Publications

We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling. © 2006 Elsevier Inc. All rights reserved.

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A differential system that models the circadian rhythm in Drosophila is analyzed with the computational singular perturbation (CSP) algorithm. Reduced nonstiff models of prespecified accuracy are constructed, the form and size of which are time-dependent. When compared with conventional asymptotic analysis, CSP exhibits superior performance in constructing reduced models, since it can algorithmically identify and apply all the required order of magnitude estimates and algebraic manipulations. A similar performance is demonstrated by CSP in generating data that allow for the acquisition of physical understanding. It is shown that the processes driving the circadian cycle are (i) mRNA translation into monomer protein, and monomer protein destruction by phosphorylation and degradation (along the largest portion of the cycle); and (ii) mRNA synthesis (along a short portion of the cycle). These are slow processes. Their action in driving the cycle is allowed by the equilibration of the fastest processes; (1) the monomer dimerization with the dimer dissociation (along the largest portion of the cycle); and (2) the net production of monomer+dimmer proteins with that of mRNA (along the short portion of the cycle). Additional results (regarding the time scales of the established equilibria, their origin, the rate limiting steps, the couplings among the variables, etc.) highlight the utility of CSP for automated identification of the important underlying dynamical features, otherwise accessible only for simple systems whose various suitable simplifications can easily be recognized. © 2006 Society for Industrial and Applied Mathematics.

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Block-structured adaptively refined meshes (SAMR) strive for efficient resolution of partial differential equations (PDEs) solved on large computational domains by clustering mesh points only where required by large gradients. Previous work has indicated that fourth-order convergence can be achieved on such meshes by using a suitable combination of high-order discretizations, interpolations, and filters and can deliver significant computational savings over conventional second-order methods at engineering error tolerances. In this paper, we explore the interactions between the errors introduced by discretizations, interpolations and filters. We develop general expressions for high-order discretizations, interpolations, and filters, in multiple dimensions, using a Fourier approach, facilitating the high-order SAMR implementation. We derive a formulation for the necessary interpolation order for given discretization and derivative orders. We also illustrate this order relationship empirically using one and two-dimensional model problems on refined meshes. We study the observed increase in accuracy with increasing interpolation order. We also examine the empirically observed order of convergence, as the effective resolution of the mesh is increased by successively adding levels of refinement, with different orders of discretization, interpolation, or filtering.

The role of chain branching in a chemical kinetic system was investigated by analyzing the eigenvalues of the system. We found that in the homogeneous ignition of the hydrogen/air and methane/air mixtures, the branching mechanism gives rise to explosive modes (eigenvalues with positive real parts) in the induction period as expected; however, in their respective premixed flames, we found none. Thus, their existence is not a necessary condition for the propagation of a premixed flame. © 2005 Elsevier Ltd.

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This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H 2-O 2 ignition under supercritical-water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one-dimensional reacting-flow simulations. The computations are used to study sensitivities to both reaction rate pre-exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first-order information to conventional sensitivity analysis, while preserving higher-order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher-order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves. © 2005 Wiley Periodicals, Inc.

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We performed calculations to investigate the classical theories of chain branching and thermal--run--away that lead to the rapid oxidation of fuels. Mathematically, both theories infer the existence of eigenvalues with positive real parts i.e., explosive modes. We found in studies of homogeneous hydrogen--air and the methane--air mixtures that when ignition is initiated by a sufficiently high initial temperature, the transient response of the system exhibits two stages. The first stage is characterized by the existence of explosive modes. The ensuing second stage consists of fast exponential decay modes that bring the system to its equilibrium point. We demonstrated with two examples that the existence of explosive modes is not a necessary condition for the existence of a premixed flame. Homogeneous ignition calculations for mixtures with an initial concentration of radical species suggest that the diffusive transport of radical species is probably responsible for the lack of explosive modes in premixed flames.