In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization.
Keywords: Infinte dimensional systems, Riccati equation, data assimilation
@article{COCV_2022__28_1_A78_0,
author = {Aussal, Matthieu and Moireau, Philippe},
title = {Kernel representation of {Kalman} observer and associated $\mathcal{H}$-matrix based discretization},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022071},
mrnumber = {4524414},
zbl = {07643481},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022071/}
}
TY - JOUR
AU - Aussal, Matthieu
AU - Moireau, Philippe
TI - Kernel representation of Kalman observer and associated $\mathcal{H}$-matrix based discretization
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2022
VL - 28
PB - EDP-Sciences
UR - https://www.numdam.org/articles/10.1051/cocv/2022071/
DO - 10.1051/cocv/2022071
LA - en
ID - COCV_2022__28_1_A78_0
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%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2022
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%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2022071/
%R 10.1051/cocv/2022071
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%F COCV_2022__28_1_A78_0
Aussal, Matthieu; Moireau, Philippe. Kernel representation of Kalman observer and associated $\mathcal{H}$-matrix based discretization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 78. doi: 10.1051/cocv/2022071
[1] , Convergence of discrete-time Kalman filter estimate to continuous time estimate. Int. J. Control 89 (2016) 668–679. | MR | Zbl | DOI
[2] and , FEM and BEM simulations with the gypsilab framework. SMAI J. Comput. Math. 4 (2018) 297–318. | MR | Zbl | DOI
[3] , Heat kernels, Technical report, ISEM course, 2005-2006.
[4] , Applied Functional Analysis, 2nd ed. Wiley (2000). | MR | Zbl
[5] , and , Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch. Comput. Methods Eng. 21 (2014) 331–358. | MR | Zbl | DOI
[6] and , Existence of -matrix approximants to the inverse FE-matrix of elliptic operators with -coefficients. Numer. Math. 95 (2003) 1–28. | MR | Zbl | DOI
[7] , Filtrage optimal des systèmes linéaires. Dunod (1971). | Zbl
[8] , Estimation and Control of Dynamical Systems. Interdisciplinary Applied Mathematics. Springer, Cham (2018). | MR | Zbl
[9] , , and , Representation and Control of Infinite Dimensional Systems, second edition. Birkhauser Verlag, Boston (2007). | MR | Zbl
[10] , volume 14 Efficient numerical methods for non-local operators: H2-matrix compression, algorithms and analysis. European Mathematical Society (2010). | MR
[11] , and , Introduction to hierarchical matrices with applications. Eng. Anal. Boundary Elem. 27 (2003) 405–422. | Zbl | DOI
[12] and , Data assimilation for the heat equation using stabilized finite element methods. Numer. Math. 139 (2018) 505–528. | MR | Zbl | DOI
[13] , and , A distributed parameter control approach to optimal filtering and smoothing with mobile sensor networks, In 17th Mediterranean Conference on Control and Automation (2009), pp. 181–186.
[14] and , Solutions and approximations to the Riccati integral equation with values in a space of compact operators. SIAM J. Control Optim. 53 (2015) 2846–2877. | MR | Zbl | DOI
[15] and , The infinite-dimensional optimal filtering problem with mobile and stationary sensor networks. Numer. Funct. Anal. Optim. 36 (2015) 181–224. | MR | Zbl | DOI
[16] , , and , Fundamental principles of data assimilation underlying the Verdandi library: applications to biophysical model personalization within euHeart. Med. Biol. Eng. Comput. (2012).
