The University of Auckland Applied Mathematics Bayesian Methods for Inverse Problems : why and how Colin Fox Tiangang Cui, Mike O Sullivan (Auckland),

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1 The University of Auckland Applied Mathematics Bayesian Methods for Inverse Problems : why and how Colin Fox Tiangang Cui, Mike O Sullivan (Auckland), Geoff Nicholls (Statistics, Oxford) fox@math.auckland.ac.nz

2 In this talk An example in geothermal model calibration Inferential solutions to inverse problems, noise and all The need to integrate: pixel-wise degraded binary images Monte Carlo integration Computation (MCMC) Output analysis for geothermal problem Next talk :: better sampling algorithms

3 An inverse problem in geothermal fields

4 Schematic of a geothermal field Geothermal Well High Den er Wat Cold sity Permeable Rock Crystalline Rock Den sity C o ld W ater High Rock of Low Permeability Temperature Contour Boiling Zone Low Density Hot Water Heat Convective Magma Given near-surface measurements of temperature, pressure, flow, want Model calibration :: rock type, porosity, fracture : heat sources Predict :: long term (50 year) power generation, land subsidence, etc Decide :: robust investment plan

5 Wairakei geothermal power generator

6 Bayesian formulation of inverse problems d = Ax + n: d, image x, measurement noise n, forward map A A-1 xml xtrue image space A forward map d dnf space π (x d, m) Pr (d x, m)pr (x m) (Bayes rule) Likelihood: measurement and noise (Physics, instrumentation, probability) Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion Inference based on posterior distribution conditioned on (computational statistics)

7 Bayesian formulation of inverse problems d = Ax + n: d, image x, measurement noise n, forward map A A-1 xml xtrue image space A forward map d dnf space π (x d, m) Pr (d x, m)pr (x m) (Bayes rule) Likelihood: measurement and noise (Physics, instrumentation, probability) Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion Inference based on posterior distribution conditioned on (computational statistics)

8 Bayesian formulation of inverse problems d = Ax + n: d, image x, measurement noise n, forward map A A-1 xml xtrue image space A forward map d dnf space π (x d, m) Pr (d x, m)pr (x m) (Bayes rule) Likelihood: measurement and noise (Physics, instrumentation, probability) Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion Inference based on posterior distribution conditioned on (computational statistics)

9 Bayesian formulation of inverse problems d = Ax + n: d, image x, measurement noise n, forward map A A-1 xml xtrue image space A forward map d dnf space π (x d, m) Pr (d x, m)pr (x m) (Bayes rule) Likelihood: measurement and noise (Physics, instrumentation, probability) Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion Inference based on posterior distribution conditioned on (computational statistics)

10 Stochastic Model true image x ^x estimate ideal forward map K loss P noise n free + + noise image recovery algorithm representation / prior density actual d=ax+n : A=PK

11 Stochastic Model true image x ^x estimate ideal forward map K loss P noise n free + + noise image recovery algorithm representation / prior density if n fn ( ) Pr(d x, m) = fn (d Ax) actual d=ax+n : A=PK

12 Stochastic Model true image x ^x estimate ideal forward map K loss P noise n free + actual d=ax+n : A=PK + noise image recovery algorithm representation / prior density if n fn ( ) Pr(d x, m) = fn (d Ax) Prior distribution πpr (x) uses physical laws, expert knowledge, stochastic modelling

13 Stochastic Model true image x ^x estimate ideal forward map K loss P noise n free + actual d=ax+n : A=PK + noise image recovery algorithm representation / prior density if n fn ( ) Pr(d x, m) = fn (d Ax) Prior distribution πpr (x) uses physical laws, expert knowledge, stochastic modelling Focus of inference is posterior distribution for x given d π (x d) Pr (d x) πpr (x)

14 Solutions to Inverse Problem = Summary Statistics Posterior distribution π (x d) encapsulates all information about x Regularized Inversion - Modes x MLE = arg max Pr (d x) x MAP = arg max π (x d) (least-squares, Moore-Penrose inverse, Tikhonov regularization, Kalman filtering, BackusGilbert, Prussian-hat cleaning) Inferential Solutions Answers are expectations over the posterior π (x d) Z Eπ [f (x)] = f (x) π (x d) dx X If f (x) = indicator function that image shows cancer E [f (x)] is posterior probability (based on measurements, prior) that patient has cancer.

