Causality and Bayesian network PDEs for multiscale representations of porous media

A hierarchical nanoporous material. Above, a hierarchical nanoporous meta-material, that arises in the context of designing energy storage devices, exhibits horizontally oriented nanotunnels through mesopores connected by a series of vertically oriented nanotubes. The porous media volume (left) consists of a periodic arrangement of unit cells (right) with pore space \(\breve{\mathcal{P}}\) and fluid-solid interface \(\breve{\Gamma}\). The parameters \(\{R, θ, d, l\}\) describing the nanopore features are constrained by the geometry of the unit cell (from [7]).

Microscopic (pore-scale) properties of porous media affect and often determine their macroscopic (continuum- or Darcy-scale) counterparts. Understanding the relationship between processes on these two scales is essential to both the derivation of macroscopic models of transport phenomena in natural porous media and the design of novel materials for energy storage. Most microscopic properties exhibit complex statistical correlations and geometric constraints, which presents challenges for the estimation of macroscopic quantities of interest (QoIs) for example in the context of global sensitivity analysis (GSA) of macroscopic QoIs with respect to miscroscopic material properties. We present a systematic way of building correlations into stochastic multiscale models through Bayesian networks. This allows us to construct the joint probability density function (PDF) of model parameters through causal relationships that emulate engineering processes in the design of hierarchical nanoporous materials. Such PDFs also serve as input for the forward propagation of parametric uncertainty; our findings indicate that the inclusion of causal relationships impacts predictions of macroscopic QoIs. To assess the impact of correlations and causal relationships between microscopic parameters on macroscopic material properties, we use a moment-independent GSA based on the differential mutual information. Our GSA accounts for the correlated inputs and complex non-Gaussian QoIs. The global sensitivity indices are used to rank the effect of uncertainty in microscopic parameters on macroscopic QoIs, to quantify the impact of causality on the multiscale model’s predictions, and to provide physical interpretations of these results for hierarchical nanoporous materials.

Bayesian network. The rich structures of the Bayesian network above represents a probabilistic model that encodes causal relationships among model components at different problem scales, \(\mathbf{\Theta}\), \(\mathbf{X}\), \(U\), and among a single problem scale, \(\mathbf{\Theta} = (\Theta_R, \Theta_l, \Theta_\theta, \Theta_d)\), arising from natural structural constraints (from [7]).
  1. Causality and Bayesian Network PDEs for multiscale representations of porous media, with K. Um, M. Katsoulakis, and D. Tartakovsky, Journal of Computational Physics, 394 (2019), p. 658–678.

Robust information divergences for model-form uncertainty in random PDE

Steady-state subsurface flow is a complex system described by a random PDE of elliptic type where the diffusion coefficient is given by a geostatistical model. The geostatistical model, with properties inferred from relevant data, represents a source of model-form uncertainty that poses a challenge for making predictions of quantities of interest. In this work, we describe a novel application of hybrid information divergences to a steady-state flow model in the context of a decision support framework. Information divergences, based on the Donsker--Varadhan variational principle from large deviations theory, have a form that balances observable and data dependent quantities. Although information divergences have been used for sensitivity analysis in other contexts, the presence of model-form uncertainty necessitates distinguishing various uncertain aspects of the system to capture essential features. The hybrid nature of the information divergences presented in this work allow us to represent, aggregate, and distinguish sources of uncertainty and in particular to quantify the propagation of model-form uncertainty. We derive tight and robust bounds for modeling errors and apply these bounds to important uncertainty quantification tasks including parametric sensitivity analysis and gauging model misspecification due to sparse data. We also explore data-informed predictions, such as quantifying the impact of incomplete data on a quantity of interest. Further, we leverage the connection between the hybrid information divergences and certain concentration inequalities for efficient computing.

Information budget. For a given information budget \(\rho\), the hybrid divergences suggests tight and computable upper and lower bounds for the weak error between a nominal model \(P\) for every alternative model \(Q\) in an infinite dimensional family that includes both parametric and nonparametric perturbations. The information budget \(\rho\) can be interpreted as an allowable level of model-form uncertainty (from [6]).
  1. Robust information divergences for model-form uncertainty arising from sparse data in random PDE, with M. Katsoulakis, SIAM/ASA Journal on Uncertainty Quantification, 6 (2018), p. 1364–1394.

Uncertainty quantification for the generalized Langevin equation

VACF for models with several different Prony modes. Small changes to the number of modes leads to qualitatively different behavior of the normalized VACF for the GLE with a harmonic confining potential (from [5]).

The generalized Langevin equation GLE is a non-Markovian stochastic dynamical system (a system with "memory") that models anomalous diffusion arising in the context of viscoelastic media such as complex biofluids. Under physically reasonable assumptions, a Markovian approximation to the GLE exists and this Markovian extended variable formulation contains many parameters that must be tuned. Although uncertainties in drift, vibration, and tracking measurements may be present in the microrheology data used to tune the GLE model, a central problem here is one of epistemic or model-form uncertainty. There is a wealth of data, but few methods that allow one to compare and evaluate the models that are suggested by this data. Therefore, a key challenge surrounds analyzing the sensitivity of the dynamics to local and global perturbations. However, well known sensitivity analysis techniques such as likelihood ratio and pathwise methods are not applicable to key parameters of interest in this context. In particular, it is relevant to understand discrete changes in the model such as perturbations to the degrees of freedom of the extended variable system.

