We consider extinction times for a class of birth-death processes commonly found in applications, where there is a control parameter which determines whether the population quickly becomes extinct, or rather persists for a long time. We give an exact expression for the discrete case and its asymptotic expansion for large values of the population. We have results below the threshold, at the threshold, and above the threshold (where there is a quasi-stationary state and the extinction time is very long.) We show that the Fokker-Planck approximation is valid only quite near the threshold. We compare our analytical results to numerical simulations for the SIS epidemic model, which is in the class that we treat. This is an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.

Uncertainty quantification in complex physical models is often challenged by the computational expense of these models. One often needs to operate under the assumption of sparsely available model simulations. This issue is even more critical when models include a large number of input parameters. This "curse of dimensionality," in particular, leads to a prohibitively large number of basis terms in spectral methods for uncertainty quantification, such as polynomial chaos (PC) methods. In this work, we implement a PC-based surrogate model construction that "learns" and retains only the most relevant basis terms of the PC expansion, using sparse Bayesian learning. This dramatically reduces the dimensionality of the problem, making it more amenable to further analysis such as sensitivity or calibration studies. The model of interest is the community land model with about 80 input parameters, which also exhibits nonsmooth input-output behavior. We enhanced the methodology by a clustering and classifying procedure that leads to a piecewise-PC surrogate thereby dealing with nonlinearity. We then obtain global sensitivity information for five outputs with respect to all input parameters using less than 10,000 model simulations-a very small number for an 80-dimensional input parameter space.

We introduce a novel statistical calibration framework for physical models, relying on probabilistic embedding of model discrepancy error within the model. For clarity of illustration, we take the measurement errors out of consideration, calibrating a chemical model of interest with respect to a more detailed model, considered as "truth" for the present purpose. We employ Bayesian statistical methods for such model-to-model calibration and demonstrate their capabilities on simple synthetic models, leading to a well-defined parameter estimation problem that employs approximate Bayesian computation. The method is then demonstrated on two case studies for calibration of kinetic rate parameters for methane air chemistry, where ignition time information from a detailed elementary-step kinetic model is used to estimate rate coefficients of a simple chemical mechanism. We show that the calibrated model predictions fit the data and that uncertainty in these predictions is consistent in a mean-square sense with the discrepancy from the detailed model data.

In this paper we present a basis selection method that can be used with 1-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets that have more terms in important dimensions and limits the number of unimportant terms that increase mutual coherence and thus degrade the performance of 1-minimization. The important features and the accuracy of basis selection are demonstrated with a number of numerical examples. Specifically, we show that for a given computational budget, basis selection produces a more accurate PCE than would be obtained if the basis is fixed a priori. We also demonstrate that basis selection can be applied with non-uniform random variables and can leverage gradient information.

We conduct a global sensitivity analysis (GSA) of the Energy Exascale Earth System Model (E3SM), land model (ELM) to calculate the sensitivity of five key carbon cycle outputs to 68 model parameters. This GSA is conducted by first constructing a Polynomial Chaos (PC) surrogate via new Weighted Iterative Bayesian Compressive Sensing (WIBCS) algorithm for adaptive basis growth leading to a sparse, high‐dimensional PC surrogate with 3,000 model evaluations. The PC surrogate allows efficient extraction of GSA information leading to further dimensionality reduction. The GSA is performed at 96 FLUXNET sites covering multiple plant functional types (PFTs) and climate conditions. About 20 of the model parameters are identified as sensitive with the rest being relatively insensitive across all outputs and PFTs. These sensitivities are dependent on PFT, and are relatively consistent among sites within the same PFT. The five model outputs have a majority of their highly sensitive parameters in common. A common subset of sensitive parameters is also shared among PFTs, but some parameters are specific to certain types (e.g., deciduous phenology). The relative importance of these parameters shifts significantly among PFTs and with climatic variables such as mean annual temperature.