Estimation of a State of the Model Variable Field
for the Burgers Equation
Yasuyoshi Horibata
Faculty of Science and Enginnering, Hosei University, Japan1Introduction

The distance function is taken as

Numerical simulation of an unsteady model is an initial/boundary value problem: given an initial conditionof the model, and boundary conditions, the model simulates the evolution of the model variables. Obviously, themore accurate the estimate of the initial conditions, thebetter the quality of the forecasts. Estimation is the process through which all the available information is used inorder to estimate as accurately as possible the state of all
the model variables (an initial condition). The available
information consists of the observation proper, and of thephysical laws that govern the evolution of the model variables. The latter are available in practice under the formof a numerical model [1].
In this paper, the Estimation is formulated as a nonlinear optimization problem with tens of thousands of variables. Problems of this size can be solved efficiently only
if the storage and computational costs of the optimization algorithm can be kept at a tolerable level. Hence, alarge-scale optimization method is employed to minimize
the objective function. At each iteration, it requires thegradient of the objective function. A method is developed
for computing the gradient efficiently. Finally, numericalexperiment is presented.
2Formulation
Consider the nonlinear Burgers equation:∂u∂u
1 ∂2u= −u+
∂t
∂x
R ∂x2

(1)

Once it has been disctretized in space using finite differences to n independent variables, the model can be written
as a set of n nonlinear coupled ordinary differential equations:dxdt
where01u1
B Cx = @ ... A
un=F(x)

(2)

0(3)

1
F1
B C
F = @ ... AFn

This is the model in differential form. Once a timedifference scheme is chosen, it becomes a set of nonlinearcoupled difference equations; this is the forward model.
A numerical solution of the model (2) starting from an
initial time t0 can be readily obtained by integrating themodel numerically using the forward model between t0 anda final time t. Let x(t0 ) denote the initial condition fromwhich the model is numerically integrated. The initial condition defines a unique solution x(t) to the model.

J=n1X

H[x(ti )] dt(4)n2where H[x(ti )] is a scalar measuring the distance between
x(ti ) and its observations available at time ti = i∆t.
The available observations are assumed to be distributed
over a limited time interval [t2 , t1 ] (t2 = n2 ∆t, t1 =n1 ∆t, t0 < t2 < t1 ). The constraint is the model equation. For a given initial condition and for the corresponding solution x(ti ) of the forward model, the distance function is evaluated Thus, the distance function is regardedas a function of x(t0 ).
Hence, estimation of x(t0 ) is formulated as an optimization problem:
min J(x(t0 ))An optimization method is used to find the value x(t0 )which minimizes J, starting from an initial guess of x(t0 ).
3

Numerical Optimization

This optimization problem has thousands or millions
of variables. Hence, L-BFGS-B [2] are used to solve thislarge-scale problem. Since the method does not require
second derivatives of the objective function, it can be applied when the Hessian is not practical to compute. It uses
the limited memory BFGS approximation to the Hessian,and so creates a quadratic model function using gradient
information in such a way that the storage required in linear in the number of the variables.
4Forward Model

The forward model integrates the Buugers equation
using the finite difference scheme; the central difference
scheme is used for the diffusion term whereas the second
order upwind difference scheme is used for the advectionterm:
8 ui−2 −4ui−1 +3ui„ «(ui > 0)<2∆x∂u=(5)∂x i : −3ui +4ui+1 −ui+2(ui < 0)2∆xwhere xi = i∆x.
5

Gradient Computation

Each iteration of numerical optimization requires thegradient of the objective function. If a finite-difference
derivative approximation is used to calculate the gradient, then it requires executing as many integrations of themodel equation as the number of the variables. Hence,huge CPU time is consumed.
We derive the tangent linear model that propagates theperturbation from t0 to t from the forward model, andthen the corresponding adjoint model [3]. The gradient iscomputed from the adjoint model.
6

Numerical Experiment1
truth
initial_guess
0.50The Burgers equation is numerically integrated between
the t0 = 0 and t1 = 1000∆t. Figure 1 - Figure 6 show theevolution of the model variable field.

2040x6080-0.5-1
11

0.5

0.5

0020

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-0.5

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-1-1Fig. 1. Time = 01

0.5

0.5

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Fig. 2. Time = 200 ∆t

10Fig. 7. Initial guess

02040x6080

-0.5

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-1-1Fig. 3. Time = 400 ∆t

Fig. 4. Time = 600 ∆t

Fig. 8. Convergence history of the optimization1truth
estimate

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0002040x60

80

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Fig. 5. Time = 800 ∆t

Fig. 6. Time = 1000 ∆t

This solution is assumed to be observations. Using theobservations between t2 = 201∆t and t1 = 1000∆t, we estimate the state of the field at t0 = 0 used for the numericalintegration. L-BFGS-B solves the large-scale nonlinear optimization problem starting an initial guess for the state.
Figure 7 shows the initial guess.
Figure 8 shows the convergence history. The objectivefunction decreases by a factor of about 106 after 500 iterations.
Figures 9 compares the estimated state and its truth.
72080Conclusions

The estimated state agrees reasonably with the initialcondition used in the numerical integration.

40x60

80

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Fig. 9. Estimated state and its truth

References[1] O. Talagrand, Assimilation of Observations, An Introduction, J. Met. Soc. Japan, Special Issue 75, 1B, pp. 191-209(1997)[2] C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, Algorithm
778: L-BFGS-B, FORTRAN subroutines for large scalebound constrained optimization, ACM Transactions onMathematical Software, 23, pp. 550-560 (1997)[3] E. Kalnay, Atmospheric Modeling, Dats Assimilation andPredictability, Cambridge University Press, Cambridge,(2003)