Download - Hierarchical Bayesian Level Set Inversion

Hierarchical Bayesian Level Set Inversion

Matt Dunlop

Mathematics Institute, University of Warwick, U.K.

Marco Iglesias (Nottingham), Andrew Stuart (Warwick)

SIAM UQ 2016,Lausanne, Switzerland,

April 7th.

EPSRC

Matt Dunlop (University of Warwick) Hierarchical Bayesian Level Set Inversion April 7th 2016 1 / 21

Outline

1 Introduction

2 Classical Level Set Inversion

3 Bayesian Level Set Inversion

4 Hierarchical Bayesian Level Set Inversion

5 Conclusions


Level Set Representation

S.Osher and J.SethianFronts propagating with curvature-dependent speed . . . .J. Comp. Phys. 79(1988), 12–49.

Piecewise constant function v defined through thresholding acontinuous level set function u. Let

�1 = c0 < c1 < · · · < cK�1 < cK = 1.

v(x) =KX

k=1

vk I{ck�1<uck}(x); v = F (u).

F : X ! Z is the level-set map, X cts fns. Z piecewise cts fns.Matt Dunlop (University of Warwick) Hierarchical Bayesian Level Set Inversion April 7th 2016 3 / 21

Example 1: Image Reconstruction

H.W. Engl, M. Hanke and A. NeubauerRegularization of Inverse ProblemsKluwer(1994)

Forward ProblemDefine K : Z ! RJ by (Kv)j = v(xj), xj 2 D ⇢ Rd . Given v 2 Z

y = Kv .

Let ⌘ 2 RJ be a realization of an observational noise.

Inverse ProblemGiven prior information v = F (u), u 2 X and y 2 RJ , find v :

y = Kv + ⌘.


Example 2: Groundwater FlowM. Dashti and A.M. StuartThe Bayesian approach to inverse problems.Handbook of Uncertainty QuantificationEditors: R. Ghanem, D.Higdon and H. Owhadi, Springer, 2017.arXiv:1302.6989

Forward Problem: Darcy Flow

Let X+ := {v 2 Z : essinfx2Dv > 0}. Given 2 X+, find y := G() 2 RJ

where yj = `j(p),V := H10 (D), `j 2 V ⇤, j = 1, . . . , J and

(�r · (rp) = f in Dp = 0 on @D.

Let ⌘ 2 RJ be a realization of an observational noise.

Inverse ProblemGiven prior information = F (u), u 2 X and y 2 RJ , find :

y = G() + ⌘.


D

el

Example 3: Electrical Impedance TomographyApply currents I` one`, ` = 1, . . . , L.Induces voltages ⇥` one`, ` = 1, . . . , L.Input is (�, I), output is (✓,⇥).We have an Ohm’s law⇥ = R(�)I.

8>>>>>>>>>><

>>>>>>>>>>:

�r · (�(x)r✓(x)) = 0 x 2 DZ

e`

�@✓

@⌫dS = I` ` = 1, . . . , L

�(x)@✓

@⌫(x) = 0 x 2 @D \

SL`=1 e`

✓(x) + z`�(x)@✓

@⌫(x) = ⇥` x 2 e`, ` = 1, . . . , L

(PDE)


Example 3: Electrical Impedance Tomography

M. M. Dunlop and A. M. StuartThe Bayesian formulation of EIT.arXiv:1509.03136Inverse Problems and Imaging, Submitted, 2015.

Forward ProblemLet X ✓ L1(D), and denote X+ := {u 2 X : essinfx2D u > 0}.Given u 2 X ,F : X ! X+ and � = F (u), find (✓,⇥) 2 H1(D)⇥ RL

solving (PDE).This gives ⇥ = R(F (u))I.

Let ⌘ 2 RL be a realisation of an observational noise.

Inverse ProblemGiven prior information � = F (u), u 2 X , and given currents I andy 2 RL, find � :

y = R(�)I + ⌘.


