Probabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations

We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the coefficients (e.g. material parameters) exhibit random, fine scale variability. The direct solution to such problems requires grids that are small enough to resolve this fine scale variability which unavoidably requires the repeated solution of very large systems of algebraic equations. We establish a physically inspired, data-driven coarse-grained model which learns a low- dimensional set of microstructural features that are predictive of the fine-grained model (FG) response. Once learned, those features provide a sharp distribution over the coarse scale effec- tive coefficients of the PDE that are most suitable for prediction of the fine scale model output. This ultimately allows to replace the computationally expensive FG by a generative proba- bilistic model based on evaluating the much cheaper CG several times. Sparsity enforcing pri- ors further increase predictive efficiency and reveal microstructural features that are important in predicting the FG response. Moreover, the model yields probabilistic rather than single-point predictions, which enables the quantification of the unavoidable epistemic uncertainty that is present due to the information loss that occurs during the coarse-graining process

A hierarchy of models for simulating experimental results from a 3D heterogeneous porous medium

In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.Comment: 27 pages, 9 Figure

Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^\alpha \ln \frac{1}{h})$ (with $\alpha \in [1/2,1)$) result in an $\mathcal{O}(h^{2-2\alpha})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^\alpha \ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).Comment: Accepted for publication in SIAM MM

Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain $D$ such that $\overline{D}$ is the union of cells $\{\overline{D_i}\}_{i\in I}$ and we introduce a two-scale representation by identifying any function $v(x)$ defined on $D$ with a bi-variate function $v(i,y)$, where $i \in I$ relates to the index of the cell containing the point $x$ and $y \in Y$ relates to a local coordinate in a reference cell $Y$. We introduce a weak formulation of the problem in a broken Sobolev space $V(D)$ using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying $V(D)$ with a tensor product space $\mathbb{R}^I \otimes V(Y)$ of functions defined over the product set $I\times Y$. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.Comment: Changed the choice of test spaces V(D) and X (with regard to regularity) and the argumentation thereof. Corrected proof of proposition 3. Corrected wrong multiplicative factor in proposition 4 and its proof (was 2 instead of 1). Added remark 6 at the end of section 2. Extended remark 7. Added references. Some minor improvements (typos, typesetting

A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media

In this paper we consider a multiscale phase-field model for capillarity-driven flows in porous media. The presented model constitutes a reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model, valid in situations where interest is restricted to dynamical and equilibrium behavior in an aggregated sense, rather than a precise description of microscale flow phenomena. The model is based on averaging of the equation of motion, thereby yielding a significant reduction in the complexity of the underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its macroscopic dynamical and equilibrium properties. Numerical results are presented for the representative 2-dimensional capillary-rise problem pertaining to two closely spaced vertical plates with both identical and disparate wetting properties. Comparison with analytical solutions for these test cases corroborates the accuracy of the presented multiscale model. In addition, we present results for a capillary-rise problem with a non-trivial geometry corresponding to a porous medium