Performing Bayesian inference via Markov chain Monte Carlo (MCMC) can be exceedingly expensive when posterior evaluations invoke the evaluation of a computationally expensive model, such as a system of partial differential equations. In recent work [Conrad et al. JASA 2016, arXiv:1402.1694], we described a framework for constructing and refining local approximations of such models during an MCMC simulation. These posterior--adapted approximations harness regularity of the model to reduce the computational cost of inference while preserving asymptotic exactness of the Markov chain. Here we describe two extensions of that work. First, we prove that samplers running in parallel can collaboratively construct a shared posterior approximation while ensuring ergodicity of each associated chain, providing a novel opportunity for exploiting parallel computation in MCMC. Second, focusing on the Metropolis--adjusted Langevin algorithm, we describe how a proposal distribution can successfully employ gradients and other relevant information extracted from the approximation. We investigate the practical performance of our strategies using two challenging inference problems, the first in subsurface hydrology and the second in glaciology. Using local approximations constructed via parallel chains, we successfully reduce the run time needed to characterize the posterior distributions in these problems from days to hours and from months to days, respectively, dramatically improving the tractability of Bayesian inference.

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

QUEST (\url {www. quest-scidac. org}) is a SciDAC Institute that is focused on uncertainty quantification (UQ) in large-scale scientific computations. Our goals are to (1) advance the state of the art in UQ mathematics, algorithms, and software; and (2) provide modeling, algorithmic, and general UQ expertise, together with software tools, to other SciDAC projects, thereby enabling and guiding a broad range of UQ activities in their respective contexts. QUEST is a collaboration among six institutions (Sandia National Laboratories, Los Alamos National Laboratory, the University of Southern California, Massachusetts Institute of Technology, the University of Texas at Austin, and Duke University) with a history of joint UQ research. Our vision encompasses all aspects of UQ in leadership-class computing. This includes the well-founded setup of UQ problems; characterization of the input space given available data/information; local and global sensitivity analysis; adaptive dimensionality and order reduction; forward and inverse propagation of uncertainty; handling of application code failures, missing data, and hardware/software fault tolerance; and model inadequacy, comparison, validation, selection, and averaging. The nature of the UQ problem requires the seamless combination of data, models, and information across this landscape in a manner that provides a self-consistent quantification of requisite uncertainties in predictions from computational models. Accordingly, more»

QUEST (\url{www.quest-scidac.org}) is a SciDAC Institute that is focused on uncertainty quantification (UQ) in large-scale scientific computations. Our goals are to (1) advance the state of the art in UQ mathematics, algorithms, and software; and (2) provide modeling, algorithmic, and general UQ expertise, together with software tools, to other SciDAC projects, thereby enabling and guiding a broad range of UQ activities in their respective contexts. QUEST is a collaboration among six institutions (Sandia National Laboratories, Los Alamos National Laboratory, the University of Southern California, Massachusetts Institute of Technology, the University of Texas at Austin, and Duke University) with a history of joint UQ research. Our vision encompasses all aspects of UQ in leadership-class computing. This includes the well-founded setup of UQ problems; characterization of the input space given available data/information; local and global sensitivity analysis; adaptive dimensionality and order reduction; forward and inverse propagation of uncertainty; handling of application code failures, missing data, and hardware/software fault tolerance; and model inadequacy, comparison, validation, selection, and averaging. The nature of the UQ problem requires the seamless combination of data, models, and information across this landscape in a manner that provides a self-consistent quantification of requisite uncertainties in predictions from computational models. Accordingly, our UQ methods and tools span an interdisciplinary space across applied math, information theory, and statistics. The MIT QUEST effort centers on statistical inference and methods for surrogate or reduced-order modeling. MIT personnel have been responsible for the development of adaptive sampling methods, methods for approximating computationally intensive models, and software for both forward uncertainty propagation and statistical inverse problems. A key software product of the MIT QUEST effort is the MIT Uncertainty Quantification library, called MUQ (\url{muq.mit.edu}).

We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis-Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler's exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the \emph{exact} posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this paper: when the likelihood has some \emph{local} regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ODE and PDE inference problems, with both synthetic and real data.

Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for non-intrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the non-adaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.

This work explores the implementation of an adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed pseudo-spectral algorithm that is based on a direct application of the Smolyak sparse grid formula and that allows the use of arbitrary admissible sparse grids. The adaptive algorithm is tested using an existing simulation database of the oceanic response to Hurricane Ivan in the Gulf of Mexico. The a priori tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling in the present setting.

