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State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

State estimation approach to nonstationary inverse problems

Hanna Katriina Pikkarainen

Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences

AIP 2007, Vancouver, Canada, June 25–29, 2007

in the collaboration with Janne M.J. Huttunen (University of Kuopio)

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Dynamical inverse problem

we are interested in the quantity X in a domain along time a model for the time evolution of X the evolution model may not be correct = ⇒ an additional source term representing possible modelling errors no direct measurements of X the quantity Y depends linearly on X = ⇒ observe the quantity Y at direct time instants an additional measurement noise in the measured values of Y calculate an estimate for X based on the measured values of Y

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

State evolution equation

D ⊂ Rd is a domain that corresponds to the object of interest X = X (t, x), x ∈ D, is the unknown distribution of the physical quantity we are interested in at time t ≥ 0 A : D(A) ⊂ L2(D) → L2(D) is defined by f →

d

• i,j=1

aij∂i∂jf +

d

• i=1

bi∂if + cf (1) where D(A) =

• f ∈ H 2(D) :

d

• i=1

βi∂if + γf    

∂D

= 0

• (2)

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

State evolution equation

Assumption Let x0 ∈ L2(D), Γ0 and Q be positive self-adjoint trace class

• perators from L2(D) to itself with trivial null spaces, and T > 0.

According to Kolmogorov’s existence theorem there exist a probability space (Ω, F, P), a Q-Wiener process W (t), t ∈ [0, T], with values in L2(D), an L2(D)-valued random variable X0 such that X0 and W (t) are independent for all t ∈ (0, T], X0 is Gaussian with mean x0 and covariance Γ0.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

State evolution equation

We assume that the time evolution of the process X can be modelled by the stochastic partial differential equation dX (t) = AX (t)dt + dW (t) (3) for every t > 0 with the initial value X (0) = X0. (4) The term dW (t) is a source term representing possible modelling errors in the time evolution model.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Observation equation

The measurement process is modelled by the equation Y (t) = C(t)X (t) + S(t) (5) for all 0 < t ≤ T where {C(t)}t∈(0,T] is a family of bounded linear

• perators from L2(D) to RL and S(t), t ∈ [0, T], is an RL-valued

stochastic process. The process S represents possible measurement errors.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Time discrete state estimation system

the measurements are made in time instants 0 < t1 < . . . < tn ≤ T we use the notation t0 := 0, ∆k+1 := tk+1 − tk, (6) Xk+1 := X (tk+1), Yk+1 := Y (tk+1), (7) Ck := C(tk), Sk+1 := S(tk+1) (8) for all k = 0, . . . , n − 1

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Time discrete state estimation system

The time discrete state estimation system is Xk+1 = U (∆k+1)Xk + Wk+1, k = 0, . . . , n − 1, (9) Yk = CkXk + Sk, k = 1, . . . , n (10) where the state noise Wk+1 is given by the formula Wk+1 = tk+1

tk

U (tk+1 − s) dW (s). (11) By the Riesz representation theorem there exist such functions ϕ(k)

p

∈ L2(D) that (Ckf )p = (f , ϕ(k)

p ) for all f ∈ L2(D), k = 1 . . . , n

and p = 1, . . . , L.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Space discretization

Definition The sequence {Vm}∞

m=1 of finite-dimensional subspaces of L2(D) is

called a sequence of appropriate discretization spaces if (i) Vm ⊂ Vm+1 for all m ∈ N, (ii) ∪Vm = L2(D). {Vm}∞

m=1 is a sequence of appropriate discretization spaces in

L2(D) {ψm

l }Nm l=1 is an orthonormal basis of Vm for all m ∈ N

Z m := ((Z, ψm

1 ), (Z, ψm 2 ) . . . , (Z, ψm Nm))T is the discretized

version of a random variable Z at the discretization level m

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Discretized state estimation system

