Bayesian inference in Inverse problems Bani Mallick - PowerPoint PPT Presentation

Bayesian inference in Inverse problems Bani Mallick bmallick@stat.tamu.edu Department of Statistics, Texas A&M University, College Station 1/20

Inverse Problems Inverse problems arise from indirect observations of a quantity of interest Observations may be limited in numbers relative to the dimension or complexity of the model space Inverse problems ill posed Classical approaches have used regularization methods to impose well-posedness ans solved the resulting deterministic problems by optimization 2/20

Bayesian approach A natural mechanism for regularization in the form of prior information Can handle non linearity, non Gaussianity Focus is on uncertainties in parameters, as much as on their best (estimated) value. Permits use of prior knowledge, e.g., previous experiments, modeling expertise, physics constraints. Model-based. Can add data sequentially 3/20

Inverse problem Inverse problems whose solutions are unknown functions: Spatial or temporal fields Estimating fields rather than parameters typically increases the ill-posedness of the inverse problem since one is recovering an infinite dimensional object from finite amounts of data Obtaining physically meaningful results requires the injection of additional information on the unknown field A standard Bayesian approach is to employ Gaussian process or Markov Random field priors 5/20

Forward Model and Inverse problem Z = F ( K ) + ǫ where F is the forward model, simulator, computer code which is non-linear and expensive to run. K is a spatial field Z is the observed response ǫ is the random error usually assumed to be Gaussian Want to estimate K with UQ This is a non-linear inverse problem 6/20

Fluid flow in porous media Studying flow of liquids (Ground water, oil) in aquifer (reservoir) Applications: Oil production, Contaminant cleanup Forward Model: Models the flow of liquid, output is the production data, inputs are physical characteristics like permeability, porosity Inverse problem: Inferring the permeability from the flow data . 1/ ??

Permeability Primary parameter of interest is the permeability field Permeability is a measure of how easily liquid flows through the aquifer at that point This permeability values vary over space Effective recovery procedures rely on good permeability estimates, as one must be able to identify high permeability channels and low permeability barriers . 2/ ??

Forward Model Darcy’s law: v j = − k rj ( S ) k f ∇ p, (1) µ j v j is the phase velocity k f is the fine-scale permeability field k rj is the relative permeability to phase j ( j =oil or water) S is the water saturation (volume fraction) p is the pressure. . 1/3

Forward Model Combining Darcy’s law with a statement of conservation of mass allows us to express the governing equations in terms of pressure and saturation equations: ∇ · ( λ ( S ) k f ∇ p ) = Q s , (2) ∂S ∂t + v · ∇ f ( S ) = 0 , (3) λ is the total mobility Q s is a source term f is the fractional flux of water v is the total velocity . 2/3

Forward Model Production (amount of oil in the produced fluid, fractional Flow or water-cut) F k f ( t ) is given by � F k f ( t ) = ∂ Ω out v n f ( S ) dl (4) where ∂ Ω out is outflow boundaries and v n is normal velocity field. . 3/3

4.5 4 3.5 3 Forward , 2.5 Permeability field Simulator 2 1.5 1 0.5 Permeability field Output

, , Forward Simulator Forward Simulator Permeability field Permeability field Output Fine-scale Permeability field

Bayesian way If p ( K ) is the prior for the spatial field K : usually Gaussian processes p ( Z | k ) is the likelihood depending on the distribution of ǫ : Gaussian, non-Gaussian Then posterior distribution: p ( K | Z ) ∝ p ( Z | K ) p ( K ) is the Bayesian solution of this inverse problem . 1/ ??

Inverse Problem Dimension reduction:Replacing K by a finite set of parameters τ Building enough structures through models and priors. Additional data: coarse-scale data Need to link data at different scales Bayesian hierarchical models have the ability to do all these things simultaneously . 2/ ??

Multiscale Data K f is the fine scale field of interest (data: well logs, cores) Additional data: from coarse scale field K c (seismic traces) Some of the observed fine-scale permeability values K o f at some spatial locations We want to infer K f conditioned on Z , K c and K o f The posterior distribution of interest: p ( K f | Z, K c , K o f ) . 1/1

K Fine−grid Coarse−grid No flow φ = Δ φ = div k x φ = 0 ( ( ) ) 0 1 f No flow 1 ∫ = Δ φ k x e e k x x e dx ( ( ) , ) ( ( ) ( ), ) c j l f j l K | | K

Dimension reduction We need to reduce the dimension of the spatial field K f This is a spatial field denoted by K f ( x , ω ) where x is for the spatial locations and ω denotes the randomness in the process Assuming K f to be a real-valued random field with finite second moments we can represent it by Kauren-Loeve (K-L) expansion 10/20

K-L expansion ∞ � � K f ( x , ω ) = θ 0 + λ l θ l ( ω ) φ l ( x ) l =1 where λ : eigen values φ ( x ) eigen functions θ : uncorrelated with zero mean and unit variance If K f is Gaussian process then θ will be Gaussian 11/20

