How do you determine whether the Earth is warming up?
Juan M. Restrepo, Uncertainty Quantification Group, Mathematics Department, University of Arizona
How does one determine whether the high summer temperatures in Moscow of a few years ago was an extreme climatic fluctuation or the result of a systematic global warming trend? How does one perform an analysis of the causes of this summer's high temperatures in the US, if climate variability is poorly constrained? It is only under exceptional circumstances that one can determine whether a climate signal belongs to a particular statistical distribution. In fact, climate signals are rarely ''statistical"; there is usually no way to obtain enough field data to produce a trend or tendency, based upon data alone. There are other challenges to obtaining a trend: inherent multi-scale manifestations, and nonlinearities and our incomplete knowledge of climate variability. We propose a trend or tendency methodology that does not make use of a parametric or a statistical assumption and is capable of dealing with multi-scale time series. The most important feature of this trend strategy is that it is defined in very precise mathematical terms.

Bayesian Approaches to the Use of Large-Scale Computer Model Output
Mark Berliner, Department of Statistics, Ohio State University
The Bayesian framework offers the opportunity to combine diverse information sources in modeling and prediction, while managing uncertainties associated with those sources. In this talk I focus on approaches for using output from very large computer models. Two illustrations are presented: use of climate system models in the context of climate change analysis and incorporating various analyses in ocean forecasting. Finally, I will review a new Bayesian heuristic for the incorporation of multi-model ensembles in the construction of stochastic models.

Uncertainty Quantification in Global Ocean State Estimation
Patrick Heimbach and Alex Kalmikov, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology
Over the last decade the consortium on "Estimating the Circulation and Climate of the Ocean" (ECCO) has produced optimal estimates of the global time-evolving circulation of the ocean using much of the available satellite and in-situ observations. These estimates form the basis for addressing various problems in climate research. At the heart of the effort is the state-of-the-art MIT ocean general circulation model (MITgcm) and its adjoint. An outstanding issue has remained the provision of formal uncertainties along with the estimates or climate diagnostics derived from them. Here, we present the development of a Hessian-based method for Uncertainty Quantification within the ECCO framework. First and second derivative codes of the MITgcm are generated via algorithmic differentiation and used to propagate uncertainties between observation, control and target variable domains, following the concept of inverse-to-forward uncertainty propagation. Propagation through the model introduces the notion of time-varying uncertainty reduction and observation impact. By way of example, the method is applied to quantify Drake Passage barotropic transport uncertainties.

Big Data Analysis: High-Performance Statistical Computation
Alexander Gray, Skytree Inc. and Georgia Institute of Technology
Virtually every area of science and engineering is either experiencing its own version of a "big data" revolution, or has such opportunities on the horizon. I will survey seven main types of data analysis methods that can be used across all areas, and give examples in astrophysics of new science that can be achieved by applying such methods to massive data sources. The major void today is in the practical computation of such analyses. I'll discuss seven main types of computational bottlenecks and how difficult they are, as well as seven main types of algorithmic concepts that can be used to tackle them, drawing from the fastest practical algorithms across many areas of applied mathematics and computer science.

Probabilistic Integration of Differential Equations for Exact Bayesian Uncertainty Quantification
Mark Girolami, Department of Statistical Science, University College London, UK
Taking a Bayesian approach to inverse problems is challenging. This talk presents general methodology to explicitly characterize the mismatch between the finite-dimensional approximation of the forward model and the infinite-dimensional solution by a well-defined probability measure in Hilbert space. Furthermore this measure provides a novel means of obtaining probabilistic solutions of the forward model that resolves issues related to characterising uncertainty in the solutions of differential equations including chaotic systems and the multiplicity of solutions in e.g. boundary value problems. The probabilistic solutions are employed in the quantification of input uncertainty thus resolving problems with optimistic estimates of system uncertainty.

This is joint work with Oksana Chkrebtii, David Campbell and Ben Calderhead.