15 Two modes over pixel image p(x d) x

16 Two modes over pixel image p(x d) x if maximum value (at mode) is twice the value of the lower local maximum

17 Two modes over pixel image p(x d) x if maximum value (at mode) is twice the value of the lower local maximum width of global mode is half the width of the lower local mode

18 Two modes over pixel image p(x d) x if maximum value (at mode) is twice the value of the lower local maximum width of global mode is half the width of the lower local mode in each of the coordinate directions x1, x2,, x

19 Two modes over pixel image p(x d) x if maximum value (at mode) is twice the value of the lower local maximum width of global mode is half the width of the lower local mode in each of the coordinate directions x1, x2,, x there is times more probability mass in the lower, broader, peak than around the mode

20 Two modes over pixel image p(x d) x if maximum value (at mode) is twice the value of the lower local maximum width of global mode is half the width of the lower local mode in each of the coordinate directions x1, x2,, x there is times more probability mass in the lower, broader, peak than around the mode Z Z π (x d) dx = π (x d) dx1 dx2 dx10000 X X

21 θ=0 θ = θ = 0.25 noisy MLE θ = true θ = θ = 0.5 Mode vs Mean MAP mean sample MPM Pixel-wise noise on binary image, Ising prior with lumping constant θ F and Nicholls Exact MAP states and expectations from perfect sampling: Greig, Porteous and Seheult revisited 2001

22 Monte Carlo integration + importance sampling If x(1),..., x(n) π ( d) n 1 X (i) f x Eπ [f (x)] n i=1 Construct x(1),..., x(n) as iterates of an ergodic map (n+1) (n) Deterministic iteration: x =M x π(x) = X M (y)=x π(y) M 0 (y) (Frobenius-Perron) Stochastic Iteration: Markov chain with transition kernel p (x, y) Z Z π (x) p (x, y) dx = X π (y) p (y, x) dx (global balance) X π (x) p (x, y) = π (y) p (y, x) (detailed balance)

23 Monte Carlo integration + importance sampling If x(1),..., x(n) π ( d) n 1 X (i) f x Eπ [f (x)] n i=1 Construct x(1),..., x(n) as iterates of an ergodic map (n+1) (n) Deterministic iteration: x =M x π(x) = X M (y)=x π(y) M 0 (y) (Frobenius-Perron) Stochastic Iteration: Markov chain with transition kernel p (x, y) Z Z π (x) p (x, y) dx = X π (y) p (y, x) dx (global balance) X π (x) p (x, y) = π (y) p (y, x) (detailed balance)

24 Sampling Algorithms in Physics Hamiltonian Dynamics Equate unknowns x to position variables q with potential energy E(q) = T log (π(x)) T log (Z) Auxiliary momentum variables p with kinetic energy K(p) = 12 kpk2. Hamiltonian H(q, p) = E(q) + K(p) has canonical distribution P (q, p) = 1 1 exp ( E(q)) exp ( K(p)) ZP ZK Hamiltonian dynamics leave P (q, p) invariant + stochastic transitions for ergodicity Langevin diffusion s2 dx = s db + log (π(x)) dt 2

25 Metropolis-Hastings algorithm 1. given state xt at time t generate candidate state x0 from a proposal distribution q (. xt ) 0 )q (x x0 ) π(x t 2. With probability α xt x0 = min 1, set Xt+1 = x0 0 π(xt )q (x xt ) otherwise Xt+1 = xt 3. Repeat q (. xt ) can be any distribution that ensures the chain is irreducible and aperiodic. Pros Provably convergent to any property, e.g. best estimate, uncertainty in estimate State space can be continuous, discrete, stochastic, or variable dimension Cons Can be (very) slow

26 Metropolis-Hastings algorithm 1. given state xt at time t generate candidate state x0 from a proposal distribution q (. xt ) 0 )q (x x0 ) π(x t 2. With probability α xt x0 = min 1, set Xt+1 = x0 0 π(xt )q (x xt ) otherwise Xt+1 = xt 3. Repeat q (. xt ) can be any distribution that ensures the chain is irreducible and aperiodic. Pros Provably convergent under mild requirements State space can be continuous, discrete, stochastic, or variable dimension Cons Can be (very) slow

27 Results for geothermal field 90 p [bar] (Wellhead) Time [day] h [kj/kg] Time [day] Wellhead pressure (top) and flowing enthalpy (bottom). Black line is estimated result, triangles and squares are observed. Tiangang Cui, Bayesian Inference for Geothermal Model Calibration MEng Auckland, 2005

28 Output analysis for geothermal field 8,000 Log-likelihood , , log(k [m2 ]) φ [-] 8, ,000 60,000 4,000 4,000 MCMC Updates Autocorrelation m [-] p0 [bar] ps [bar] 50 8, ,000 4, , ,000 Lag Srl [-] Sls ps [bar] p0 [bar] log 10 (k [m2 ]) Sv0 0.3 Autocorrelation and MCMC output trace. Histogram for the marginal distribution of parameters. Scatter plot of the joint marginal distribution for log(k), p0 and ps. Histogram for the marginal distribution of Sv0 and Sls and scatter plot of their joint marginal distribution. Tiangang Cui, Bayesian Inference for Geothermal Model Calibration MEng Auckland, 2005

29 Inferred iso-temperature surface

30 Conclusions 1. Bayesian methods can solve substantial real-world inverse problems 2. Time to convergence depends on efficient simulation of the forward map, fast MCMC algorithms, good proposal distributions 3. Computation required is feasible

Statistical Estimation of the Parameters of a PDE

PIMS-MITACS 2001 1 Statistical Estimation of the Parameters of a PDE Colin Fox, Geoff Nicholls (University of Auckland) Nomenclature for image recovery Statistical model for inverse problems Traditional