EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
A Mathematical Framework
For Data Assimilation
Andrew Stuart1
1 Mathematics Institute and
Centre for Scientific Computing
University of Warwick
Lorentz Center
Leiden, March 4th 2009
Collaboration with
S.L.Cotter, M. Dashti, J.C. Robinson (Warwick).
Funded by EPSRC, ERC, ONR
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
The Lorenz Equations
Consider the Lorenz equations for v (t) = x(t), y (t), z(t) :
ẋ = σ(y − x),
ẏ = rx − y − xz,
ż = xy − bz.
Find u = v (0)
Given noisy observations
yj = v (tj ) + ηj ,
j = 1, · · · , J.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Data Assimilation in Fluid Mechanics
Consider the Navier-Stokes equation:
dv
+ νAv + B(v , v ) = f ,
dt
v (0) = u
Find u (or (u, f )).
Given noisy Eulerian observations
yj,k = v (xj , tk ) + ηj,k
Or given noisy Lagrangian observations
yj,k = zj (tk ) + ηj,k
żj = v (zj , t),
zj (0) = zj,0
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Nuclear Waste Management
Darcy’s law and mass conservation:
v = − exp(u)∇p,
−∇ · v = 0,
u = g,
x ∈Ω
x ∈ Ω,
x ∈ ∂Ω.
Find log permeability u(x).
Given noisy observations
yj = p(xj ) + ηj .
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Oil Recovery
Consider the equations for transport in a porous medium:
v = −λ(S) exp(u)∇p,
−∇ · v = h, x ∈ Ω
∂S
= −v · ∇f (S) + ν4S,
∂t
x ∈Ω
x ∈ Ω.
Find log permeability u(x).
Given noisy observations
R
yj = 1 −
∂Ωout
f S(x, tj ) vn d`
R
+ ηj .
∂Ωout vn d`
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Common Structure
Find u ∈ X from (a noisy version of) the equation
y = G(u).
The function G is the observation operator.
Problem is ill-posed: no or many solutions, sensitive
dependence.
there exist K > 0, p > 0 such that, for all u ∈ X ,
|G(u)| ≤ K 1 + kukpX ;
for every r > 0 there is K (r ) > 0 such that, for all
u1 , u2 ∈ X with max{ku1 kX , ku2 kX } < r ,
|G(u1 ) − G(u2 )| ≤ K (r )ku1 − u2 kX .
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Classical Approach to Inverse Problems
Replace by least squares problem
min Φ(u; y ),
u∈X
Φ(u; y ) =
1
ky − G(u)k2Y .
2
Can have Φ(un ; y ) → 0 but un does not converge in X .
Regularize to obtain
min I(u; y ),
u∈E
1
I(u; y ) = Φ(u; y ) + ku − m0 k2E .
2
How should k · kY , k · kE and m0 be chosen?
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Least Squares
9000
"x"
8000
7000
6000
5000
4000
3000
2000
1000
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Figure: Least Squares Objective Function – Lorenz Equations
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Bayesian Approach to Inverse Problems
Prior on u : µ0 = N (m0 , C0 ).
Given data y :
y = G(u) + η, η ∼ N (0, Γ).
Bayes rule:
P(u|y )
dµy
=
(u) ∝ exp −Φ(u; y ) ∝ P(y |u).
P(u)
dµ0
Choose k · kY , C0 and m0 so that this makes sense for
− 12
functions u. Determines k · kE := kC0
· k.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Well-Posed Inverse Problem
Theorem
Assume that µ0 (X ) = 1. Then
dµy
(u) ∝ exp −Φ(u; y )
dµ0
makes sense.
Furthermore µy is Lipschitz in the data y : there is C = C(r ) > 0
such that, for all y , y 0 with max{ky kY , ky 0 kY } ≤ r ,
0
dTV (µy , µy ) ≤ Cky − y 0 kY .
Similar perturbation results with respect to numerical
approximation of the observation operator G.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Probability Maximizers
We define probability maximizers as minimizers of the
functional
I(u; y ) :=
1
ku − m0 k2E + Φ(u; y ).
2
Theorem
Assume that µ0 (X ) = 1. There exists u ∈ E such that
I(u) = I := inf{I(u) : u ∈ E}.
Furthermore, if {un } is a minimizing sequence satisfying
I(un ) → I(u) then there is a subsequence {un0 } that converges
strongly to u in E.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Weak Constraint 4DVAR for NSE
Consider the Navier-Stokes equation:
dv
+ νAv + B(v , v ) = f ,
dt
√ dW
df
+ Rf = 2Λ
.
dt
dt
v (0) = u,
Equation for f forms the prior on model error.
f (0) chosen to make f statistically stationary.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Weak Constraint 4DVAR for NSE
Define
k(u, f )k2E = kAα/2 uk2 + kf (0)k2R −1 Λ + kf (T )k2R −1 Λ
Z T df 2
2
+
+
kRf
k
Λ dt.
dt Λ
0
Theory gives constraints on R, Λ and α which make weak
constraint 4DVAR well-posed.
