Observability Gramian and Its Role in the Placement of Observations in Dynamic Data Assimilation

Lakshmivarahan, S.; Lewis, John M.; Maryada, Sai Kiran R.

doi:10.1007/978-3-030-77722-7_9

Cited by 2 publications

(6 citation statements)

References 23 publications

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“…We first examine the fine structure of the adjoint gradient and its dependence on the observability Gramian using the forward sensitivity analysis [24,25]. Let xfalse(tfalse) and xfalse¯false(tfalse) be the solution of equation (2.1) starting from c=false(x0,αfalse)T and cfalse¯=false(x¯0,αfalse¯false)T, respectively, where the perturbations in the initial conditions and the parameter are given by δx0=x¯0−x0,1emδα=αfalse¯−α, δc=false(δx0,δαfalse)T.The induced first variation in the solution of equation (2.1) resulting from the initial perturbation δc in the control c is defined as follows: δxfalse(kfalse)=xfalse¯false(kfalse)−xfalse(kfalse).The foreca...…”

Section: Observability Gramian and Adjoint Gradientmentioning

confidence: 99%

“…In Lakshmivarahan & Lewis [52], we first proved the equivalence between FSM and the four-dimensional VAR method. In Lakshmivarahan et al [24,25], by relating the observability Gramian to the forward sensitivities, we derived an intrinsic expression for the adjoint sensitivity in terms of the observability Gramian. By exploiting the structure of the observability Gramian, we then derived a strategy to place observations where the squares of the forward sensitivities attain maximum values.…”

Section: Appendix a Proof Of The Second Advantage Of The Fsm Observat...mentioning

confidence: 99%

“…In Lakshmivarahan et al [24,25], a strategy for answering this third level of question within the four-dimensional VAR framework was provided for the first time. An analysis based on the forward sensitivity method (FSM) has shown that the observability Gramian, G, is a good approximation to the Hessian of the cost function (it is exact for the linear case).…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the observability Gramian admits an additive decomposition: G=∑i=1NGi, where Gi is the contribution to the overall Gramian at the ith observation time and N is the number of observations. While each of these components is symmetric and positive semi-definite, it is shown in [24,25] that we can indeed force the Gramian G to be positive definite by placing the N=false(n+pfalse) observations where the squares of the forward sensitivities of the model solution (with respect to n components of the initial condition and p parameters including the boundary conditions) attain their maxima. This strategy avoids flat patches in the cost functional by bounding the norm of the adjoint gradient away from zero.…”

Section: Introductionmentioning

confidence: 99%

“…This quantity is particularly important when the given dynamic model exhibits high sensitivity to errors in the control. In this paper, we prove that the sensitivity of the estimates with respect to the observations attains its minimum value when we place the observations where the squares of the forward sensitivities are maximized, as in [24,25]. The positive definite Gramian resulting from the proposed strategy for observation placement determines the control estimates that exhibit the smallest possible sensitivity to observations.…”

Section: Introductionmentioning

confidence: 99%

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On the dual advantage of placing observations through forward sensitivity analysis

et al. 2023

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The four-dimensional variational data assimilation methodology for assimilating noisy observations into a deterministic model has been the workhorse in forecasting centres for over three decades. While this method provides a computationally efficient framework for dynamic data assimilation, it is largely silent on the important question concerning the minimum number and placement of observations. To answer this question, we demonstrate the dual advantage of placing the observations where the square of the sensitivity of the model solution with respect to the unknown control variables, called forward sensitivities, attains its maximum. By following this approach, we can force the observability Gramian to be of full rank, which in turn guarantees efficient recovery of the optimal values of the control variables, which is the first of the two advantages of this strategy. We further show that the proposed strategy of placing observations has another inherent optimality: the square of the sensitivity of the optimal estimates of the control with respect to the observations (used to obtain these estimates) attains its minimum value, a second advantage that is a direct consequence of the above strategy for placing observations. Our analytical framework and numerical experiments on linear and nonlinear systems confirm the effectiveness of our proposed strategy.

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