Model Calibration with Processed Data Products
Habib Najm, Sandia National Laboratories
Uncertainty quantification in computational models requires a probabilistic specification of uncertain model parameters/inputs. These are generally available from published experimental literature, based on empirical data fitting, in the form of summary statistics, typically given as nominal values and bounds. However, this processed data, while informative, is not fully representative of the raw data behind the measurement, and does not explicitly provide information on the joint probability density function on parameters of interest. There is a need for methods that allow the use of such processed data products to provide an adequate probabilistic representation of model input parameters. We have developed such an algorithm, relying on maximum entropy and approximate Bayesian computation methods. The algorithm pursues the maximum entropy joint posterior on the data-parameter space. It employs available information as constraints that allow the sampling of consistent data sets, and associated parameter posteriors. Marginalizing over the data space provides the marginalized MaxEnt posterior on the parameters. I will describe this construction and illustrate its application in the context of fitting chemical kinetic rate parameters.

Defining and using information in inverse problems
Wolfgang Bangerth, Department of Mathematics, Texas A&M University and
Bart van Bloemen Waanders, Sandia National Laboratory
Colloquially, we frequently refer to the concept of "information" when talking about regularization, the choice of a mesh size, or the degree of ill-posedness of an inverse problem. However, we rarely put in formulas what we mean by that. We also never really make systematic use of it. In this work, we try to be more systematic in defining an "information density" when solving inverse problems, and in using it for regularization, mesh refinement, and optimal experimental design.

Optimal experimental design of large-scale ill posed problems with simultaneous sources
Lior Horesh, Numerical Analysis and Optimization, IBM T J Watson Research Center
With the development of unprecedented levels of sensing technologies, a broad range of parameter estimation problems involve the collection of intrusively large number of observations N. Typically, each such observation involves the excitation of the domain through admission of energy at a set of pre-defined sites and consequent recording of the domain response at another set of positions. It has been observed that similar results can often be obtained by considering a far smaller number K of multiple linear superpositions of experiments with K << N. As the solution to the inverse problem is at best scale linearly with the number of sources, appropriate selection of weighted superimposed simultaneous source would enable computation of the solution to the inverse problem in time O(K) instead of O(N). Devising such a procedure would entail a drastic reduction in data acquisition and processing time and associated costs. The question we attempt to rigorously investigate in this study is: what are the optimal weights? We formulate the problem as an optimal experimental design problem and show that by leveraging techniques from this field an answer is readily available. Obviously, designing optimal experiments requires a consistent statistical framework and inherently its choice plays a major role in the selection of the weights.

Data Assimilation by a Multi-scale Ensemble Kalman Filter
Henning Omre, Department of Mathematical Sciences, Norwegian University of Science & Technology, Trondheim, Norway
The EnKF provides an approximate Monte Carlo solution to forecasts in hidden Markov chain models. The EnKF will be presented and its characteristics will be discussed--with focus on uncertainty quantification.

The number of ensemble members is critical for the performance of the filter. For large-scale problems with computer demanding forward models, computer efficient simplified surrogate models are often used in order to allow larger ensemble sizes. The introduction of these simplified models does not come without bias and precision costs, however. The multi-scale EnKF is designed to quantify these costs, and to provide more representative forecasts and forecast intervals.

Iterative Gaussian samplers in finite precision
Colin Fox, Department of Physics, Electronics Research Group, The University of Otago, Auckland, New Zealand and
Al Parker, Montana State University, Bozeman
Iterative linear solvers and iterative (Gibbs) samplers for Gaussian distributions are equivalent (in exact arithmetic). That is, the convergence theory is the same and the methods have the same reduction in error per iteration. Results for iterative solvers give a sampler stopping criterion, and the number of sampler iterations required for convergence. Interestingly, samplers that are equivalent in infinite precision perform differently in finite precision. We present a state-of-the-art CG-Chebyshev-SSOR Gaussian sampler, and show that in finite precision convergence to N(0, A^{-1}) implies convergence to N(0,A), but the converse is not true.