VACF vs epsilon. The optimal coupling eliminates the dependence of the variance of the estimator on the bias, asymptotically, in the case of a convex potential (from [5]).

In this work we present efficient finite difference estimators for goal-oriented sensitivity indices. These easily implemented estimators are formed by coupling the nominal and perturbed dynamics appearing in the finite difference through a common driving noise, or common random path. After developing a general framework for variance reduction via coupling, we demonstrate the optimality of the common random path coupling in the sense that it produces a minimal variance surrogate for the difference estimator relative to sampling dynamics driven by independent paths. In order to build intuition for the common random path coupling, we evaluate the efficiency of the proposed estimators for a comprehensive set of examples of interest in particle dynamics. These reduced variance difference estimators are also a useful tool for performing global sensitivity analysis and for investigating non-local perturbations of parameters, such as discrete changes to the degrees of freedom in the extended variable system.

  1. Uncertainty quantification for generalized Langevin dynamics, with M. Katsoulakis and L. Rey-Bellet, Journal of Chemical Physics, 145 (2016), p. 224108.

Computable estimates for random PDE with rough stochastic coefficients

In the geophysics literature, elliptic PDE with uncertain data arise in the study of time-independent groundwater flow on the local scale (on the order of 100's of meters). In application, prescribing the conductivity requires more information than is reasonably possible to acquire; a common feature of groundwater flow at the local scale is the spatial heterogeneity of the medium. Such uncertainty in the problem data is incorporated by modeling the conductivity as a random field. In applications to subsurface flow, the law of the hydraulic conductivity is assumed to be a normal field with Lipschitz covariance.

Rough, lognormal field. Displacements. One sample of a rough lognormal conductivity (top) and the corresponding pathwise finite element approximation (bottom) (from [4]). For conductivities arising in groundwater flow problems the typically short correlation lengths involved motivate the use of MC finite element methods over stochastic Galerkin methods.

In this work we give a computable estimate for observables of the Galerkin error committed in standard piecewise linear finite element approximations. For this model, the standard a posteriori analysis for the dual weighted residual does not give a reliable estimate. In contrast to the case of a smooth conductivity, we see that for a rough lognormal conductivity the residual contains non-negligible high frequency content.

Figure showing Galerkin error vs low frequency content for a smooth conductivity. Figure showing Galerkin error vs low frequency content for a lognormal conductivity. Galerkin error vs low frequency content for a smooth conductivity (top) and a rough lognormal conductivity (bottom).

Analyzing the frequency content for the 1-dimensional problem suggests that this high frequency contribution can be approximated by low frequency content. A related assumption on scales yields a computable estimator of the Galerkin error observable based on local error indicators. We also obtain estimators for the expected quadrature error committed in the finite element approximations as, in contrast to the case of a smooth conductivity, the quadrature error is observed to be on the same order as the Galerkin error in this setting. These estimates, derived using easily validated assumptions, can be computed at a relatively low cost and fill a much needed gap by providing an important and novel computational tool for PDE with rough stochastic coefficients.

  1. Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data, with H. Hoel, M. Sandberg, A. Szepessy, R. Tempone, SIAM Journal on Scientific Computing, 38 (2016), p. A3773–A3807.

Accelerated finite difference schemes for SPDE

Filtering problems pertain to a stochastic dynamical system, possibly nonlinear, modeled by a signal process that cannot be sampled directly and an observation process that yields partial (noisy) information about the signal. The goal of the filtering problem is to estimate the density of a functional of the signal process at a given time conditioned on knowledge of the observation process. The evolution of such a density can be described by the solution to the Cauchy problem for the Zakai equation, a second order linear SPDE of parabolic type. In applications, such as satellite tracking and guidance, solutions are desired in real time and information about the error is required in a stochastically strong, or pathwise, manner. Since analytic solutions are unavailable, there is a demand for accurate and effective numerical schemes for approximating the solutions to these problems. My PhD research focuses on the analysis of finite difference approximations for a class of parabolic SPDEs that includes the Zakai equation. In these works, I give sufficient conditions on when Richardson's method can be used to obtain higher order spatial approximations. I also extend these results to the case of degenerate parabolic SPDE, a significant contribution as one is not guaranteed the strong parabolicity condition in physically relevant applications.

  1. Higher order spatial approximations for degenerate parabolic stochastic partial differential equations, SIAM Journal on Mathematical Analysis, 45 (2013), p. 2071–2098.
  2. Accelerated spatial approximations for time descretized stochastic partial differential equations, SIAM Journal on Mathematical Analysis, 44 (2012), p. 3162–3185.
  3. Accelerated Numerical Schemes for Deterministic and Stochastic Partial Differential Equations of Parabolic Type, Ph.D. thesis, University of Edinburgh, 2013.