Publications

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Uncertainty quantification in complex climate models is challenged by the sparsity of available climate model predictions due to the high computational cost of model runs. Another feature that prevents classical uncertainty analysis from being readily applicable is bifurcative behavior in climate model response with respect to certain input parameters. A typical example is the Atlantic Meridional Overturning Circulation. The predicted maximum overturning stream function exhibits discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO2 forcing. We outline a methodology for uncertainty quantification given discontinuous model response and a limited number of model runs. Our approach is two-fold. First we detect the discontinuity with Bayesian inference, thus obtaining a probabilistic representation of the discontinuity curve shape and location for arbitrarily distributed input parameter values. Then, we construct spectral representations of uncertainty, using Polynomial Chaos (PC) expansions on either side of the discontinuity curve, leading to an averaged-PC representation of the forward model that allows efficient uncertainty quantification. The approach is enabled by a Rosenblatt transformation that maps each side of the discontinuity to regular domains where desirable orthogonality properties for the spectral bases hold. We obtain PC modes by either orthogonal projection or Bayesian inference, and argue for a hybrid approach that targets a balance between the accuracy provided by the orthogonal projection and the flexibility provided by the Bayesian inference - where the latter allows obtaining reasonable expansions without extra forward model runs. The model output, and its associated uncertainty at specific design points, are then computed by taking an ensemble average over PC expansions corresponding to possible realizations of the discontinuity curve. The methodology is tested on synthetic examples of discontinuous model data with adjustable sharpness and structure.

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Uncertainty quantification in climate models is challenged by the prohibitive cost of a large number of model evaluations for sampling. Another feature that often prevents classical uncertainty analysis from being readily applicable is the bifurcative behavior in the climate data with respect to certain parameters. A typical example is the Meridional Overturning Circulation in the Atlantic Ocean. The maximum overturning stream function exhibits a discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO2 forcing. In order to propagate uncertainties from model parameters to model output we use polynomial chaos (PC) expansions to represent the maximum overturning stream function in terms of the uncertain climate sensitivity and CO2 forcing parameters. Since the spectral methodology assumes a certain degree of smoothness, the presence of discontinuities suggests that separate PC expansions on each side of the discontinuity will lead to more accurate descriptions of the climate model output compared to global PC expansions. We propose a methodology that first finds a probabilistic description of the discontinuity given a number of data points. Assuming the discontinuity curve is a polynomial, the algorithm is based on Bayesian inference of its coefficients. Markov chain Monte Carlo sampling is used to obtain joint distributions for the polynomial coefficients, effectively parameterizing the distribution over all possible discontinuity curves. Next, we apply the Rosenblatt transformation to the irregular parameter domains on each side of the discontinuity. This transformation maps a space of uncertain parameters with specific probability distributions to a space of i.i.d standard random variables where orthogonal projections can be used to obtain PC coefficients. In particular, we use uniform random variables that are compatible with PC expansions based on Legendre polynomials. The Rosenblatt transformation and the corresponding PC expansions for the model output on either side of the discontinuity are applied successively for several realizations of the discontinuity curve. The climate model output and its associated uncertainty at specific design points is then computed by taking a quadrature-based integration average over PC expansions corresponding to possible realizations of the discontinuity curve.

Conventional methods for uncertainty quantification are generally challenged in the 'tails' of probability distributions. This is specifically an issue for many climate observables since extensive sampling to obtain a reasonable accuracy in tail regions is especially costly in climate models. Moreover, the accuracy of spectral representations of uncertainty is weighted in favor of more probable ranges of the underlying basis variable, which, in conventional bases does not particularly target tail regions. Therefore, what is ideally desired is a methodology that requires only a limited number of full computational model evaluations while remaining accurate enough in the tail region. To develop such a methodology, we explore the use of surrogate models based on non-intrusive Polynomial Chaos expansions and Galerkin projection. We consider non-conventional and custom basis functions, orthogonal with respect to probability distributions that exhibit fat-tailed regions. We illustrate how the use of non-conventional basis functions, and surrogate model analysis, improves the accuracy of the spectral expansions in the tail regions. Finally, we also demonstrate these methodologies using precipitation data from CCSM simulations.

Uncertainty quantificatio in climate models is challenged by the sparsity of the available climate data due to the high computational cost of the model runs. Another feature that prevents classical uncertainty analyses from being easily applicable is the bifurcative behavior in the climate data with respect to certain parameters. A typical example is the Meridional Overturning Circulation in the Atlantic Ocean. The maximum overturning stream function exhibits discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO{sub 2} forcing. We develop a methodology that performs uncertainty quantificatio in the presence of limited data that have discontinuous character. Our approach is two-fold. First we detect the discontinuity location with a Bayesian inference, thus obtaining a probabilistic representation of the discontinuity curve location in presence of arbitrarily distributed input parameter values. Furthermore, we developed a spectral approach that relies on Polynomial Chaos (PC) expansions on each sides of the discontinuity curve leading to an averaged-PC representation of the forward model that allows efficient uncertainty quantification and propagation. The methodology is tested on synthetic examples of discontinuous data with adjustable sharpness and structure.

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Recent advances in high frame rate complementary metal-oxide-semiconductor (CMOS) cameras coupled with high repetition rate lasers have enabled laser-based imaging measurements of the temporal evolution of turbulent reacting flows. This measurement capability provides new opportunities for understanding the dynamics of turbulence-chemistry interactions, which is necessary for developing predictive simulations of turbulent combustion. However, quantitative imaging measurements using high frame rate CMOS cameras require careful characterization of the their noise, non-linear response, and variations in this response from pixel to pixel. We develop a noise model and calibration tools to mitigate these problems and to enable quantitative use of CMOS cameras. We have demonstrated proof of principle for image de-noising using both wavelet methods and Bayesian inference. The results offer new approaches for quantitative interpretation of imaging measurements from noisy data acquired with non-linear detectors. These approaches are potentially useful in many areas of scientific research that rely on quantitative imaging measurements.

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