Bert J. Debusschere

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include {Taylor} series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.

Spectral stochastic uncertainty quantification in chemical systems

Uncertainty quantification (UQ) in the computational modelling of physical systems is important for scientific investigation, engineering design, and model validation. We have implemented an ‘intrusive’ UQ technique in which (1) model parameters and field variables are modelled as stochastic quantities, and are represented using polynomial chaos (PC) expansions in terms of Hermite polynomial functions of Gaussian random variables, and (2) the deterministic model equations are reformulated using Galerkin projection into a set of equations for the time evolution of the field variable PC mode strengths. The mode strengths relate specific parametric uncertainties to their effects on model outputs. In this work, the intrusive reformulation is applied to homogeneous ignition using a detailed chemistry model through the development of a reformulated pseudospectral chemical source term. We present results analysing the growth of uncertainty during the ignition process. We also discuss numerical issues pertaining to the accurate representation of uncertainty with truncated PC expansions, and ensuing stability of the time integration of the chemical system.

Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation

This paper presents a model for two-dimensional electrochemical microchannel flow including the propagation of uncertainty from model parameters to the simulation results. For a detailed representation of electroosmotic and pressure-driven microchannel flow, the model considers the coupled momentum, species transport, and electrostatic field equations, including variable zeta potential. The chemistry model accounts for pH-dependent protein labeling reactions as well as detailed buffer electrochemistry in a mixed finite-rate/equilibrium formulation. Uncertainty from the model parameters and boundary conditions is propagated to the model predictions using a pseudo-spectral stochastic formulation with polynomial chaos (PC) representations for parameters and field quantities. Using a Galerkin approach, the governing equations are reformulated into equations for the coefficients in the PC expansion. The implementation of the physical model with the stochastic uncertainty propagation is applied to protein-labeling in a homogeneous buffer, as well as in two-dimensional electrochemical microchannel flow. The results for the two-dimensional channel show strong distortion of sample profiles due to ion movement and consequent buffer disturbances. The uncertainty in these results is dominated by the uncertainty in the applied voltage across the channel.

A multigrid solver for two-dimensional stochastic diffusion equations

Steady and unsteady diffusion equations, with stochastic diffusivity coefficient and forcing term, are modeled in two dimensions by means of stochastic spectral representations. Problem data and solution variables are expanded using the Polynomial Chaos system. The approach leads to a set of coupled problems for the stochastic modes. Spatial finite-difference discretization of these coupled problems results in a large system of equations, whose dimension necessitates the use of iterative approaches in order to obtain the solution within a reasonable computational time. To accelerate the convergence of the iterative technique, a multigrid method, based on spatial coarsening, is implemented. Numerical experiments show good scaling properties of the method, both with respect to the number of spatial grid points and the stochastic resolution level.

Uncertainty Quantification in MD Simulations. Part II: Bayesian Inference of Force-Field Parameters

This paper explores the inference of small-scale, atomistic parameters, based on the specification of large, or macroscale, observables. Specifically, we focus on estimating a set of force-field parameters for the four-site, TIP4P, water model, based on a synthetic problem involving isothermal, isobaric molecular dynamics (MD) simulations of water at ambient conditions. We exploit the polynomial chaos (PC) expansions developed in Part I as surrogate representations of three macroscale observables, namely density, self-diffusion, and enthalpy, as a function of the force-field parameters. We analyze and discuss the use of two different PC representations in a Bayesian framework for the inference of atomistic parameters, based on synthetic observations of three macroscale observables. The first surrogate is a deterministic PC representation, constructed in Part I using nonintrusive spectral projection (NISP). An alternative strategy exploits a nondeterministic PC representation obtained using Bayesian infere...

Sources of Cell-to-cell Variability in Canonical Nuclear Factor-κB (NF-κB) Signaling Pathway Inferred from Single Cell Dynamic Images

The canonical nuclear factor-κB (NF-κB) signaling pathway controls a gene network important in the cellular inflammatory response. Upon activation, NF-κB/RelA is released from cytoplasmic inhibitors, from where it translocates into the nucleus, subsequently activating negative feedback loops producing either monophasic or damped oscillatory nucleo-cytoplasmic dynamics. Although the population behavior of the NF-κB pathway has been extensively modeled, the sources of cell-to-cell variability are not well understood. We describe an integrated experimental-computational analysis of NF-κB/RelA translocation in a validated cell model exhibiting monophasic dynamics. Quantitative measures of cellular geometry and total cytoplasmic concentration and translocated RelA amounts were used as priors in Bayesian inference to estimate biophysically realistic parameter values based on dynamic live cell imaging studies of enhanced GFP-tagged RelA in stable transfectants. Bayesian inference was performed on multiple cells simultaneously, assuming identical reaction rate parameters, whereas cellular geometry and initial and total NF-κB concentration-related parameters were cell-specific. A subpopulation of cells exhibiting distinct kinetic profiles was identified that corresponded to differences in the IκBα translation rate. We conclude that cellular geometry, initial and total NF-κB concentration, IκBα translation, and IκBα degradation rates account for distinct cell-to-cell differences in canonical NF-κB translocation dynamics.

Uncertainty Quantification in MD Simulations. Part I: Forward Propagation

This work focuses on quantifying the effect of intrinsic (thermal) noise and parametric uncertainty in molecular dynamics (MD) simulations. We consider isothermal, isobaric MD simulations of TIP4P (or four-site) water at ambient conditions,

Ovis-2: A robust distributed architecture for scalable RAS

T=298

Natural Convection in a Closed Cavity under Stochastic Non-Boussinesq Conditions

K and

Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. I. Sensitivity to physical parameters of the pore

P=1