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4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Optimal state estimation for numerical weather prediction using reduced order models

A.S. Lawless1, C. Boess1, N.K. Nichols1 and

• A. Bunse-Gerstner2

1 School of Mathematical and Physical Sciences, University of Reading, U.K. 2 ZeTeM, Universtaet Bremen, Germany. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Outline

1 4D-Var and Gauss-Newton 2 Model reduction in data assimilation 3 Extension to unstable systems 4 Numerical results 5 Conclusions

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

4-Dimensional Variational Data Assimilation (4D-Var)

Aim To find the best estimate of the true state of a system consistent with observations distributed in time and with the system dynamics.

Time x xb tN t0 A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

4-Dimensional Variational Data Assimilation (4D-Var)

Aim To find the best estimate of the true state of a system consistent with observations distributed in time and with the system dynamics.

Time x xb xa tN t0 A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Nonlinear least squares problem

We consider a dynamical system xi+1 = Mi(xi) yi = H(xi) + ηi with Mi Model dynamics xi State vector O(107 − 108) yi Observations at time ti O(106 − 107) ηi Observational noise Hi Observation operator

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Nonlinear least squares problem

Then the data assimilation problem is to minimize J [x0] = 1 2(x0−xb)TB−1

0 (x0−xb)+1

2

N

• i=0

(Hi[xi]−yi)TR−1

i

(Hi[xi]−yi) subject to the dynamical system, where xb A priori (background) estimate B0 Background error covariance matrix Ri Observation error covariance matrix

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

This can be written as a standard nonlinear least squares problem J [x] = 1 2||f(x)||2

2

with f(x0) =        B

1 2

0 (x0 − xb)

R

1 2

0 (H0[x0] − y0)

. . . R

1 2

N(HN[xN] − yN)

      

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Gauss-Newton method (Incremental 4D-Var)

For the nonlinear least squares problem J [x] = 1 2||f(x)||2

2

the Gauss-Newton method is equivalent to the following iteration Start with iterate x(0) For k = 0, . . . , K − 1

Solve minδx ||J(x(k))δx + f(x(k))||2

2

Update x(k+1) = x(k) + δx

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

For the data assimilation problem the inner cost function is ˜ J (k)[δx(k)

0 ]

= 1 2(δx(k) − [xb − x0(k)])TB−1

0 (δx(k)

− [xb − x0(k)]) + 1 2

N

• i=0

(Hiδx(k)

i

− d(k)

i

)TR−1

i

(Hiδx(k)

i

− d(k)

i

) with di = yi − Hi[xi], subject to the linear dynamical system δxi+1 = Miδxi di = Hiδxi.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

In practice the inner problem can only be solved approximately. Common approximations include

1 Truncate iterative solution, so that exact minimum not found. 2 Use approximate linear model. 3 Solve the problem in a restricted space (usually a lower spatial

resolution). Gratton, Lawless and Nichols (SIOPT, 2007) proved conditions for convergence of Gauss-Newton under approximations 1 & 2. In this work we consider approximation 3.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Low resolution G-N in data assimilation

We introduce linear restriction operators UT

i ∈ Rr×n such that

δˆ xi = UT

i δxi ∈ Rr, r < n.

We also define linear prolongation operators Vi ∈ Rn×r, with UT

i Vi = Ir and ViUT i

a projection.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Low resolution G-N in data assimilation

We introduce linear restriction operators UT

i ∈ Rr×n such that

δˆ xi = UT

i δxi ∈ Rr, r < n.

We also define linear prolongation operators Vi ∈ Rn×r, with UT

i Vi = Ir and ViUT i

a projection. We can then define a restricted dynamical system in Rr δˆ xi+1 = ˆ Miδˆ xi, ˆ di = ˆ Hiδˆ xi, where Vi ˆ MiUT

i

approximates Mi and ˆ HiUT

i

approximates Hi.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

We solve the minimization problem in Rr ˆ J (k)[δˆ x(k)

0 ]

= 1 2(δˆ x(k) − UT

0 [xb − x0(k)])T

׈ B−1

0 (δˆ

x(k) − UT

0 [xb − x0(k)])

+ 1 2

N

• i=0

(ˆ Hiδˆ x(k)

i

− d(k)

i

)TR−1

i

(ˆ Hiδˆ x(k)

i

− d(k)

i

), subject to the restricted system, and then update the G-N iterate using δx(k) = V0δˆ x(k)

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

In practice the restricted system is chosen to be a low resolution version of the full system. This does not take into account the dynamics or observations. Can we do better using ideas of model reduction?