Outline

1 Introduction




5 Conclusions


Tikhonov Regularization

G : Z ! RJ and ⌘ 2 RJ a realization of an observational noise.

Inverse ProblemFind u 2 X , given y 2 RJ satisfying y = G � F (u) + ⌘.

Model-Data Misfit

The model-data misfit is �(u; y) = 12

��12�y � G � F (u)

��2.

Tikhonov RegularizationMinimize, for some E compactly embedded into X ,

I(u; y) := �(u; y) +12kuk2

E .


Issue 1: Discontinuity of Level Set Map

F (·) is continuous at u F (·) is discontinuous at u

v = F (u) := v+ I{u�0}(x) + v� I{u<0}(x).

Causes problems in classical level set inversion.


Issue 2: Length-Scale Matters

Figure demonstrates role of length-scale in level-set function.Classical Tikhonov-Phillips regularization does not allow for directcontrol of the length-scale.

New ideas needed.


Issue 3: Amplitude Matters

Y. LuProbabilistic Analysis of Interface ProblemsPhD Thesis (2016), Warwick University

Consider case of K = 2, c1 = 0.For contradiction assume u? is a minimizer of I(u; y).Now define the sequence u✏ = ✏u?. Then if 0 < ✏ < 1,

�(u✏; y) = �(u?; y), ku✏kE = ✏ku?kE ) I(u✏; y) < I(u?; y).

Hence I(u✏; y) is an infimizing sequence. But u✏ ! 0 andI(0; y) > I(u✏; y) for ✏ ⌧ 1.Thus infimum cannot be attained.This issue caused by thresholding at 0. But idea generalizes.

Choice of threshold matters.


Outline

1 Introduction




5 Conclusions


Bayes’ Formula

PriorProbabilistic information about u before data is collected:

µ0(du)

LikelihoodSince y = G(u) + ⌘, if ⌘ ⇠ N(0, �), then y |u ⇠ N(G(u), �). Themodel-data misfit � is the negative log-likelihood:

P(y |u) / exp��(u; y)

�, �(u; y) =

12

��12�y � G(u)

��2.

PosteriorProbabilistic information about u after data is collected:

µy (du) / exp��(u; y)

�µ0(du).


Rigorous StatementM. Iglesias, Y. Lu and A. M . StuartA level-set approach to Bayesian geometric inverse problems.arXiv:1504.00313

M. IglesiasA regularizing iterative ensemble Kalman method for PDE constrained inverse problems.arXiv:1505.03876

L2⌫(X ;S) = {f : X ! S : E⌫kf (u)k2

S < 1}.

Theorem (Iglesias, Lu and S)Let µ0(du) = P(du) and µy (du) = P(du|y). Assume that u 2 X µ0�a.s.Then for Examples 1–3 (and more) µy ⌧ µ0 and y 7! µy (du) isLipschitz in the Hellinger metric. Furthermore, if S is a separableBanach space and f 2 L2

µ0(X ;S), then

��Eµy1 f (u)� Eµy2 f (u)��

S C|y1 � y2|.

Key idea in proof is that F : X ! Z is continuous µ0� a.s.


Whittle-Matern Priors

c(x , y) = �2 12⌫�1�(⌫)

(⌧ |x � y |)⌫K⌫(⌧ |x � y |).

⌫ controls smoothness – draws from Gaussian fields with thiscovariance have ⌫ fractional Sobolev and Hölder derivatives.⌧ is an inverse length-scale.� is an amplitude scale.

Ignoring boundary conditions, corresponding covariance operatorgiven by

CWM(�,⌧) / �2⌧2⌫(⌧2I �4)�⌫� d2 .


Example 2: Groundwater Flow

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Figure: (Row 1, ⌧ = 15) Logarithm of the true hydraulic conductivity. (Row 2⌧ = 10, 30, 50, 70, 90) samples of F (u). (Row 3) F (E(u)). (Row 4) E(F (u)).