Structure-Exploiting scalable methods for large-scale UQ problems in high dimensional parameter spaces
Tan Bui-Thanh, Omar Ghattas, James, Martin, Georg Stadler, Institute for Computational Engineering and Sciences, The University of Texas at Austin; Carsten Burstedde, University of Bonn; Lucas Wilcox, Naval Postgraduate School
We review our recent efforts in developing scalable approaches to large-scale Bayesian statistical inverse problems. At the heart of our approach is the exploitation of Hessian information, which captures the local curvature and correlations of the posterior pdf, in turn greatly improving efficiency and scalability to high dimensions. First, we present a Gaussian approximation of the posterior in which we exploit the low rank nature of the misfit Hessian to devise a scalable strategy to approximate infinite dimensional Bayesian inverse problems. Second, we propose a complementary scaled stochastic Newton method that eliminates the dependence on variance in number of iterations required by the chain. Finally, we present our current work on scalable Maximum Randomized Likelihood method for MCMC that is capable of generating approximate independent samples for Bayesian posterior in high dimensional parameter spaces. We demonstrate our methods on large-scale full wave form seismic inversion, inverse thermal fin design, and inverse Helmholtz equation.

Big data meets big models in carbon cycle science: Spatiotemporal tools for constraining the CO2 budget from atmospheric observations
Anna M. Michalak, Department of Global Ecology, Carnegie Institution for Science, Stanford, California
Predicting future changes to the global carbon cycle (and therefore climate) and quantifying anthropogenic emissions of carbon dioxide both require an understanding of net carbon sources and sinks, and their variability, across a variety of spatial and temporal scales. This need highlights the importance of understanding the spatial and temporal scale-dependence of parameters controlling this variability, and developing methods for using data collected at multiple scales to infer carbon fluxes.

This presentation will describe ongoing work examining the carbon cycle from an atmospheric perspective in three major areas: spatiotemporal inverse modeling tools for characterizing uptake and emissions of carbon dioxide at the Earth surface using atmospheric observations of CO2; regression approaches for understanding the scale-dependence of processes controlling carbon flux variability; and mapping tools for massive datasets used for obtaining global atmospheric CO2 distributions from satellite observations. Common challenges across these applications include: the multiscale and nonstationary space-time variability of the signal; the lack of direct observations of CO2 exchange at the Earth surface at the scales of highest interest; the diffusive nature of atmospheric transport, which links the unobserved CO2 exchange at the Earth surface with observations of atmospheric CO2; the large to massive data volumes; and the large to massive size of the state space.

Supervised learning algorithms for construction of likelihood and prior densities for Bayesian inverse medium problems
George Biros, Institute for Computational Engineering and Sciences, The University of Texas at Austin
Remote sensing, image analysis, subsurface characterization, and inverse scattering are examples of problems in which we seek to reconstruct a spatial field ("the medium"). In Bayesian inverse problems we need to construct likelihood and prior probability density functions. I will present a methodology that combines supervised learning with classical Gibbs measures for likelihood and smoothness priors to construct likelihood and prior functions. I will discuss the computational challenges in working with such probability functions and will present experimental results for an image segmentation problem.

Representing Uncertainty due to Inaccurate Models
Robert D. Moser and Todd Oliver, Institute for Computational Engineering and Sciences, The University of Texas at Austin
In the modeling and simulation of complex physical systems, it is common to use models of some phenomena that are approximate or even ad hoc. In assessing the uncertainty in the results of such simulations, it is necessary to account for the inaccuracies of the models. Representations of this model inadequacy are generally statistical and must be formulated to allow the resulting uncertainty to be propagated to the quantities being predicted. The parameters of this inadequacy representation along with physical model parameters need to be calibrated from available observation data in a statistical inverse problem which is made challenging by the fact that the forward problem is now stochastic and commonly high-dimensional. The challenges of formulated and calibrating such inadequacy models will be discussed.

Maintained by: Sue Rodriguez
Modified on: May 20, 2013