Chen, X.; Navon, I. M.; Fang, F. A dual-weighted trust-region adaptive POD 4D-VAR applied to a finite-element shallow-water equations model. (English) Zbl 1428.76145 Int. J. Numer. Methods Fluids 65, No. 5, 520-541 (2011). Summary: We consider a limited-area finite-element discretization of the shallow-water equations model. Our purpose in this paper is to solve an inverse problem for the above model controlling its initial conditions in presence of observations being assimilated in a time interval (window of assimilation). We then attempt to obtain a reduced-order model of the above inverse problem, based on proper orthogonal decomposition (POD), referred to as POD 4-D VAR. Different approaches of POD implementation of the reduced inverse problem are compared, including a dual-weighed method for snapshot selection coupled with a trust-region POD approach. Numerical results obtained point to an improved accuracy in all metrics tested when dual-weighing choice of snapshots is combined with POD adaptivity of the trust-region type. Results of ad-hoc adaptivity of the POD 4-D VAR turn out to yield less accurate results than trust-region POD when compared with high-fidelity model. Directions of future research are finally outlined. Cited in 9 Documents MSC: 76M21 Inverse problems in fluid mechanics 76M10 Finite element methods applied to problems in fluid mechanics 35Q35 PDEs in connection with fluid mechanics Keywords:proper orthogonal decomposition; 4-D VAR; shallow water equations; dual weighting; trust-region method; inverse problem Software:4D-VAR; GQTPAR; FEUDX PDFBibTeX XMLCite \textit{X. Chen} et al., Int. J. Numer. Methods Fluids 65, No. 5, 520--541 (2011; Zbl 1428.76145) Full Text: DOI References: [1] Fahl M Trust-region methods for flow control based on reduced order modeling 2000 [2] Arian E Fahl M Sachs EW Trust-region proper orthogonal decomposition for flow control 2000 [3] Conn, Trust-region Methods (2000) · Zbl 0958.65071 [4] Toint, Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space, IMA Journal of Numerical Analysis 8 (2) pp 231– (1988) · Zbl 0698.65043 [5] Nocedal, Springer Series in Operations Research and Financial Engineering, in: Numerical Optimization (2006) [6] Bui-Thanh, Goal-oriented model constrained for reduction of large-scale systems, Journal of Computational Physics 224 (2) pp 880– (2007) · Zbl 1123.65081 [7] Daescu, A dual-weighted approach to order reduction in 4DVAR data assimilation, Monthly Weather Review 136 (3) pp 1026– (2008) [8] Cao, Reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition, Computers and Mathematics with Applications 52 (8-9) pp 1373– (2006) · Zbl 1161.86002 [9] Cao, A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition, International Journal for Numerical Methods in Fluids 53 (10) pp 1571– (2007) · Zbl 1370.86002 [10] Fang, An adaptive mesh adjoint data assimilation method applied to free surface flows, International Journal for Numerical Methods in Fluids 47 (8-9) pp 995– (2005) · Zbl 1134.86004 [11] Fang, Adjoint data assimilation into a 3d unstructured mesh coastal finite element model, Ocean Modelling 15 (1-2) pp 3– (2006) [12] Fang, Reduced order modelling of an adaptive mesh ocean model, International Journal for Numerical Methods in Fluids 59 (8) pp 827– (2008) · Zbl 1155.86004 [13] Fang, A POD reduced order 4d-var adaptive mesh ocean modelling approach, International Journal for Numerical Methods in Fluids 60 (7) pp 709– (2009) · Zbl 1163.86002 [14] Altaf, Inverse shallow water flow modeling using model reduction, International Journal for Multiscale Computational Engineering 7 (6) pp 577– (2009) [15] Tan, Shallow-water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow-water Equations (1992) [16] Vreugdenhil, Numerical Methods for Shallow-water Flow (1994) · Zbl 0829.47021 [17] Galewsky, An initial-value problem for testing numerical models of the global shallow-water equations, Tellus 56 (5) pp 429– (2004) [18] Chen, Optimal control of a finite-element limited-area shallow-water equations model, Studies in Informatics and Control 18 (1) pp 41– (2009) [19] Kreiss, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24 pp 199– (1972) [20] Lumley, Atmospheric Turbulence and Radio Wave Propagation pp 166– (1967) [21] Berkooz, The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics 25 (1) pp 539– (1993) [22] Sirovich, Turbulence and the dynamics of coherent structures, part III: dynamics and scaling, Quarterly of Applied Mathematics 45 (3) pp 583– (1987) · Zbl 0676.76047 [23] Holmes, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (1996) [24] Karhunen, Zur Spektraltheorie stochastischer Prozesse, Annales Academiae Scientiarum Fennicae Series A1-Mathematica Physica pp 34– (1946) [25] Loeve, Fonctions Aleatoires de Second Ordre, Comptes Rendus de l’Academie des