Data assimilation of time under-sampled measurements using observers, the wave-like equation example

Ĉındea, Nicolae; Impériale, Alexandre; Moireau, Philippe

doi:10.1051/cocv/2014042

Cited by 11 publications

(25 citation statements)

References 37 publications

(80 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We now end this section by finally justifying (18). Indeed, by differentiating (16) 1 around x =x − n+1 and y =x…”

Section: Proposition 10mentioning

confidence: 87%

“…Mirroring the continuous setting, H n is a C 1 (R N obs )-mapping and dH n is bounded uniformly with respect to n. Note that, in general, measurement procedures are in essence time-sampled and the observation discrete-time model should be more directly defined than the observation continuous-time model. Then the latter can be regenerated by interpolation from the sampled measurements with a measurement error ω incorporating some interpolation error [18]. Eventually, we can consider the two frameworks independently or assuming that for all t ∈ [t n , t n+1 ], we have H(x, t) − H n (x) = O(∆t)).…”

Section: Models Settingsmentioning

confidence: 99%

See 1 more Smart Citation

A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer

Moireau

2018

ESAIM: COCV

View full text Add to dashboard Cite

Abstract. In this work, we seek exact formulations of the optimal estimator and filter for a nonlinear framework, as the Kalman filter is for a linear framework. The solution is well established with the Mortensen filter in a continuous-time setting, but we seek here its counterpart in a discrete-time context. We demonstrate that it is possible to pursue at the discrete-time level an exact dynamic programming strategy and we find an optimal estimator combining a prediction step using the model and a correction step using the data. This optimal estimator reduces to the discrete-time Kalman estimator when the operators are in fact linear. Furthermore, the strategy that consists of discretizing the least square criterion and then finding the exact estimator at the discrete level allows to determine a new time-scheme for the Mortensen filter which is proven to be consistent and unconditionally stable, with also a consistent and stable discretization of the underlying HamiltonJacobi-Bellman equation.1991 Mathematics Subject Classification. 34K35, 93E24, 93B52, 49J21 , 49L20.The dates will be set by the publisher.

show abstract

“…We now end this section by finally justifying (18). Indeed, by differentiating (16) 1 around x =x − n+1 and y =x…”

Section: Proposition 10mentioning

confidence: 87%

Section: Models Settingsmentioning

confidence: 99%

A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer

Moireau

2018

ESAIM: COCV

View full text Add to dashboard Cite

show abstract

“…This is the case of the discrete-time Kalman Filter considered as a discretization of the continuous-time Kalman Filter [61]. This was also proposed for Luenberger observers in [23]. Moreover, we emphasize that in our case a major advantage lies in the fact that the splitting step is explicit as is already the case for the non-linear reaction term.…”

Section: Numerical Discretizationmentioning

confidence: 87%

“…and we can see that the observer correction term in (12) is nothing but −λ∇Jû with our particular choice of α function given in (23), namely, this is a gradient descent term quite similar to gradient projection methods in image processing [90].…”

Section: Proposition 3 Assuming That We Havementioning

confidence: 94%

“…For instance in cardiac electrophysiology, ECGI devices typically produce isochrones with a time-sampling of T ∈ [1, 10] ms, whereas the simulation time step t can typically reach 0.1 ms [26,70]. In order to handle the data time-sampling, two strategies are conceivable, namely, we can either use the data only when they are available, or we can rely on some time-interpolation [23]. In the latter case, as the fronts data do not easily lend themselves to interpolation we propose to follow [63] and interpolate (linearly) the corrections, leading to MÛ n, 3 …”

Section: Numerical Discretizationmentioning

confidence: 99%

See 1 more Smart Citation

A Luenberger observer for reaction–diffusion models with front position data

Collin

Chapelle

Moireau

2015

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We propose a Luenberger observer for reaction-diffusion models with propagating front features, and for data associated with the location of the front over time. Such models are considered in various application fields, such as electrophysiology, wild-land fire propagation and tumor growth modeling. Drawing our inspiration from image processing methods, we start by proposing an observer for the eikonal-curvature equation that can be derived from the reaction-diffusion model by an asymptotic expansion. We then carry over this observer to the underlying reaction-diffusion equation by an "inverse asymptotic analysis", and we show that the associated correction in the dynamics has a stabilizing effect for the linearized estimation error. We also discuss the extension to joint stateparameter estimation by using the earlier-proposed ROUKF strategy. We then illustrate and assess our proposed observer method with test problems pertaining to electrophysiology modeling, including with a realistic model of cardiac atria. Our numerical trials show that state estimation is directly very effective with the proposed Luenberger observer, while specific strategies are needed to accurately perform parameter estimation -as is usual with Kalman filtering used in a nonlinear setting -and we demonstrate two such successful strategies.

show abstract

Data assimilation of time under-sampled measurements using observers, the wave-like equation example

Cited by 11 publications

References 37 publications

A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer

A discrete-time optimal filtering approach for non-linear systems as a stable discretization of the Mortensen observer

A Luenberger observer for reaction–diffusion models with front position data

Discrete-time formulations as time discretization strategies in data assimilation

Contact Info

Product

Resources

About