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Model reduction

Given a time-invariant linear system δxi+1 = Mδxi + Gui di = Hδxi where ui ∼ N(0, W), find projection matrices U, V with UTV = Ir and r << n, such that the reduced order system δˆ xi+1 = UTMViδˆ xi + UTGui ˆ di = HViδˆ xi approximates the full system with δxi ≈ Vδˆ xi.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

We seek matrices U, V such that we minimize lim

i→∞ E

• ˆ

di − di T R−1 ˆ di − di

• ver all inputs of normalized unit length.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

We seek matrices U, V such that we minimize lim

i→∞ E

• ˆ

di − di T R−1 ˆ di − di

• ver all inputs of normalized unit length.

Bernstein et al. (1986) derive necessary conditions for a minimum, but it is not practicable to obtain the optimal matrices. The method of balanced truncation finds an approximation solution, with error bounded in terms of the Hankel singular values.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Balanced truncation

Balanced truncation removes states which are least affected by inputs and which have least effect on outputs. There are 2 steps

1 Balancing - Transform the system into one in which these

states are the same.

2 Truncation - Truncate states related to the smallest singular

values of the transformed covariance matrices (Hankel singular values).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Step 1: The balancing transformation simultaneously diagonalizes the state covariance matrices P and Q associated with the inputs and outputs. This requires the solution of the Stein equations P = MPMT + GGT, Q = MTQM + HTR−1H. The transformation Ψ, given by the matrix of eigenvectors of PQ, is used to transform the system into balanced form.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Step 1: The balancing transformation simultaneously diagonalizes the state covariance matrices P and Q associated with the inputs and outputs. This requires the solution of the Stein equations P = MPMT + GGT, Q = MTQM + HTR−1H. The transformation Ψ, given by the matrix of eigenvectors of PQ, is used to transform the system into balanced form. Step 2: The balanced system is truncated using UT = [Ir, 0] Ψ−1, V = Ψ Ir

• .

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Application to data assimilation

In data assimilation we have the (time-invariant) stochastic dynamical system δxi+1 = Mδxi di = Hδxi with stochastic initial condition δx0 ∼ N(0, B0). To apply model reduction we need to write this in the form δxi+1 = Mδxi + Gui di = Hδxi

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

We write our original system in the form δxi+1 = Mδxi + ˜ G˜ ui di = Hδxi with ˜ G = [G In], ˜ ui = ui vi

• and

˜ ui ∼ N(0, ˜ W), ˜ W = W B0

• .

For 4D-Var we have G = 0.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Reduced order cost function

The reduced order minimization problem is then min ˆ J (k)[δˆ x(k)

0 ]

= 1 2(δˆ x(k) − UT[xb − x0(k)])T × (UTB0U)−1(δˆ x(k) − UT[xb − x0(k)]) + 1 2

N

• i=0

(HVδˆ x(k)

i

− d(k)

i

)TR−1(HVδˆ x(k)

i

− d(k)

i

), subject to δˆ xi+1 = UTMVδˆ xi and set δx(k) = Vδˆ x(k)

0 .

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Extension to unstable systems

Previously we have shown the benefit of using reduced order models in data assimilation when the system matrix is stable (Lawless et. al, 2008, Mon. Wea. Rev.; Lawless et. al, 2008, Int.

• J. Numer. Methods in Fluids.)

But How can we choose the projection operators UT and V when the system matrix is unstable?

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

We consider 3 methods:

1 Choose UT as a low resolution spatial restriction. 2 Balanced truncation, with standard splitting for unstable

systems.

3 New α-bounded balanced truncation for unstable systems. A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Standard extension of balanced truncation

Divide the system S into its stable and unstable parts S = S+ + S− Apply the usual balanced truncation method to the stable part S+ − → (S+)red Combine the reduced stable part with the unchanged unstable part ˆ S = (S+)red + S−

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Standard extension of balanced truncation

Divide the system S into its stable and unstable parts S = S+ + S− Apply the usual balanced truncation method to the stable part S+ − → (S+)red Combine the reduced stable part with the unchanged unstable part ˆ S = (S+)red + S− Problem: Only works if the number of unstable poles is small.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

New approach for unstable systems I

Boess, Bunse-Gerstner and Nichols (2010) introduce an extension

• f balanced truncation for α-bounded systems, i.e. systems whose

eigenvalues lie in a circle of radius α > 1.