Outline

1 Introduction




5 Conclusions


Hierarchical Whittle-Matern Priors (The Problem)

Prior:P(u, ⌧) = P(u|⌧)P(⌧).

P(u|⌧) = N(0,CWM(�,⌧)).

Recall CWM(�,⌧) / �2⌧2⌫(⌧2I �4)�⌫� d2 .

TheoremFor fixed � the family of measures N(0,CWM(�,·)) are mutually singular.

Algorithms which attempt to move between singular measures performbadly under mesh refinement (since they fail completely in the fullyresolved limit).


Hierarchical Whittle-Matern Priors (The Solution)M. Dunlop, M. Iglesias and A. M . StuartHierarchical Bayesian Level Set InversionarXiv:1601.03605

Hence choose � = ⌧�⌫ and define: C⌧ / (⌧2I �4)�⌫� d2 . Prior:

P(u, ⌧) = P(u|⌧)P(⌧).

P(u|⌧) = N(0,C⌧ ).

TheoremThe family of measures N(0,C⌧ ) are mutually equivalent.

Suggests need to scale thresholds in F by ⌧. Let

�1 = c0 < c1 < · · · < cK�1 < cK = 1.

v(x) =KX

k=1

vk I{ck�1<u⌧⌫ck}(x); v = F (u, ⌧).


Hierarchical Posterior

Prior

µ0(du, d⌧) = µ0(du | ⌧)⇡0(d⌧)

where µ0( · |⌧) = N(0,C⌧ ), ⇡0 is hyperprior on ⌧ .

Likelihood

Likelihood is given by P(y |u, ⌧) / exp⇣��(u, ⌧ ; y)

⌘. where the

model-data misfit is now �(u, ⌧ ; y) = 12

��12�y � G � F (u, ⌧)

��2.

PosteriorProbabilistic information about u after data is collected:

µy (du, d⌧) / exp��(u, ⌧ ; y)

�µ0(du, d⌧).


MCMC

We implement a Metropolis-within-Gibbs algorithm to generate aposterior-invariant Markov chain (uk , ⌧k ):

Propose vk+1 from a P(u|⌧k )�reversible kernel.uk+1 = vk+1 w.p. 1 ^ exp(�(uk , ⌧k )� �(vk+1, ⌧k )),uk+1 = uk otherwise.Propose tk+1 from a P(⌧)�reversible kernel.⌧k+1 = tk+1 w.p. 1 ^ exp(�(uk+1, ⌧k )� �(uk+1, tk+1))w(⌧k , tk+1),⌧k+1 = ⌧k otherwise.

Here w(⌧, t) is density of N(0,Ct) with respect to N(0,C⌧ ).


Example 2: Groundwater Flow

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

Figure: (Row 1, ⌧ = 15) Logarithm of the true hydraulic conductivity (middle,top). (Row 2 ⌧ = 10, 30, 50, 70, 90) samples of F (u, ⌧). (Row 3)F (E(u),E(⌧)). (Row 4) E(F (u, ⌧)).


Outline

1 Introduction




5 Conclusions


SummaryLast 5 years development of a theoretical and computational frameworkfor infinite dimensional Bayesian inversion with wide applicability:

1 Framework allows for noisy data and uncertain prior information.2 Probabilistic well-posedness.3 Theory clearly delineates (and links) analysis and probability.4 Theory leads to new algorithms (defined on Banach space).5 Grid-independent convergence rates for MCMC.

The methodology has been extended to solve interface problems. This isachieved via the level set representation.

1 Level set method becomes well-posed in this setting.2 Discontinuity in level set map is a probability zero event.3 Hierarchical choice of length-scale improves performance.4 Amplitude scale is linked to length-scale via measure equivalence.5 Algorithms which reflect this link are mesh-invariant.

Potential for many future developments for both applications and theory:1 Applications: subsurface imaging, medical imaging.2 Theory: inhomogenous length-scales, new hierarchies.


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