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Published in: Journal of Scientific Computing 2/2014

01-11-2014

A Scalable Approach for Variational Data Assimilation

Authors: Luisa D’Amore, Rossella Arcucci, Luisa Carracciuolo, Almerico Murli

Published in: Journal of Scientific Computing | Issue 2/2014

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Abstract

Data assimilation (DA) is a methodology for combining mathematical models simulating complex systems (the background knowledge) and measurements (the reality or observational data) in order to improve the estimate of the system state (the forecast). The DA is an inverse and ill posed problem usually used to handle a huge amount of data, so, it is a large and computationally expensive problem. Here we focus on scalable methods that makes DA applications feasible for a huge number of background data and observations. We present a scalable algorithm for solving variational DA which is highly parallel. We provide a mathematical formalization of this approach and we also study the performance of the resulted algorithm.

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Footnotes
1
It was shown [28] that incremental 3D-Var is equivalent to a GaussNewton method [10] (i.e. an approximation to a Newton iteration, in which the second-order terms of the Hessian are neglected) applied to the full nonlinear regularization functional.
 
2
The idea of using the L-BFGS method in variational data assimilation is not new because of its modest storage requirements and its high performance on large-scale unconstrained convex minimization [39, 46]. L-BFGS method is a Quasi–Newton method that can be viewed as extension of conjugate-gradient methods in which the addition of some modest storage serves to accelerate the convergence rate. The L-BFGS update formula generates the matrices approximating the Hessian using information from the last \(m\) Quasi-Newton iterations, where \(m\) is determined by the user (generally \(3 \le m \le 30\)). After having used the \(m\) vector storage locations for \(m\) quasi-Newton updates, the approximation of the Hessian matrix is updated by dropping the oldest information and replacing it by the newest information. Hence, time complexity of the L-BFGS increases linearly with the size of the \(m\) vectors [30].
 