[17] , , and , A Galerkin strategy with proper orthogonal decomposition for parameter-dependent problems - analysis, assessments and applications to parameter estimation. ESAIM: Math. Model. Numer. Anal. 47 (2013) 1821–1843. | MR | Zbl | Numdam | DOI
[18] , Nonlinear Least Squares for Inverse Problems. Springer (2010). | MR | DOI
[19] , A survey of infinite-dimensional filtering. SIAM Rev. 17 (1975) 395–411. | MR | Zbl | DOI
[20] , and , The Hilbert-Schmidt property of feedback operators. J. Math. Anal. Appl. 329 (2007) 1145–1160. | MR | Zbl | DOI
[21] and , An introduction to infinite-dimensional linear systems theory. Vol. 21 of Texts in Applied Mathematics. Springer-Verlag, New York (1995). | MR | Zbl | DOI
[22] and , Peano’s kernel theorem for vector-valued functions and some applications. Numer. Funct. Anal. Optim. 17 (1996) 57–64. | MR | Zbl | DOI
[23] , and , Regularization of inverse problems. Vol. 375 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1996). | MR | Zbl
[24] , On the semigroup approach to stochastic evolution equations. Stoch. Anal. Appl. 10 (1992) 181–203. | MR | Zbl | DOI
[25] , and , Operator Theory and Numerical Methods, Studies in Mathematics and Its Applications. North Holland (2001). | MR | Zbl
[26] , and , Galerkin approximation for optimal linear filtering of infinite-dimensional linear systems. SIAM J. Control Optim. 26 (1988) 1287–1305. | MR | Zbl | DOI
[27] , and , Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Comput. Arch. Sci. Comput. 70 (2003) 121–165. | MR | Zbl
[28] and , Singular optimal control for a transport-diffusion equation. Commun. Partial Differ. Equ. 32 (2007) 1813–1836. | MR | Zbl | DOI
[29] , A sparse matrix arithmetic based on -matrices. I. Introduction to -matrices. Comput. Arch. Sci. Comput. 62 (1999) 89–108. | MR | Zbl
[30] , Hierarchical Matrices: Algorithms and Analysis, Springer Publishing Company, Incorporated, 1st edition (2015). | MR | Zbl
[31] , A new approach to linear filtering and prediction problems. J. Basic Eng. 82 (1960) 35–45. | MR | DOI
[32] and , New results in linear filtering and prediction theory. J. Basic Eng. 83 (1961) 95–108. | MR | Zbl | DOI
[33] , Fractional powers of dissipative operators. J. Math. Soc. Jpn. 13 (1961) 246–274. | MR | Zbl | DOI
[34] and , Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lecture Notes in Control and Information Sciences. Springer, Berlin, Heidelberg (1991). | MR | Zbl | DOI
[35] and , Low-rank numerical approximations for high-dimensional Lindblad equations. Phys. Rev. A 87 (2013) 022125. | DOI
[36] , Optimal control for data assimilation in meteorology, In Control theory of distributed parameter systems and applications (Shanghai, 1990). Vol. 159 of Lecture Notes in Control and Inform. Sci. Springer, Berlin (1991), pp. 51–60. | MR | Zbl
[37] and , Variational algorithms for analysis and assimilation of meteorological observation: theoretical aspects. Tellus 38 (1986) 97–110. | DOI
[38] , , and , A Kalman filter powered by H2-matrices for quasi-continuous data assimilation problems. Water Resour. Res. 50 (2014) 3734–3749. | DOI
[39] , Controle optimal de systèmes gouvernés par des équations aux dérivées partielles, Avant propos de P. Lelong. Dunod, Paris (1968). | MR | Zbl
[40] , Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. | MR | Zbl | DOI
[41] , Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs. J. Math. Soc. Jpn. 14 (1962) 233–241. | MR | Zbl | DOI
[42] , , and , A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics. Int. J. Numer. Methods Eng. 102 (2014) 933–965. | MR | Zbl | DOI
[43] and , Fast three-dimensional temperature reconstruction. Comput. Methods Appl. Mech. Eng. 199 (2010) 3169–3178. | MR | Zbl | DOI
[44] , and , Efficient state/parameter estimation in nonlinear unsteady PDEs by a reduced basis ensemble Kalman filter. SIAM/ASA J. Uncert. Quantif. 5 (2017) 890–921. | MR | Zbl | DOI
[45] , Semigroups of linear operators and applications to partial differential equations. Vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York (1983). | MR | Zbl
[46] , and , A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst. 16 (1998) 323–340. | DOI
[47] and , Introduction de la notion d’écart entre sous-espaces vectoriels en analyse de données. RAIRO: Oper. Res. 14 (1980). | MR | Zbl | Numdam | DOI
[48] , Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches. Wiley-Interscience (2006). | DOI
[49] , Equations of evolution. Vol. 6 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass.-London (1979). | MR | Zbl
[50] , Sur l’équation de Riccati associée à des opérateurs non bornés, en dimension infinie. J. Funct. Anal. 7 (1971) 85–115. | MR | Zbl | DOI
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