A reduced basis Kalman filter for parametrized partial differential equations

Dihlmann, Markus; Haasdonk, Bernard

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 625-669.

The Kalman filter is a widely known tool in control theory for estimating the state of a linear system disturbed by noise. However, when applying the Kalman filter on systems described by parametrerized partial differential equations (PPDEs) the calculation of state estimates can take an excessive amount of time and real-time state estimation may be infeasible. In this work we derive a low dimensional representation of a parameter dependent Kalman filter for PPDEs via the reduced basis method. Thereby rapid state estimation, and in particular the rapid estimation of a linear output of interest, will be feasible. We will also derive a posteriori error bounds for evaluating the quality of the output estimations. Furthermore we will show how to verify the stability of the filter using an observability condition. We will demonstrate the performance of the reduced order Kalman filter and the error bounds with a numerical example modeling the heat transfer in a plate.

A. Aalto, Spatial discretization error in Kalman filtering for discrete-time infinite dimensional systems. Technical report (2014).

A.V. Balakrishnan, Stochastic Optimization Theory in Hilbert Spaces – 1. Appl. Math. Opt. 1 (1974) 97–120. | DOI | MR | Zbl

P. Binev, A. Cohen, P. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 42 (2011) 1457–1472. | DOI | MR | Zbl

S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen and A.T. Patera, A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin coefficient. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3187–3206. | DOI | MR | Zbl

S. Boyaval, C. Le Bris, T. Lelièvre, Y. Maday, N.C. Nguyen and A.T. Patera, Reduced basis techniques for stochastic problems. Arch. Comput. Method E 17 (2010) 435–454. | DOI | MR | Zbl

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2011). | MR | Zbl

A. Buffa, Y. Maday, A.T. Patera, C. Prud’Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: M2AN 46 (2012) 595–603. | DOI | Numdam | MR | Zbl

Y. Cao, J. Zhu, I.M. Navon and Z. Luo, A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Int. J. Numer. Meth. Fluids 53 (2006) 1571–1583. | DOI | Zbl

J. Chandrasekar, I.S. Kim, D.S. Bernstein and A.J. Ridley, Cholesky-Based Reduced-Rank Square-Root Kalman Filtering. In 2008 American Control Conference Seattle, USA (2008).

C.K. Chui and G. Chen, Kalman Filtering with Real-Time Applications. Springer (2009). | Zbl

S.E. Cohn, Dynamics of short-term univariate forecast error covariances. Mon. Weather Rev. 121 (1993) 3123–3149. | DOI

S.E. Cohn and D.P. Dee, Observability of discretized partial differential equations. SIAM J. Numer. Anal. 25 (1988) 586–617. | DOI | MR | Zbl

S.E. Cohn and R. Todling, Approximate Data Assimilation Schemes for Stable and Unstable Dynamics. J. Meteorol. Soc. Jpn 74 (1996) 63–75. | DOI

R. Curtain, A survey of infinite-dimensional filtering. SIAM Rev. 17 (1975) 395–411. | DOI | MR | Zbl

R.F. Curtain, Stochastic Evolution Equations with General White Noise Disturbance. J. Math. Anal. Appl. 60 (1977) 570–595. | DOI | MR | Zbl

D.N. Daescu and I.M. Navon, A Dual-Weighted Approach to Order Reduction in 4DVAR Data Assimilation. Mon. Weather Rev. 136 (2007) 1026–1041. | DOI

M. Dihlmann, Adaptive Reduced Basis Method for Parameter Optimization and State Estimation of Parameterized Evolution Problems. Ph.D. thesis, University of Stuttgart (2015).

M. Dihlmann, M. Drohmann and B. Haasdonk, Model reduction of parametrized evolution problems using the reduced basis method with adaptive time-partitioning. In Proc. of ADMOS 2011 (2011).

M. Drohmann and K. Carlberg, The ROMES method for statistical modeling of reduced-order-model error. Technical report, Sandia National Labs (2014). | MR

J.L. Eftang, D.J. Knezevic and A.T. Patera, An hp certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. 17 (2011) 395–422. | DOI | MR | Zbl

P.L. Falb, Infinite Dimensional Filtering: The Kalman Bucy-Filter in Hilbert Space. Inform. Control. 11 (1967) 103–138. | MR | Zbl

B.F. Farrell and P.J. Ioannou, State estimation using a reduced-order Kalman filter. J. Atmos. Sci. 58 (2001) 3666–3680. | DOI

R.J. Fitzgerald, Divergence of the Kalman Filter. IEEE T. Automat. Contr. 16 (1971) 736–747. | DOI

A. Germani, L. Jetto and M. Piccioni, Galerkin approximation of optimal linear filtering of infinite-dimensional linear systems. SIAM J. Control. Optim. 26 (1988) 1287–1305. | DOI | MR | Zbl

M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157–181. | DOI | Numdam | MR | Zbl

B. Haasdonk, Convergence rates of the POD-Greedy method. ESAIM: M2AN 47 (2013) 859–873. | DOI | Numdam | MR | Zbl

B. Haasdonk, M. Dihlmann and M. Ohlberger, A training set and multiple basis generation approach for parametrized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. 17 (2012) 423–442. | DOI | MR | Zbl

B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277–302. | DOI | Numdam | MR | Zbl

B. Haasdonk, K. Urban and B. Wieland, Reduced Basis Methods for Parameterized Partial Differential Equations with Stochastic Influences Using the Karhunen-Loéve Expansion. SIAM J. Uncertainty Quantification 1 (2013) 79–105. | DOI | MR | Zbl

E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations. Springer Ser. Comput. Math. Springer (2009).