Literature
1.
go back to reference Antonelli, L., Carracciuolo, L., Ceccarelli, M., D’Amore, L., Murli, A.: Total Variation Regularization for Edge Preserving 3D SPECT Imaging in High Performance Computing Environments. Lecture Notes in Computer Science (LNCS), vol. 2330, pp. 171–180, Springer-Verlag, Berlin, Heidelberg (2002) Antonelli, L., Carracciuolo, L., Ceccarelli, M., D’Amore, L., Murli, A.: Total Variation Regularization for Edge Preserving 3D SPECT Imaging in High Performance Computing Environments. Lecture Notes in Computer Science (LNCS), vol. 2330, pp. 171–180, Springer-Verlag, Berlin, Heidelberg (2002)
2.
go back to reference Auroux, D., Blum, J.: Back and forth nudging algorithm for data assimilation problems. C. R. Acad. Sci. Ser. I 340(340), 873–878 (2005)MathSciNetCrossRefMATH Auroux, D., Blum, J.: Back and forth nudging algorithm for data assimilation problems. C. R. Acad. Sci. Ser. I 340(340), 873–878 (2005)MathSciNetCrossRefMATH
3.
go back to reference Blum, J., Le Dimet, F.X., Navon, I.M.: Data Assimilation for Geophysical Fluids, Volume XIV of Handbook of Numerical Analysis, chapter 9. Elsevier, Amsterdam (2005) Blum, J., Le Dimet, F.X., Navon, I.M.: Data Assimilation for Geophysical Fluids, Volume XIV of Handbook of Numerical Analysis, chapter 9. Elsevier, Amsterdam (2005)
4.
go back to reference Carracciuolo, L., D’Amore, L., Murli, A.: Towards a parallel component for imaging in PETSc programming environment: a case study in 3-D echocardiography. Parallel Comput. 32, 67–83 (2006)MathSciNetCrossRef Carracciuolo, L., D’Amore, L., Murli, A.: Towards a parallel component for imaging in PETSc programming environment: a case study in 3-D echocardiography. Parallel Comput. 32, 67–83 (2006)MathSciNetCrossRef
5.
go back to reference Delahaies, S., Roulstone, L., Nichols, N.K.: Regularization of a Carbon-Cycle Model-Data Fusion Problem. Preprint University of, Reading, MPS-2013-10 (2013) Delahaies, S., Roulstone, L., Nichols, N.K.: Regularization of a Carbon-Cycle Model-Data Fusion Problem. Preprint University of, Reading, MPS-2013-10 (2013)
6.
go back to reference D’Amore, L., Arcucci, R., Marcellino, L., Murli, A.: A parallel three-dimensional variational data assimilation scheme. In: Numerical Analysis and Applied Mathematics, AIP C.P. vol. 1389, pp. 1829–1831 (2011) D’Amore, L., Arcucci, R., Marcellino, L., Murli, A.: A parallel three-dimensional variational data assimilation scheme. In: Numerical Analysis and Applied Mathematics, AIP C.P. vol. 1389, pp. 1829–1831 (2011)
7.
go back to reference D’Amore, L., Arcucci, R., Marcellino, L., Murli, A.: HPC computation issues of the incremental 3D variational data assimilation scheme in OceanVar software. J. Numer. Anal. Ind. Appl. Math. 7(3–4), 91–105 (2012)MathSciNet D’Amore, L., Arcucci, R., Marcellino, L., Murli, A.: HPC computation issues of the incremental 3D variational data assimilation scheme in OceanVar software. J. Numer. Anal. Ind. Appl. Math. 7(3–4), 91–105 (2012)MathSciNet
8.
go back to reference D’Amore, L., Arcucci, R., Carracciuolo, L., Murli, A.: OceanVAR software for use with NEMO: documentation and test guide. In: CMCC Research Papers Issue RP0173 (2012) D’Amore, L., Arcucci, R., Carracciuolo, L., Murli, A.: OceanVAR software for use with NEMO: documentation and test guide. In: CMCC Research Papers Issue RP0173 (2012)
9.
go back to reference D’Elia, M., Perego, M., Veneziani, A.: A variational data assimilation procedure for the incompressible Navier–Stokes equations in hemodynamics. J. Sci. Comput. 1–20 (2011) D’Elia, M., Perego, M., Veneziani, A.: A variational data assimilation procedure for the incompressible Navier–Stokes equations in hemodynamics. J. Sci. Comput. 1–20 (2011)
10.
go back to reference Dennis, J.E. Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall Series in Computational Mathematics. Prentice Hall Inc., Englewood Cliffs, NJ (1983) Dennis, J.E. Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall Series in Computational Mathematics. Prentice Hall Inc., Englewood Cliffs, NJ (1983)
11.
go back to reference Dobricic, S., Pinardi, N.: An oceanographic threedimensional variational data assimilation scheme. Ocean Modell. 22, 89–105 (2008)CrossRef Dobricic, S., Pinardi, N.: An oceanographic threedimensional variational data assimilation scheme. Ocean Modell. 22, 89–105 (2008)CrossRef
12.
go back to reference Freitag, M., Budd, C.J., Nichols, N.K.: Tikhonov regularization for large inverse problems. In: 17th ILAS Conference. Braunschweig, Germany (2011) Freitag, M., Budd, C.J., Nichols, N.K.: Tikhonov regularization for large inverse problems. In: 17th ILAS Conference. Braunschweig, Germany (2011)
13.
go back to reference Fox, G.C., Williams, R.D., Messina, P.C.: Parallel. Computing Works!. Morgan Kaufmann Publishers Inc., Los Altos, CA (1994) Fox, G.C., Williams, R.D., Messina, P.C.: Parallel. Computing Works!. Morgan Kaufmann Publishers Inc., Los Altos, CA (1994)