1 Find α ∈ R+ such that all eigenvalues of the system matrix M

lie inside an α-circle around the origin.

2 Choose a reduction order r ≪ n. 3 Apply model reduction to the α-scaled system

δx(α)

i+1

= Mαδx(α)

i

+ ˜ Gαui d(α)

i

= Hαδx(α)

i

with Mα := M/α, ˜ Gα := ˜ G/√α, Hα := H/√α.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

New approach for unstable systems II

This yields projection matrices Uα, Vα ∈ Rn×r and a low

• rder system:

δˆ x(α)

i+1

= ˆ Mαδˆ x(α)

i

+ ˜ Gαui ˆ d(α)

i

= ˆ Hαδˆ x(α)

i

with ˆ Mα := UT

α MαVα, ˆ

Gα := UT

α ˜

Gα and ˆ Hα := HαVα.

4 The scaling α is then reversed to give the reduced system

δˆ xi+1 = ˆ Mδˆ xi + ˜ Gui ˆ di = ˆ Hδˆ xi with ˆ M = α ˆ Mα, ˜ G = √α˜ Gα and ˆ H = √αˆ Hα.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Reduced order cost function

The reduced order minimization problem is now min ˆ J (k)[δˆ x(k)

0 ]

= 1 2(δˆ x(k) − UT

α [xb − x0(k)])T

× (UT

α B0Uα)−1(δˆ

x(k) − UT

α [xb − x0(k)])

+ 1 2

N

• i=0

(HVαδˆ x(k)

i

− d(k)

i

)TR−1(HVαδˆ x(k)

i

− d(k)

i

), subject to the reduced system. We then set δx(k) = Vαδˆ x(k)

0 .

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Numerical results - 2D Eady model

The Eady model is a 2D x − z model describing the evolution of a buoyancy b on the boundaries and a potential vorticity q in the interior. We have the evolution equations ∂ ∂t + z ∂ ∂x

• b = ∂ψ

∂x ,

• n z = ±1

2, x ∈ [0, X], and ∂ ∂t + z ∂ ∂x

• q = 0,

in z ∈ [−1 2, 1 2], x ∈ [0, X], where the streamfunction ψ satisfies ∂2ψ ∂x2 + ∂2ψ ∂z2 = q, in z ∈ [−1 2, 1 2], x ∈ [0, X], with boundary conditions ∂ψ ∂z = b,

• n z = ±1

2, x ∈ [0, X].

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Methodology

Define the ‘true’ solution at the initial time. Run the model to generate synthetic observations at times t0, . . . , t5 (12h window). Define a linear least squares problem using these observations and a background field. Solve the reduced order least squares problem using

low resolution model; reduced order system using standard balanced truncation; reduced order system using new method.

Lift solution to full resolution and compare with solution to full linear least squares problem.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Experiment 1: Zero PV

When the interior potential voriticity is zero, the only evolution is

• n the upper and lower boundaries. We consider

20 horizontal grid points (system size 40). True state is the eigenvector associated with the largest eigenvalue of the model. Zero background state with inverse Laplacian covariance matrix of variance 1. Observation error covariance matrix is identity. Observations taken of buoyancy on lower boundary only. Reduced problem solved with size 20.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

2 4 6 8 10 12 14 16 18 20 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 vector components solution Eady40, lower buoyancy, α= 1.12: Solution plot x0 x0

(low), order 20

x0

(std), order 20

x0

(alpha), order 20

2 4 6 8 10 12 14 16 18 20 10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 vector components error Eady40, lower buoyancy, α= 1.12: Error plot e1=x0−x0

(low), order 20

e2=x0−x0

(std), order 20

e3=x0−x0

(alpha), order 20

Figure: Lower boundary solution (top) and error (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

22 24 26 28 30 32 34 36 38 40 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 vector components solution Eady40, upper buoyancy, α= 1.12: Solution plot x0 x0

(low), order 20

x0

(std), order 20

x0

(alpha), order 20

22 24 26 28 30 32 34 36 38 40 10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 vector components error Eady40, upper buoyancy, α= 1.12: Error plot e1=x0−x0