This work presents Drake, a dynamic executive for temporal plans with choice. Dynamic plan execution strategies allow an autonomous agent to react quickly to unfolding events, improving the robustness of the agent. Prior work developed methods for dynamically dispatching Simple Temporal Networks, and further research enriched the expressiveness of the plans executives could handle, including discrete choices, which are the focus of this work. However, in some approaches to date, these additional choices induce significant storage or latency requirements to make flexible execution possible. Drake is designed to leverage the low latency made possible by a preprocessing step called compilation, while avoiding high memory costs through a compact representation. We leverage the concepts of labels and environments, taken from prior work in Assumption-based Truth Maintenance Systems (ATMS), to concisely record the implications of the discrete choices, exploiting the structure of the plan to avoid redundant reasoning or storage. Our labeling and maintenance scheme, called the Labeled Value Set Maintenance System, is distinguished by its focus on properties fundamental to temporal problems, and, more generally, weighted graph algorithms. In particular, the maintenance system focuses on maintaining a minimal representation of non-dominated constraints. We benchmark Drakes performance on random structured problems, and find that Drake reduces the size of the compiled representation by a factor of over 500 for large problems, while incurring only a modest increase in run-time latency, compared to prior work in compiled executives for temporal plans with discrete choices.

We introduce a novel task execution capability that enhances the ability of in-situ crew members to function independently from Earth by enabling safe and efficient interaction with automated systems. This task execution capability provides the ability to (1) map goal-directed commands from humans into safe, compliant, automated actions, (2) quickly and safely respond to human commands and actions during task execution, and (3) specify complex motions through teaching by demonstration. Our results are applicable to future surface robotic systems, and we have demonstrated these capabilities on JPL's All-Terrain Hex-Limbed Extra-Terrestrial Explorer (ATHLETE) robot.

Dynamic plan execution strategies allow an autonomous agent to respond to uncertainties while improving robustness and reducing the need for an overly conservative plan. Executives have improved this robustness by expanding the types of choices made dynamically, such as selecting alternate methods. However, in methods to date, these additional choices introduce substantial run-time latency. This paper presents a novel system called Drake that makes steps towards executing an expanded set of choices dynamically without significant latency. Drake frames a plan as a Disjunctive Temporal Problem and executes it with a fast dynamic scheduling algorithm. Prior work demonstrated an efficient technique for dynamic execution of one special type of DTPs by using an off-line compilation step to find the possible consistent choices and compactly record the differences between them. Drake extends this work to handle a more general set of choices by recording the minimal differences between the solutions which are required at run-time. On randomly generated structured plans with choice, we show a reduction in the size of the solution set of over two orders of magnitude, compared to prior art.

An essential quality of a good partner is her responsiveness to other team members. Recent work in dynamic plan execution exhibits elements of this quality through the ability to adapt to the temporal uncertainties of others agents and the environment. However, a good teammate also has the ability to adapt on-the-fly through task assignment. We generalize the framework of dynamic execution to perform plan execution with dynamic task assignment as well as scheduling.This paper introduces Chaski, a multi-agent executive for scheduling temporal plans with online task assignment. Chaski enables an agent to dynamically update its plan in response to disturbances in task assignment and the schedule of other agents. The agent then uses the updated plan to choose, schedule and execute actions that are guaranteed to be temporally consistent and logically valid within the multi-agent plan. Chaski is made efficient through an incremental algorithm that compactly encodes all scheduling policies for all possible task assignments. We apply Chaski to perform multi-manipulator coordination using two Barrett Arms within the authors' hardware testbed. We empirically demonstrate up to one order of magnitude improvements in execution latency and solution compactness compared to prior art.

We present a comparison of two methods for color based road segmentation. The first was implemented using a neural network, while the second approach is based on support vector machines. A large number of training images were used with varying road conditions including roads with snow, dirt or gravel surfaces, and asphalt. We experimented with grouping the training images by road condition and generating a separate model for each group. The system would automatically select the appropriate one for each novel image. Those results were compared with creating a single model with all images. In another set of experiments, we added the image coordinates of each point as an additional feature in the models. Finally, we compared the results and the efficiency of neural networks and support vector machines of segmentation with each combination of feature sets and image groups.