14.
go back to reference Foster, I.: Designing and Building Parallel Programs. Addison-Wesley, Reading, MA (1995)MATH Foster, I.: Designing and Building Parallel Programs. Addison-Wesley, Reading, MA (1995)MATH
15.
go back to reference Ghil, M., Malanotte-Rizzoli, P.: Data Assimilation in Meteorology and Oceanography, Advances in Geophysics, vol. 33. Academic Press, New York (1991) Ghil, M., Malanotte-Rizzoli, P.: Data Assimilation in Meteorology and Oceanography, Advances in Geophysics, vol. 33. Academic Press, New York (1991)
16.
go back to reference Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979) Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979)
17.
go back to reference Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of the 3DVAR Data Assimilation Problem, Mathematics Report 3/2009. Department of Mathematics, University of Reading (2009) Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of the 3DVAR Data Assimilation Problem, Mathematics Report 3/2009. Department of Mathematics, University of Reading (2009)
18.
go back to reference Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of Incremental Variational Data Assimilation, with Application to the Met Office System. Preprint University of Reading, MPS-2010-22 (2010) Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of Incremental Variational Data Assimilation, with Application to the Met Office System. Preprint University of Reading, MPS-2010-22 (2010)
19.
go back to reference Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning and preconditioning of the variational data assimilation. Comput. Fluids 46, 252–256 (2011)CrossRefMATH Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning and preconditioning of the variational data assimilation. Comput. Fluids 46, 252–256 (2011)CrossRefMATH
20.
go back to reference Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of incremental variational data assimilation, with application to the Met Office system. Tellus A 63, 782–792 (2011) Haben, S.A., Lawless, A.S., Nichols, N.K.: Conditioning of incremental variational data assimilation, with application to the Met Office system. Tellus A 63, 782–792 (2011)
21.
go back to reference Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82(Series D), 35–45 (1960) Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82(Series D), 35–45 (1960)
22.
go back to reference Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge, MA (2003) Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge, MA (2003)
23.
go back to reference Keyes, D.E.: How scalable is domain decomposition in practice? In: Proceedings of the International Conference Domain Decomposition Methods, Greenwich, pp. 286–297 (1998) Keyes, D.E.: How scalable is domain decomposition in practice? In: Proceedings of the International Conference Domain Decomposition Methods, Greenwich, pp. 286–297 (1998)
24.
go back to reference Koivunen, A.C., Kostinski, A.B.: The feasibility of data whitening to improve performance of weather radar. J. Appl. Meteorol. 38, 741–749 (1999)CrossRef Koivunen, A.C., Kostinski, A.B.: The feasibility of data whitening to improve performance of weather radar. J. Appl. Meteorol. 38, 741–749 (1999)CrossRef
25.
go back to reference Kostinski, A.B., Koivunen, A.C.: On the condition number of Gaussian sample-covariance matrices. IEEE Trans. Geosci. Remote Sens. 38, 329–332 (2000)CrossRef Kostinski, A.B., Koivunen, A.C.: On the condition number of Gaussian sample-covariance matrices. IEEE Trans. Geosci. Remote Sens. 38, 329–332 (2000)CrossRef
26.
go back to reference Johnson, C., Hoskins, B.J., Nichols, N.K.: A singular vector perspective of 4-DVar: filtering and interpolation. Q. J. R. Meteorol. Soc. 131, 120 (2005a)CrossRef Johnson, C., Hoskins, B.J., Nichols, N.K.: A singular vector perspective of 4-DVar: filtering and interpolation. Q. J. R. Meteorol. Soc. 131, 120 (2005a)CrossRef
27.
go back to reference Johnson, C., Nichols, N.K., Hoskins, B.J.: Very large inverse problems in atmosphere and ocean modelling. Int. J. Numeric. Methods Fluids 47, 759771 (2005b)MathSciNet Johnson, C., Nichols, N.K., Hoskins, B.J.: Very large inverse problems in atmosphere and ocean modelling. Int. J. Numeric. Methods Fluids 47, 759771 (2005b)MathSciNet
28.
go back to reference Lawless, A.S., Gratton, S., Nichols, N.K.: Approximate iterative methods for variational data assimilation. Int. J. Numer. Meth. Fluids 47, 1129–1135 (2005)MathSciNetCrossRefMATH Lawless, A.S., Gratton, S., Nichols, N.K.: Approximate iterative methods for variational data assimilation. Int. J. Numer. Meth. Fluids 47, 1129–1135 (2005)MathSciNetCrossRefMATH
29.
go back to reference Le Dimet, F.X., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–110 (1986)CrossRef Le Dimet, F.X., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–110 (1986)CrossRef
30.
31.
