Publications

Abstract not provided.

Abstract not provided.

Abstract not provided.

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.

Abstract not provided.

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.

Abstract not provided.

Abstract not provided.

Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a low-dimensional approximation (or sparse approximation) for some quantity of interest in a model, where just a few orthogonal basis polynomials are required. In this work, we consider linear dynamical systems consisting of ordinary differential equations with random variables. The aim of this paper is to explore methods for producing low-dimensional approximations of the quantity of interest further. We investigate two numerical techniques to compute a low-dimensional representation, which both fit the approximation to a set of samples in the time domain. On the one hand, a frequency domain analysis of a stochastic Galerkin system yields the selection of the basis polynomials. It follows a linear least squares problem. On the other hand, a sparse minimization yields the choice of the basis polynomials by information from the time domain only. An orthogonal matching pursuit produces an approximate solution of the minimization problem. Finally, we compare the two approaches using a test example from a mechanical application.

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.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

We propose an algorithm for recovering sparse orthogonal polynomial expansions via collocation. A standard sampling approach for recovering sparse polynomials uses Monte Carlo sampling, from the density of orthogonality, which results in poor function recovery when the polynomial degree is high. Our proposed approach aims to mitigate this limitation by sampling with respect to the weighted equilibrium measure of the parametric domain and subsequently solves a preconditioned'1-minimization problem, where the weights of the diagonal preconditioning matrix are given by evaluations of the Christoffel function. Our algorithm can be applied to a wide class of orthogonal polynomial families on bounded and unbounded domains, including all classical families. We present theoretical analysis to motivate the algorithm and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest. Numerical examples are also provided to demonstrate that our proposed algorithm leads to comparable or improved accuracy even when compared with Legendre- and Hermite-specific algorithms.