K-L expansion If the covariance kernel is C then we obtain them by solving � C ( x 1 , x 2 ) φ l ( x 2 ) d x 2 = λ l φ l ( x 1 ) and can express C as ∞ � C ( x 1 , x 2 ) = λ l φ l ( x 1 ) φ l ( x 2 ) l =1 12/20

Spatial covariance We assume the correlation structure � − | x 1 − y 1 | 2 − | x 2 − y 2 | 2 � C ( x , y ) = σ 2 exp . 2 l 2 2 l 2 1 2 where, l 1 and l 2 are correlation lengths. For an m -term KLE approximation m � � K m = θ 0 + λ i θ i Φ i , f i =1 B ( l 1 , l 2 , σ 2 ) θ, (say) = (1) 13/20

Existing methods The energy ratio of the approximation is given by e ( m ) := E � k m P m f � 2 i =1 λ i E � k f � 2 = i =1 λ i . P ∞ Assume correlation length l 1 , l 2 and σ 2 are known. We treat all of them as model parameters, hence τ = ( θ, σ 2 , l 1 , l 2 , m ) . 14/20

Hierarchical Bayes’ model P ( θ, l 1 , l 2 , σ 2 | Z, k c , k o P ( z | θ, l 1 , l 2 , σ 2 ) P ( k c | θ, l 1 , l 2 , σ 2 ) f ) ∝ P ( k o f | θ, l 1 , l 2 , σ 2 ) P ( θ ) P ( l 1 , l 2 ) P ( σ 2 ) P ( z | θ, l 1 , l 2 , σ 2 ) : Likelihood P ( k c | θ, l 1 , l 2 , σ 2 ) : Upscale model linking fine and coarse scales P ( k o f | θ, l 1 , l 2 , σ 2 ) : Observed fine scale model P ( θ ) P ( l 1 , l 2 ) P ( σ 2 ) : Priors 15/20

Likelihood The likelihood can be written as follows: F [ B ( l 1 , l 2 , σ 2 ) θ ] + ǫ f Z = F 1 ( θ, l 1 , l 2 , σ 2 ) + ǫ f = where, ǫ f ∼ MV N (0 , σ 2 f I ) . 16/20

Likelihood calculations Z = F ( τ ) + ǫ For Gaussian model the likelihood will be Exp ( − [ Z − F ( τ )] 2 1 P ( Z | τ ) = √ ) 2 σ 2 2 πσ 1 1 where σ 2 1 is the variance of ǫ . 17/20

Likelihood Calculations It is like a black-box likelihood which we can’t write analytically, although we do have a code F that will compute it. We need to run F to compute the likelihood which is expensive. Hence, no hope of having any conjugacy in the model, other than for the error variance in the likelihood. Need to be somewhat intelligent about the update steps during MCMC so that do not spend too much time computing likelihoods for poor candidates. 18/20

Upscale model The Coarse-scale model can be written as follows. k c = L 1 ( k f ) + ǫ c L 1 ( θ, l 1 , l 2 , σ 2 ) + ǫ c = where, ǫ c ∼ MV N (0 , σ 2 c I ) . i.e k c | θ, l 1 , l 2 , σ 2 , σ 2 c ∼ MV N ( L 1 ( θ, l 1 , l 2 , σ 2 ) , σ 2 c I ) . L 1 is the upsacling operator It could be as simple as average It could be more complex where you need to solve the original system on the coarse grid with boundary conditions 19/20

K Fine−grid Coarse−grid No flow φ = Δ φ = div k x φ = 0 ( ( ) ) 0 1 f No flow 1 ∫ = Δ φ k x e e k x x e dx ( ( ) , ) ( ( ) ( ), ) c j l f j l K | | K

Observed fine scale model We assume the model k o f = k o p + ǫ k where, ǫ k ∼ MV N (0 , σ 2 k ) . k o p is the spatial field obtained from K-L the expansion at the observed well locations. So here we assume, k o f | θ, l 1 , l 2 , σ 2 , σ 2 k ∼ MV N ( k o p , σ 2 k ) , 20/20

  l l , 1 2 o K f K.L.  Covariance Matrix Expansion Upscaling  K K f c Forward Solve z F (.)

Inverse problem We can show that the posterior measure is Lipschitz continuous with respect to the data in the total variation distance It guaranties that this Bayesian inverse problem is well-posed   z Say, y is the total dataset, i.e, y = k c     k 0 f g ( τ, y ) is the likelihood and π 0 ( τ ) is the prior . 1/2

Inverse problem Theorem 0.1. ∀ r > 0 , ∃ C = C ( r ) such that the posterior measures π 1 and π 2 for two different data sets y 1 and y 2 with max ( � y 1 � l 2 , � y 2 � l 2 ) ≤ r , satisfy � π 1 − π 2 � TV ≤ C � y 1 − y 2 � l 2 , . 2/2

MCMC computation Metropolis-Hastings (M-H) Algorithm to generate the parameters. Reversible jump M-H algorithm when the dimension m of the K-L expansion is treated as model unknown. Two step MCMC or Langevin can accelerate our computation. 21/20