go back to reference Marx, B. A., Potthast, R.W.E.: On Instabilities in Data Assimilation Algorithms. University of Reading, Department of Mathematics and Statistics, Preprint MPS-2012-06 (2012) Marx, B. A., Potthast, R.W.E.: On Instabilities in Data Assimilation Algorithms. University of Reading, Department of Mathematics and Statistics, Preprint MPS-2012-06 (2012)
32.
go back to reference Miyoshi, T.: Computational Challenges in Big Data Assimilation with Extreme-Scale Simulations, Talk at BDEC Workshop. Charleston, SC (2013) Miyoshi, T.: Computational Challenges in Big Data Assimilation with Extreme-Scale Simulations, Talk at BDEC Workshop. Charleston, SC (2013)
33.
go back to reference Mogensen, K., Alonso Balmaseda, M., Weaver, A.: The NEMOVAR ocean data assimilation system as implemented in the ECMWF ocean analysis for System 4. Research Department, CERFACS, Toulouse (2012) Mogensen, K., Alonso Balmaseda, M., Weaver, A.: The NEMOVAR ocean data assimilation system as implemented in the ECMWF ocean analysis for System 4. Research Department, CERFACS, Toulouse (2012)
34.
35.
go back to reference Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Wien. Springer, New York (1984)CrossRef Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Wien. Springer, New York (1984)CrossRef
36.
go back to reference Murli, A., D’Amore, L., Carracciuolo, L., Ceccarelli, M., Antonelli, L.: High performance edge-preserving regularization in 3D SPECT imaging. Parallel Comput. 34(2), 115–132 (2008) Murli, A., D’Amore, L., Carracciuolo, L., Ceccarelli, M., Antonelli, L.: High performance edge-preserving regularization in 3D SPECT imaging. Parallel Comput. 34(2), 115–132 (2008)
38.
go back to reference Navon, I.M.: Data assimilation for numerical weather prediction: a review. In: Park, S.K., Xu, L. (eds.) Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications. Springer, Berlin (2009) Navon, I.M.: Data assimilation for numerical weather prediction: a review. In: Park, S.K., Xu, L. (eds.) Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications. Springer, Berlin (2009)
39.
go back to reference Navon, I.M., Legler, D.M.: Conjugate gradient methods for large-scale minimization in metereology. Mon. Weather Rev. 115, 1479–1502 (1987) Navon, I.M., Legler, D.M.: Conjugate gradient methods for large-scale minimization in metereology. Mon. Weather Rev. 115, 1479–1502 (1987)
40.
go back to reference Nerger, L., Hiller, W. and Schrter, J. The Parallel Data Assimilation Framework PDAF—a flexible software framework for ensemble data assimilation, EGU General Assembly, April 23–27, 2012, Vienna, Austria (Geophysical Research Abstracts, vol. 14, EGU2012-1885) Nerger, L., Hiller, W. and Schrter, J. The Parallel Data Assimilation Framework PDAF—a flexible software framework for ensemble data assimilation, EGU General Assembly, April 23–27, 2012, Vienna, Austria (Geophysical Research Abstracts, vol. 14, EGU2012-1885)
41.
go back to reference Nocedal, J., Byrd, R.H., Lu, P., Zhu, C.: L-BFGS-B: fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560 (1997)MathSciNetCrossRefMATH Nocedal, J., Byrd, R.H., Lu, P., Zhu, C.: L-BFGS-B: fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560 (1997)MathSciNetCrossRefMATH
43.
go back to reference Stewart, L.M., Dance, S., Nichols, N.K.: Data assimilation with correlated observation errors: experiments with a 1-D shallow water model. Tellus A. 65, 19546 (2013) Stewart, L.M., Dance, S., Nichols, N.K.: Data assimilation with correlated observation errors: experiments with a 1-D shallow water model. Tellus A. 65, 19546 (2013)
44.
go back to reference Tikhonov, A.N.: Regularization of incorrectly posed problems. Sov. Math. Dokl. 4, 1624–1627 (1963)MATH Tikhonov, A.N.: Regularization of incorrectly posed problems. Sov. Math. Dokl. 4, 1624–1627 (1963)MATH
45.
go back to reference Vanoye, A., Mendoza, A.: Application of direct regularization techniques and bounded-variable least squares for inverse modeling of an urban emissions inventory. Athmosferic Pollut. Res. 5 (2014) Vanoye, A., Mendoza, A.: Application of direct regularization techniques and bounded-variable least squares for inverse modeling of an urban emissions inventory. Athmosferic Pollut. Res. 5 (2014)
46.
go back to reference Zou, X., Navon, I.M., Berger, M., Phua, K.H., Schlick, T. LeDimet, F.X.: Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM J. Optim. 3, 582–608 (1993) Zou, X., Navon, I.M., Berger, M., Phua, K.H., Schlick, T. LeDimet, F.X.: Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM J. Optim. 3, 582–608 (1993)
Metadata
Title
A Scalable Approach for Variational Data Assimilation
Authors
Luisa D’Amore
Rossella Arcucci
Luisa Carracciuolo
Almerico Murli
Publication date
01-11-2014
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 2/2014
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9824-2

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