Information Geometry and Model ReductionSorin Mitran1

1Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA

Reconstruction of Medium Permeability

INTRODUCTION

• Projective model reduction. System description

• Reduced description

• Galerkin projection

• Concept of best approximant

• Presupposes L2 norm and choice of basis is relevant to problem• Can be extended to stochastic phenomena with considerable computational effort by introducing basis for random space (e.g., XFEM)

• L2 projection is computationally efficient• Transport along curved geodesics would require solving a boundary value problem• Remarkably (Amari, 2001) a dual-affine structure can be defined for the most commonly encountered PDF families (exponential, mixture) that restores the computational efficiency of L2 projection

• Define multiple affine connections between points on a manifold•

•Manifold of PDFs•Metric•Geodesic equation

•PDF encapsulates known information about system:• Information• Relative information• Infinitesimal relative entropy (Fisher metric)

•Reconstruction of medium permeability as characterized by Reynolds-Forchheimer relationship using single-mode model via

•Single-mode model yields accurate representation in comparison to empirical relationship (left)

•The permeability, described by Reynolds number as a stochastic process indexed by the forcing, Re(Rc) is characterized by stochastics (marginal PDFs) of all orders corresponding to centered multivariate normal distributions

•May reduce dimensionality by n(n+1)/2 (correlation parameters), gain tractability yielded by statistical independence, by determining closest uncorrelated normal distribution

RESULTS & DISCUSSION

CONCLUSIONS

•Direct flow simulations carried out using a three-dimensional, 19-velocity vector, multi-relaxation-time Lattice Boltzmann scheme.

– Recovers N-S equations to second order– Valid within Non-Darcy Regime– Flows generated sequence of Reynolds

numbers such that Re<120.•N=60 Lattice-Boltzmann simulations of fluid flow at m=21 forcings using surrogate porous media for various domain sizes.•Spherical inclusions representing isotropic case with equal radii and uniformly-distributed centers used to generate each medium•Domains with volumes representing 1/60, 2/60, 4/60, 8/60, 16/60, and 20/60 of a posited representative elementary volume

Direct Numerical Simulation

•The first principal component of the stochastic response explains 99.5 percent of the variance and captures most of the L2 norm of

•A good representation of the process P is obtained by a single stochastic mode, discretely approximated as

•The single-mode model yields accurate reconstruction of the Forchheimer-Reynolds relationship yielded by

•Decay in variance over associated with first mode predicts a variance predictive of deterministic behavior at the posited domain size corresponding to a posited Representative Elementary Volume for single-phase, incompressible flows through isotropic porous media

•Methods of Information Geometry facilitate stochastic model reduction by identifying uncorrelated distributions closest to true distributions by nonlinear projection

RESULTS & DISCUSSIONMETHODS – Stochastic Resistance Model

• Information geometry approach. Consider a manifold of probability density functions (PDFs) for the m system variables with n parameters

• Choice of the PDF family induces a non-Euclidean structure on the space• Basic idea: use the inherent nonlinearity to obtain a more efficient (more “compressive”) model reduction• Example: 2-parameter family of normal distributions

• Closeness of two distributions given by Fisher information metric

• Information geometry model reduction.• Construct PDF manifold (n parameters)• Reduced model obtained by geodesic transport onto submanifold (m

parameters)• Difference in information content of PDFs quantified by divergences

• Kullback-Leibler divergence

• Induced Fisher metric

•Approach:• View momentum resistance (inverse of permeability) as stochastic

process with Karhunen-Loeve Expansion

• Discretely approximate process via PCA

– Using approximate eigenfunctions by eigenmodes of correlation matrix of flow rate data

•Principal Modes determined via PCA of DNS data

– Exhibit rapidly decaying variance with mode number (top left)

– Yield basis with respect to which process mean

– is rapidly convergent (bottom)

Information-Geometric Perspective

•Decaying variance and projection of process mean on principal component basis suggest single mode model

•First principal mode maintains form over all domain sizes, suggesting low-dimensional characterization of permeability using direct numerical simulations of small domains

•Variance of associated with first mode (blue points) decreases with increasing size of simulated domain

– Suggestive of approach to deterministic limit for isotropic surrogate porous media.

• Exponential fit (blue curve) and 95% confidence intervals suggest negligible variance at posited Representative Elementary Volume (1500 inclusions)

QP

R

e-geodesic (submanifold)

m-geodesic

m-geodesic

R

P

Q

R S

•Use Information Projection may to find the closest uncorrelated normal distribution in terms of Kullback-Leibler divergence.

•Figure: Given point P on statistical manifold corresponding to correlated multivariate normal distributions, determine closest point Q lying on submanifold (blue) of uncorrelated multivariate distributions through orthogonality of e-geodesics (dark blue and magenta curves) and m-geodesic (tan curve) at point Q

BACKGROUND – Information geometry