Publications

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Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.

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We present a preliminary investigation of the use of Multi-Layer Perceptrons (MLP) and Recurrent Neural Networks (RNNs) as surrogates of parameter-to-prediction maps of computational expensive dynamical models. In particular, we target the approximation of Quantities of Interest (QoIs) derived from the solution of a Partial Differential Equations (PDEs) at different time instants. In order to limit the scope of our study while targeting a relevant application, we focus on the problem of computing variations in the ice sheets mass (our QoI), which is a proxy for global mean sea-level changes. We present a number of neural network formulations and compare their performance with that of Polynomial Chaos Expansions (PCE) constructed on the same data.

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

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The recovery of approximately sparse or compressible coefficients in a polynomial chaos expansion is a common goal in many modern parametric uncertainty quantification (UQ) problems. However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized t1 minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform (the theoretical guarantees hold for all compressible signals and compressible corruptions vectors), and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.

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Uncertainty pervades the representation of systems in the water–environment–agriculture cross-sector. Successful methods to address uncertainties have largely focused on standard mathematical formulations of biophysical processes in a single sector, such as partial or ordinary differential equations. More attention to integrated models of such systems is warranted. Model components representing the different sectors of an integrated model can have less standard, and different, formulations to one another, as well as different levels of epistemic knowledge and data informativeness. Thus, uncertainty is not only pervasive but also crosses boundaries and propagates between system components. Uncertainty assessment (UA) cries out for more eclectic treatment in these circumstances, some of it being more qualitative and empirical. Here in this paper, we discuss the various sources of uncertainty in such a cross-sectoral setting and ways to assess and manage them. We have outlined a fast-growing set of methodologies, particularly in the computational mathematics literature on uncertainty quantification (UQ), that seem highly pertinent for uncertainty assessment. There appears to be considerable scope for advancing UA by integrating relevant UQ techniques into cross-sectoral problem applications. Of course this will entail considerable collaboration between domain specialists who often take first ownership of the problem and computational methods experts.

We consider the utilization of a computational model to guide the optimal acquisition of experimental data to inform the stochastic description of model input parameters. Our formulation is based on the recently developed consistent Bayesian approach for solving stochastic inverse problems, which seeks a posterior probability density that is consistent with the model and the data in the sense that the push-forward of the posterior (through the computational model) matches the observed density on the observations almost everywhere. Given a set of potential observations, our optimal experimental design (OED) seeks the observation, or set of observations, that maximizes the expected information gain from the prior probability density on the model parameters. We discuss the characterization of the space of observed densities and a computationally efficient approach for rescaling observed densities to satisfy the fundamental assumptions of the consistent Bayesian approach. Numerical results are presented to compare our approach with existing OED methodologies using the classical/statistical Bayesian approach and to demonstrate our OED on a set of representative partial differential equations (PDE)-based models.

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We formulate, and present a numerical method for solving, an inverse problem for inferring parameters of a deterministic model from stochastic observational data on quantities of interest. The solution, given as a probability measure, is derived using a Bayesian updating approach for measurable maps that finds a posterior probability measure that when propagated through the deterministic model produces a push-forward measure that exactly matches the observed probability measure on the data. Our approach for finding such posterior measures, which we call consistent Bayesian inference or push-forward based inference, is simple and only requires the computation of the push-forward probability measure induced by the combination of a prior probability measure and the deterministic model. We establish existence and uniqueness of observation-consistent posteriors and present both stability and error analyses. We also discuss the relationships between consistent Bayesian inference, classical/statistical Bayesian inference, and a recently developed measure-theoretic approach for inference. Finally, analytical and numerical results are presented to highlight certain properties of the consistent Bayesian approach and the differences between this approach and the two aforementioned alternatives for inference.

Within the SEQUOIA project, funded by the DARPA EQUiPS program, we pursue algorithmic approaches that enable comprehensive design under uncertainty, through inclusion of aleatory/parametric and epistemic/model form uncertainties within scalable forward/inverse UQ approaches. These statistical methods are embedded within design processes that manage computational expense through active subspace, multilevel-multifidelity, and reduced-order modeling approximations. To demonstrate these methods, we focus on the design of devices that involve multi-physics interactions in advanced aerospace vehicles. A particular problem of interest is the shape design of nozzles for advanced vehicles such as the Northrop Grumman UCAS X-47B, involving coupled aero-structural-thermal simulations for nozzle performance. In this paper, we explore a combination of multilevel and multifidelity forward and inverse UQ algorithms to reduce the overall computational cost of the analysis by leveraging hierarchies of model form (i.e., multifidelity hierarchies) and solution discretization (i.e., multilevel hierarchies) in order of exploit trade offs between solution accuracy and cost. In particular, we seek the most cost effective fusion of information across complex multi-dimensional modeling hierarchies. Results to date indicate the utility of multiple approaches, including methods that optimally allocate resources when estimator variance varies smoothly across levels, methods that allocate sufficient sampling density based on sparsity estimates, and methods that employ greedy multilevel refinement.

The recovery of approximately sparse or compressible coefficients in a polynomial chaos expansion is a common goal in many modern parametric uncertainty quantification (UQ) problems. However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized \ell1 minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform (the theoretical guarantees hold for all compressible signals and compressible corruptions vectors) and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter and empirically produces superior results. Our numerical results test our framework on several medium to high dimensional examples of solutions to parameterized differential equations and demonstrate the effectiveness of our approach.

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