Sciences, Paris pp 220– (1945) [26] Kosambi, Statistics in function space, Journal of the Indian Mathematical Society 7 (1) pp 76– (1943) · Zbl 0063.03317 [27] Pearson, On lines and planes of closest fit to systems of points in space, Philosophical Magazine 2 (1) pp 559– (1901) · JFM 32.0246.07 [28] Hotelling, Analysis of a complex of statistical variables into principal components, Journal of Educational Psychology 24 (1) pp 417– (1933) · JFM 59.1182.04 [29] Barone, Stable Galerkin reduced order models for linearized compressible flow, Journal of Computational Physics 228 (6) pp 1932– (2009) · Zbl 1162.76025 [30] Rowley, Model reduction for compressible flows using POD and Galerkin projection, Physica D 189 (1-2) pp 115– (2004) · Zbl 1098.76602 [31] Aquino, Generalized finite element method using proper orthogonal decomposition, International Journal for Numerical Methods in Engineering 79 (7) pp 887– (2009) · Zbl 1171.76416 [32] Navon, Variational data assimilation with an adiabatic version of the NMC spectral mode, Monthly Weather Review 120 (7) pp 1435– (1992) [33] Kunish, Proper orthogonal decomposition for optimality systems, Mathematical Modelling and Numerical Analysis 42 (1) pp 1– (2008) · Zbl 1141.65050 [34] Vermeulen, Model-reduced variational data assimilation, Monthly Weather Review 134 (10) pp 2888– (2006) [35] Kunisch, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik 90 (1) pp 117– (2001) · Zbl 1005.65112 [36] Kunisch, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM Journal on Numerical Analysis 40 (2) pp 492– (2002) · Zbl 1075.65118 [37] Levenberg, A method for the solution of certain problems in least squares, Quarterly Journal on Applied Mathematics 2 pp 164– (1944) · Zbl 0063.03501 [38] Morrison, Proceedings of the Seminar on Tracking Programs and Orbit Determination (1960) [39] Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM Journal on Applied Mathematics 11 pp 431– (1963) · Zbl 0112.10505 [40] Winfield D Function and Functional Optimization by Interpolation in Data Tables 1969 146 [41] Powell, Numerical Methods for Nonlinear Algebraic Equations pp 115– (1970) [42] Powell MJD A FORTRAN subroutine for unconstrained minimization, requiring first derivatives of the objective function 1970 [43] Dennis, Proceedings of Symposia in Applied Mathematics Series, in: Numerical Analysis pp 19– (1978) [44] More, Computing a trust region step, SIAM Journal on Scientific and Statistical Computing 4 (3) pp 553– (1983) · Zbl 0551.65042 [45] Celis, Numerical Optimization pp 71– (1985) [46] Bergmann, Optimal control of the cylinder wake in the laminar regime by trust-region methods and pod reduced-order models, Journal of Computational Physics 227 (16) pp 7813– (2008) · Zbl 1388.76073 [47] Bergmann, Enablers for robust pod models, Journal of Computational Physics 228 (2) pp 516– (2009) · Zbl 1409.76099 [48] Grammeltvedt, A survey of finite-difference schemes for the primitive equations for a barotropic fluid, Monthly Weather Review 97 (5) pp 384– (1969) [49] Zienkiewicz, The Finite Element Method for a Barotropic Fluid (2005) [50] Zhu, Variational data assimilation with a variable resolution finite-element shallow-water equations model, Monthly Weather Review 122 (5) pp 946– (1994) [51] Navon, Finite-element simulation of the shallow-water equations model on a limited area domain, Applied Mathematical Modelling 3 (1) pp 337– (1979) · Zbl 0438.76017 [52] Navon, FEUDX: a two-stage, high-accuracy, finite-element FORTRAN program for solving shallow-water equations, Computers and Geosciences 13 (3) pp 255– (1987) [53] Wang, Numerical solutions of the one-dimensional primitive equations using Galerkin approximation with localized basic functions, Monthly Weather Review 100 (10) pp 738– (1972) [54] Payne NA Irons BM 1963 [55] Huebner, The Finite-element Method for Engineers (2001) [56] Polak, Note sur la convergence de directions conjugues, Revue Francaise Informat. Recherche Operationelle 3 (16) pp 35– (1969) [57] Daescu, Efficiency of a POD-based reduced second-order adjoint model in 4d-var data assimilation, International Journal for Numerical Methods in Fluids 53 (10) pp 985– (2007) · Zbl 1370.76122 [58] Rowley, Model reduction for fluids using balanced proper orthogonal decomposition, International Journal of Bifurcation and Chaos 15 (3) pp 997– (2005) · Zbl 1140.76443 [59] Noack, System reduction strategy for Galerkin models of fluid flows, International Journal for Numerical Methods in Engineering (2009) · Zbl 1423.76349 [60] Noack, The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows, Journal of Fluid Mechanics 523 pp 339– (2005) · Zbl 1065.76102 [61] Tadmor, System reduction mean field Galerkin models for the natural and actuated cylinder wake flow, Physics of Fluids (2009) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.