A.H. Jazwinski, Stochastic Processes and Filtering Theory. Academic Press (1970). | Zbl

R.E. Kalman, A new approach to linear filtering and prediction problems. J. Basic Eng-T ASME 82 (1960) 35–45. | DOI | MR

I. Kim, J. Chandrasekar, H.J. Palanthandalam, A. Ridley and D.S. Bernstein. State estimation for large-scale systems based on reduced-order error-covariance propagation. In Amer. Contr. Conf. New York (2007).

L.B. Koralov and Y.G. Sinai, Theory of Probability and Random Processes. Springer (2007). | MR | Zbl

K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. | DOI | MR | Zbl

W. Lee, D. Mcdougall and A.M. Stuart, Kalman filtering and smoothing for linear wave equations with model error. Inverse Probl. 27 (2011) 095008. | DOI | MR | Zbl

Z. Li and I.M. Navon, Optimality of variational data assimilation and its relationship with the Kalman filter and smoother. Q. J. R. Meteorol. Soc. 127 (2001) 661–683. | DOI

M. Loève, Probability Theory I. Springer (1987). | MR | Zbl

Y. Maday and O. Mula, A generalized empirical interpolation method: application of reduced basis techniques to data assimilation. In Anal. Numer. Partial Differ. Equ. Springer (2013). | MR | Zbl

Y. Maday, A.T. Patera, J.D. Penn and M. Yano, A Parametrized-Background Data-Weak approach to variational data assimilation: Formulation, analysis, and application to acoustics. Int. J. Numer. Meth. Engng. 102 (2015) 933–965. | DOI | MR | Zbl

R.K. Mehra, On the Identification of Variances and Adaptive Kalman Filtering. IEEE T. Automat. Contr. 15 (1970) 175–184. | DOI | MR

P. Moireau and D. Chapelle, Reduced-order unscented Kalman filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2010) 380–405. | Numdam | MR | Zbl

P. Moireau, D. Chapelle and P. Le Tallec, Joint state and parameter estimation for distributed mechanical systems. Comput. Methods. Appl. Mech. Engng. 197 (2008) 659–677. | DOI | MR | Zbl

M. Morf, G.S. Sighu and T. Kailath, Some New Algortihms for Recursive Estimation in Constant, Linear, Discrete-Time Systems. IEEE Trans. Automat. Contr. 19 (1974) 315–323. | DOI | Zbl

K.M. Nagpal, R.E. Helmick and C.S. Sims, Reduced-order estimation Part 1. Filtering. Int. J. Control 45 (1987) 1867–1888. | DOI | MR | Zbl

C.F. Price, An Analysis of the Divergence Problem in the Kalman Filter. IEEE T. Automat. Contr. 13 (1968) 699–702. | DOI | MR

B Ristic, S. Arulampalam and N. Gordon, Beyond the Kalman Filter. Particle Filters for Tracking Applications. Artech House (2004). | Zbl

E.M. Rønquist and A.T. Patera, Regression on parametric manifolds: Estimation of spatial fields, functional outputs, and parameters from noisy data. C.R. Acad. Sci. Paris, Ser I 350 (2012) 543–547. | DOI | MR | Zbl

G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced Basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Meth. Eng. 15 (2008) 229–275. | DOI | MR | Zbl

D. Simon, Optimal State Estimation, Kalman, H${}_{\infty }$, and Nonlinear Approaches. John Wiley&Sons (2006).

H.W. Sorenson, On Applications of Kalman Filtering. IEEE T. Automat. Contr. 28 (1983) 254–255. | DOI

R. Todling and S.E. Cohn, Suboptimal Schemes for Atmospheric Data Assimilation Based on the Kalman filter. Monthly Weather Review 122 (1994) 2530–2557. | DOI

K. Urban and A.T. Patera, An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83 (2014) 1599–1615. | DOI | MR | Zbl

M. Verlaan and A.W. Heemink, Tidal flow forecasting using reduced rank square root filters. Stoch. Hydrol. Hydraul. 11 (1997) 349–368. | DOI | Zbl

K. Veroy, C. Prud’homme, D.V. Rovas and A.T. Patera,A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proc. of 16th AIAA computational fluid dynamics conference (2003) 2003–3847.

D. Xiu, Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010). | MR | Zbl

K. Yosida, Functional Analysis. Springer (1991). | MR