(low), order 20

e2=x0−x0

(std), order 20

e3=x0−x0

(alpha), order 20

Figure: Upper boundary solution (top) and error (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Eigenvalues of α−reduced M, α = 1.12 real axis imaginary axis eigs M eigs Mα unit circle 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Eigenvalues of low resolution M real axis imaginary axis eigs M eigs Mlow unit circle 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Eigenvalues of standard reduced M real axis imaginary axis eigs M eigs Mstd unit circle

Figure: Comaprison of eigenvalues with full system for α-reduced (top left), standard reduced (top right) and low resolution (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Experiment 2: Non-zero PV

We now consider a case when there is evolution of the PV in the interior. The full system size is now 1040 (80 horizontal grid points). True solution is non-modal growing mode. Observations are taken at every second grid point. Background error variances are 10−5 on the boundaries and 1 in the interior. Reduced problem solved with size 520.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

−2e−006 −2e−006 2 e − 6 2e−006 −4e−006 4 e − 6 6e−006 −6e−006 8e−006 −8e−006 Contour plot of true solution 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

2 e − 6 2 e − 6 −2e−006 − 2 e − 6 4e−006 −4e−006 6 e − 6 − 6 e − 6 Contour plot of lifted low res solution, k=520 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

Figure: True solution (top) and low resolution (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

−2e−006 −2e−006 2 e − 6 2e−006 −4e−006 4 e − 6 6e−006 −6e−006 8e−006 −8e−006 Contour plot of true solution 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

2 e − 7 − 2 e − 7 − 2 e − 7 4e−007 2e−007 2 e − 7 −4e−007 −2e−007 −4e−007 6 e − 7 Contour plot of lifted mod red solution using standard BT, k=520 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

Figure: True solution (top) and standard BT (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

−2e−006 −2e−006 2 e − 6 2e−006 −4e−006 4 e − 6 6e−006 −6e−006 8e−006 −8e−006 Contour plot of true solution 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

2e−006 2e−006 −2e−006 −2e−006 4e−006 −4e−006 6e−006 − 6 e − 6 8e−006 −8e−006 Contour plot of lifted mod red solution using α BT, α = 1.13, k=520 10 20 30 40 50 60 70 80 1 2 3 4 5 6 7 8 9 10 11 −8 −6 −4 −2 2 4 6 8 x 10

−6

Figure: True solution (top) and α BT (bottom).

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Conclusions

In many applications dynamical systems may be unstable over finite time intervals.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Conclusions

In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Conclusions

In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var. Numerical results show a more accurate solution than using low resolution models or standard model reduction.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Conclusions

In many applications dynamical systems may be unstable over finite time intervals. A new method of α-bounded balanced truncation has been incorporated into 4D-Var. Numerical results show a more accurate solution than using low resolution models or standard model reduction. In part this can be explained by preservation of the model eigenvalues.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Remaining issues

In order to be implemented in real systems the method still requires some further work:

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Remaining issues

In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Remaining issues

In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Remaining issues

In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems. Update of reduced order system within iterative process.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

Remaining issues

In order to be implemented in real systems the method still requires some further work: Extension to time-varying systems. Efficient reduction methods for large-scale systems. Update of reduced order system within iterative process. Nevertheless results indicate that this is research that is worth pursuing.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

References I

Boess, C., Nichols, N.K. and Bunse-Gerstner, A. (2010), Model reduction for discrete unstable control systems using a balanced truncation approach. Preprint MPS 2010 06, Department of Mathematics, University of Reading. Gratton, S., Lawless, A.S. and Nichols, N.K. (2007), Approximate Gauss-Newton methods for nonlinear least squares problems. SIAM J. on Optimization, 18, 106-132. Lawless, A.S., Nichols, N.K., Boess, C. and Bunse-Gerstner, A. (2008), Using model reduction methods within incremental 4D-Var. Mon. Wea. Rev., 136, 1511-1522.

A.S. Lawless et al. Optimal state estimation using reduced order models

4D-Var Model reduction in DA Unstable systems Numerical results Conclusions

References II

Lawless, A.S., Nichols, N.K., Boess, C. and Bunse-Gerstner, A. (2008), Approximate Gauss-Newton methods for optimal state estimation using reduced order models. Int. J. Numer. Methods in Fluids, 56, 1367-1373. Boess, C., Lawless, A.S., Nichols, N.K. and Bunse-Gerstner, A. (2011), State estimation using model order reduction for unstable systems. Comput. Fluids, 46, 155-160.

A.S. Lawless et al. Optimal state estimation using reduced order models