The functional I attains its infimum in E :
I(u, f ) =
1
1
k(u, f )k2E + |y − G(u, f )|2Γ .
2
2
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Sampling on Function Space
Same framework allows construction of MCMC methods
on function space.
Generate Markov chain {u m } which is µy invariant.
Given u m propose
1
v = 1 − β 2 ) 2 u m + βw,
w ∼ µ0 .
Set u m+1 = v with probability α(u m , v ) where
α(u, v ) = 1 ∧ exp Φ(u; y ) − Φ(v ; y ) .
Otherwise u m+1 = u m .
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Posterior – Eulerian Data
Probability Density Functions With T=0.5
8
9 Stations
25 Stations
49 Stations
64 Stations
81 Stations
100 Stations
400 Stations
900 Stations
Re(u00,1)
7
Probability Density
6
5
4
3
2
1
0
−2
−1.5
−1
−0.5
Re(un0,1)
0
0.5
1
1.5
Figure: Posterior on Fourier mode for increasing number of
observations.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Posterior – Eulerian Data
Probability Denisty Functions with T=0.5
100
9 Paths
25 Paths
49 Paths
64 Paths
81 Paths
100 Paths
400 Paths
900 Paths
Re(un0,1)
90
80
Probability Density
70
60
50
40
30
20
10
0
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
Re(u0,1)
−0.2
−0.1
0
0.1
Figure: Posterior on Fourier mode for increasing number of
observations.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Outline
1
EXAMPLES OF INVERSE PROBLEMS
2
COMMON STRUCTURE
3
VARIATIONAL METHODS
4
METROPOLIS-HASTINGS METHODS
5
CONCLUSIONS
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
What We Have Shown
We have shown that:
Applications: Many inverse problems in differential
equations can be formulated in the framework of Bayesian
statistics on function space.
Common Structure: These problems share a common
mathematical structure leading to well-posed inverse
problems for measures.
Algorithms I: Variational methods (MAP estimators) have
a solution on function space: uses methods from optimal
control.
Algorithms II: MCMC methods can be defined on function
space: algorithms robust to discretization.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
What Remains Open
Algorithms: improve MAP and MCMC methods by
combining, use of adjoints, low-rank approximation etc.
Applications: are numerous in physics, data assimilation,
signal processing and econometrics.
Evaluation: of methods introduced here in comparison
with other methods.
Statistics: incoporate into (Gibbs) sampler to estimate
parameters as well as functions.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
References
S.L.Cotter, M. Dashti, J.C.Robinson, A.M.Stuart. ”Data
assimilation problems in fluid mechanics: Bayesian
formualtion in function space.” Submitted, Inverse
Problems.
A. Beskos, G.O. Roberts, A.M. Stuart and J. Voss. ”An
MCMC Method for diffusion bridges.” To appear,
Stochastics and Dynamics.
A. Beskos and A.M. Stuart. ”MCMC Methods for Sampling
Function Space”. To appear, proceedings of ICIAM 2007.
A. Beskos, G.O. Roberts and A.M. Stuart. ”Optimal
scalings for local Metropolis-Hastings chains on
non-product targets in high dimensions.” To appear, Ann.
Appl. Prob.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
References (Continued)
M. Hairer, A.M.Stuart and J. Voss. ”Sampling the posterior:
an approach to non-Gaussian data assimilation.”
PhysicaD, 230(2007), 50–64.
M. Hairer, A.M.Stuart, P. Wiberg and J. Voss. ”Analysis of
SPDEs Arising in Path Sampling. Part 1: The Gaussian
Case.” Comm. Math. Sci. 3(2005), 587–603
M. Hairer, A.M.Stuart and J. Voss. ”Analysis of SPDEs
Arising in Path Sampling. Part 2: The Nonlinear Case.”
Ann. Appl. Prob. 17(2007), 1657–1706.
For all papers see:
http : //www.maths.warwick .ac.uk / ∼ masdr /sample.html
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
Introductory Example
Sample x ∈ [3, 4] from the pdf
π(x) ∝ exp(−x 2 /2)I(3,4) (x).
MCMC methods a very flexible tool.
Extend to sample π in infinite (n 1) dimensions.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
The Heat Equation
Consider the heat equation:
dv
+ Av = 0,
dt
v (0) = u
Find u.
Given noisy observations
y = v (T ) + η.
university-logo
EXAMPLES OF INVERSE PROBLEMS COMMON STRUCTURE VARIATIONAL METHODS METROPOLIS-HASTINGS METHOD
The Heat Equation
Because the observation operator G is linear, the posterior
measure µy is Gaussian N (m, C).
Assume A, C and Γ all commute for simplicity.
If we define K = I + exp(−2AT )Γ−1 C0 then
m = m0 + exp(−AT )Γ−1 C0 K −1 y − exp(−AT )m0
C = C0 K −1 .
Formally, in the limit Γ → 0, we recover
m → exp(AT )y
C → 0.
university-logo