By using the time discrete state estimation equation (9)–(10) we are able to give a state estimation system for the finite-dimensional processes {X m

k }n k=0 and {Yk}n k=1. The discretized state estimation

system is X m

k+1 = Am k+1X m k + ǫm k+1 + W m k+1,

k = 0, . . . , n − 1, (12) Yk = C m

k X m k + νm k + Sk,

k = 1, . . . , n. (13) The matrices Am

k+1 and C m k

are given by (Am

k )ij := (U (∆k)ψm j , ψm i ) and (C m k )pj := (ψm j , ϕ(k) p ) for all

i, j = 1, . . . , Nm, k = 1, . . . , n and p = 1, . . . , L.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Discretization errors

The discrete stochastic process ǫm

k+1 := ((Xk, (I − Pm)U ∗(∆k)ψm 1 ), . . . ,

. . . , (Xk, (I − Pm)U ∗(∆k)ψm

Nm))T

(14) represent the discretization error in the state evolution equation, W m

k

= ((Wk, ψm

1 ), . . . , (Wk, ψm Nm))T

(15) is the state noise vector and νm

k := ((Xk, (I − Pm)ϕ(k) 1 ), . . . , (Xk, (I − Pm)ϕ(k) L ))T

(16) represents the discretization error in the observation equation for all k = 1, . . . , n.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Discretized filtering problem

the measurements Dk := (Y T

k , Y T k−1, . . . , Y T 1 )T

the measured data dk := (yT

k , yT k−1, . . . , yT 1 )T

we are interested in a real-time monitoring for the quantity X = ⇒ we want to solve the filtering problem µX m

k |Dk ( · | dk)

the conditional distribution µX m

k | Dk ( · | dk) of X m

k

respect to the measurements dk is uniquely defined by the formula P((X m

k )−1(G) ∩ D−1 k (F)) =

• F

µX m

k |Dk (G | dk) µDk (ddk)

for all RNm-Borel sets G and RkL-Borel sets F where µDk := P ◦ D−1

k

is the distribution of Dk

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

State noise vector

the state noise Wk+1 is a Gaussian random variable for all k = 1, . . . , n = ⇒ the state noise vector W m

k+1 is normal

W m

k+1 is independent of X m l

and ǫm

l+1 for all l ≤ k and

k = 0, . . . , n − 1 the state noise vectors W m

k

and W m

l

are mutually independent for all k = l

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Observation noise vector

Assumption The observation noise vectors Sk are chosen such a way that they are normal random variables, the mean ESk is zero, Sk is independent of X0 for all k = 1, . . . , n. In addition, Sk and Sl are mutually independent for all k = l, Sk and W m

l

are mutually independent for all k, l = 1, . . . , n.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Conditional distribution

The joint distribution of X m

k

and Dk is normal for all k = 1, . . . , n. Hence the solution to the discretized filtering problem µX m

k |Dk ( · | dk)

is the normal distribution with the density function πX m

k |Dk (xk | dk) ∝ exp

• −1

2(xk − ¯ ηk)T ¯ Σ−1

k (xk − ¯

ηk)

• (17)

where ¯ ηk = EX m

k + Cor(X m k , Dk) Cov(Dk)−1(dk − EDk),

(18) ¯ Σk = Cov X m

k − Cor(X m k , Dk) Cov(Dk)−1 Cor(X m k , Dk)T

(19) for all k = 1, . . . , n.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Prediction

We denote ˜ µk := µX m

k |Dk−1( · | dk−1) and ¯

µk := µX m

k |Dk ( · | dk).

Theorem The expectation and the covariance matrix of the conditional distributions ˜ µk+1 are given by ˜ ηk+1 = Am

k+1¯

ηk + Edk (ǫm

k+1),

(20) ˜ Σk+1 = Am

k+1 ¯

Σk(Am

k+1)T + Covdk (ǫm k+1) + Cov(W m k+1)

+ Am

k+1 Cordk (X m k , ǫm k+1) + Cordk (ǫm k+1, X m k )(Am k+1)T (21)

for all k = 0, . . . , n − 1.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Filtering

Theorem The expectation and the covariance matrix of the conditional distribution ¯ µk+1 are given by ¯ ηk+1 = ˜ ηk+1 + K m

k+1

• yk+1 − C m

k+1˜

ηk+1 − Edk (νm

k+1)

• ,

(22) ¯ Σk+1 = ˜ Σk+1 − K m

k+1

• C m

k+1 ˜

Σk+1 + Cordk (νm

k+1, X m k+1)

