to be performed efficiently. When a
simulation output does not depend smoothly on its inputs, the error and
convergence rate of many approximation methods deteriorate substantially. This
paper details a method for efficiently localizing discontinuities in the input
parameter domain, so that the model output can be approximated as a piecewise
smooth function. The approach comprises an initialization phase, which uses
polynomial annihilation to assign function values to different regions and thus
seed an automated labeling procedure, followed by a refinement phase that
adaptively updates a kernel support vector machine representation of the
separating surface via active learning. The overall approach avoids structured
grids and exploits any available simplicity in the geometry of the separating
surface, thus reducing the number of model evaluations required to localize the
discontinuity. The method is illustrated on examples of up to eleven
dimensions, including algebraic models and ODE/PDE systems, and demonstrates
improved scaling and efficiency over other discontinuity localization
approaches
Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems
One of the major challenges in the Bayesian solution of inverse problems
governed by partial differential equations (PDEs) is the computational cost of
repeatedly evaluating numerical PDE models, as required by Markov chain Monte
Carlo (MCMC) methods for posterior sampling. This paper proposes a data-driven
projection-based model reduction technique to reduce this computational cost.
The proposed technique has two distinctive features. First, the model reduction
strategy is tailored to inverse problems: the snapshots used to construct the
reduced-order model are computed adaptively from the posterior distribution.
Posterior exploration and model reduction are thus pursued simultaneously.
Second, to avoid repeated evaluations of the full-scale numerical model as in a
standard MCMC method, we couple the full-scale model and the reduced-order
model together in the MCMC algorithm. This maintains accurate inference while
reducing its overall computational cost. In numerical experiments considering
steady-state flow in a porous medium, the data-driven reduced-order model
achieves better accuracy than a reduced-order model constructed using the
classical approach. It also improves posterior sampling efficiency by several
orders of magnitude compared to a standard MCMC method
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