• (23)

for all k = 0, . . . , n − 1 where the matrix K m

k+1 is

K m

k+1 =

• ˜

Σk+1(C m

k+1)T + Cordk (X m k+1, νm k+1)

• (24)

×

• C m

k+1 ˜

Σk+1(C m

k+1)T + Covdk (νm k+1) + Cov(Sk+1)

+ C m

k+1 Cordk (X m k+1, νm k+1) + Cordk (νm k+1, X m k+1)(C m k+1)T−

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

One-dimensional model case

The time evolution of the concentration distribution X is modelled by the stochastic initial value problem

• dX (t) = AX (t)dt + dW (t),

t > 0, X (0) = X0 (25) where the operator A is defined by A : H 2(R) → L2(R), f → κ d2 dx 2 f − v d dx f (26) and κ, v > 0.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Measurements

Measurements are made at time instants 0 < t1 < . . . < tn ≤ T and are described by the observation equation Yk = CXk + Sk (27) for all k = 1, . . . , n. The operator C : L2(R) → RL is defined by (Cf )p = ∞

−∞

f (x)ϕp(x) dx = (f , ϕp) (28) for all p = 1, . . . , L with ϕp(x) = 1 2wp exp

• −|x − xp|

wp

• (29)

for all x ∈ R where xp ∈ R is a measurement position and 0 < wp < 1.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Wiener process and the initial state

The mean of the initial state X0 is chosen to be x0(x) =

• x0

if |x| ≤ M , x0e−(|x|−M) if |x| > M (30) where x0, M > 0. The covariance operators of the initial state X0 and the Wiener process W are Γ0 = σ2

0Γ and Q = σ2Γ where

σ0, σ > 0 and Γ is the integral operator with kernel w(x)γ(x − y)w(y) with γ(x) = (1 + |x|)e−|x|/4 and w(x) =

• 1

if |x| < N , e−(|x|−N) if |x| ≥ N , (31) for some N > 0.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Observation noise and discretization

The observation noise vectors Sk are chosen such a way that Sk is normal with the mean ESk = 0 and the covariance matrix Cov(Sk) = σ2

SI for all k = 1, . . . , n where σS > 0 and I ∈ RL×L is

the identity matrix. A family of appropriate discretization spaces in L2(R) is defined by Vm := span √mχ[ i−1

m −m, i m −m], i = 1, . . . , 2m2

for all m ∈ N.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Computation

Parameter values for simulations: Coefficient Value Coefficient Value κ 1 v 0.5 T 10 n 5 L 10 wp 0.25 x0 1 M 2 σ0 0.1 σ 0.04 σS 0.1 m 3 tk = 2k and xp = −2 + 4p − 1 L − 1

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Energy of the error terms

1 1.5 2 2.5 3 3.5 4 4.5 5 0.05 0.1 0.15 0.2 t 1 1.5 2 2.5 3 3.5 4 4.5 5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 t

Figure: Left: The norm of the expectation of ǫm

k (−) and νm k (··). Right:

The square-root of the trace of the covariance of ǫm

k (−), νm k (··) and W m k

(−·).

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Comparison of different filtering methods

!3 !2 !1 1 2 3 0.2 0.4 0.6 0.8 1 Exact conditional expectation !3 !2 !1 1 2 3 0.2 0.4 0.6 0.8 1 Presented method !3 !2 !1 1 2 3 0.2 0.4 0.6 0.8 1 Kalman filter

Figure: The computed point estimates for the final state t = 10 (black lines) with the S.D. error intervals (··). A gray line corresponds to the exact solution.

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Overview

1

Introduction

2

State estimation system

3

Space discretization

4

Discretized filtering problem

5

Numerical example

6

Conclusions

State estimation approach to nonstationary inverse problems H.K. Pikkarainen Introduction State estimation system Space discretization Discretized filtering problem Numerical example Conclusions

Conclusions

In this talk we have presented state estimation approach to nonstationary inverse problems, the space discretization of an infinite-dimensional state estimation system, the distributions of the discretization errors, the solution to the finite-dimensional filtering problem taking into account the discretization errors, numerical simulations of a one-dimensional model case.