## A new selection operator for the discrete empirical interpolation method – improved a priori error bound and extensions.(English)Zbl 1382.65193

### MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 15A12 Conditioning of matrices 15A23 Factorization of matrices 93B11 System structure simplification 93B40 Computational methods in systems theory (MSC2010)

### Software:

LAPACK; ScaLAPACK; kappa_SQ; toms/782; RRQR
Full Text:

### References:

 [1] E. Anderson et al., LAPACK Users’ Guide, 3rd ed., SIAM, Philadelphia, 1999. · Zbl 0755.65028 [2] H. Antil, S. E. Field, F. Herrmann, R. H. Nochetto, and M. Tiglio, Two-step greedy algorithm for reduced order quadratures, J. Sci. Comput., 57 (2013), pp. 604–637. · Zbl 1292.65024 [3] P. Astrid, S. Weiland, K. Willcox, and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automat. Control, 53 (2008), pp. 2237–2251. · Zbl 1367.93110 [4] Z. Bai and D. Skoogh, A projection method for model reduction of bilinear dynamical systems, Linear Algebra Appl., 415 (2006), pp. 406–425. · Zbl 1107.93012 [5] M. Barrault, N. C. Nguyen, Y. Maday, and A. T. Patera, An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667–672. · Zbl 1061.65118 [6] P. Benner and T. Breiten, Interpolation-based $$\mathcal{H}_2$$-model reduction of bilinear control systems, SIAM J. Matrix Anal. Appl., 33 (2012), pp. 859–885. · Zbl 1256.93027 [7] P. Benner and T. Breiten, Two-sided projection methods for nonlinear model order reduction, SIAM J. Sci. Comput., 37 (2015), pp. B239–B260. · Zbl 1312.93016 [8] P. Benner, S. Gugercin, and K. Willcox, A Survey of Model Reduction Methods for Parametric Systems, Technical report MPIMD/13-14, Max Planck Institute Magdeburg, Magdeburg, Germany, 2013. · Zbl 1339.37089 [9] G. Berkooz, P. Holmes, and J. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25 (1993), pp. 539–575. [10] C. H. Bischof and G. Quintana-Orti, Algorithm 782: Codes for rank–revealing QR factorizations of dense matrices, ACM Trans. Math. Software, 24 (1998), pp. 254–257. · Zbl 0932.65034 [11] C. H. Bischof and G. Quintana-Orti, Computing rank–revealing QR factorizations of dense matrices, ACM Trans. Math. Software, 24 (1998), pp. 226–253. · Zbl 0932.65033 [12] L. S. Blackford et al., ScaLAPACK User’s Guide, SIAM, Philadelphia, 1997. [13] P. A. Businger and G. H. Golub, Linear least squares solutions by Householder transformations, Numer. Math., 7 (1965), pp. 269–276. · Zbl 0142.11503 [14] K. Carlberg, C. Farhat, J. Cortial, and D. Amsallem, The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys., 242 (2013), pp. 623–647. · Zbl 1299.76180 [15] S. Chandrasekaran and I. C. F. Ipsen, On rank–revealing factorisations, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 592–622. · Zbl 0796.65030 [16] Y. Chen, Model Order Reduction for Nonlinear Systems, Master’s thesis, Massachusetts Institute of Technology, Cambridge, MA, 1999. [17] M. Condon and R. Ivanov, Model reduction of nonlinear systems, COMPEL, 23 (2004), pp. 547–557. · Zbl 1064.94564 [18] E. de Sturler, S. Gugercin, M. Kilmer, S. Chaturantabut, C. Beattie, and M. O’Connell, Nonlinear parametric inversion using interpolatory model reduction, SIAM J. Sci. Comput., 37 (2015), pp. B495–B517. · Zbl 1433.65194 [19] Z. Drmač, A global convergence proof for cyclic Jacobi methods with block rotations, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1329–1350. · Zbl 1201.65052 [20] Z. Drmač and Z. Bujanović, On the failure of rank revealing QR factorization software – a case study, ACM Trans. Math. Software, 35 (2008), pp. 1–28. [21] R. Everson and L. Sirovich, The Karhunen-Loève Procedure for Gappy Data, J. Opt. Soc. Amer., 12 (1995), pp. 1657–1664. [22] D. Faddeev, V. Kublanovskaya, and V. Faddeeva, Solution of linear algebraic systems with rectangular matrices, Proc. Steklov Inst. Math., 96 (1968), pp. 93–111. · Zbl 0208.40001 [23] G. Flagg and S. Gugercin, Multipoint Volterra series interpolation and $$\mathcal{H}_2$$ optimal model reduction of bilinear systems, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 549–579. · Zbl 1315.93036 [24] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $$l_∞$$-error bounds, Internat. J. Control, 39 (1984), pp. 1115–1193. · Zbl 0543.93036 [25] S. Goreinov, E. Tyrtyshnikov, and N. Zamarashkin, A theory of pseudoskeleton approximations, Linear Algebra Appl., 261 (1997), pp. 1–21. · Zbl 0877.65021 [26] C. Gu, QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems, IEEE Trans. Comput. Aided Des. Integrated Circuits Syst., 30 (2011), pp. 1307–1320. [27] S. Gugercin, A. C. Antoulas, and C. Beattie, $$\mathcal{H}_2$$ model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 609–638. · Zbl 1159.93318 [28] M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, in Dimension Reduction of Large-Scale Systems, Springer, Berlin, 2005, pp. 261–306. · Zbl 1079.65533 [29] H. Hotelling, Analysis of a complex of statistical variables with principal components, J. Educ. Psych., 24 (1933), pp. 417–441, 498–520. · JFM 59.1183.01 [30] I. C. F. Ipsen and T. Wentworth, The effect of coherence on sampling from matrices with orthonormal columns, and preconditioned least squares problems, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1490–1520. · Zbl 1359.65063 [31] W. Kahan, Numerical linear algebra, Canad. Math. Bull., 9 (1965), pp. 757–801. · Zbl 0236.65025 [32] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), pp. 492–515. · Zbl 1075.65118 [33] C. L. Lawson and R. J. Hanson, Solving Least Squares Problems, Prentice–Hall, Englewood Cliffs, NJ, 1974. · Zbl 0860.65028 [34] M. Loéve, Probability Theory, D. Van Nostrand, New York, 1955. [35] J. Lumley, The Structures of Inhomogeneous Turbulent Flow, in Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, 1967, pp. 166–178. [36] B. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), pp. 17–32. · Zbl 0464.93022 [37] C. Mullis and R. Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans. Circuits Syst., 23 (1976), pp. 551–562. · Zbl 0342.93066 [38] S. S. Chaturantabut and D. Sorensen, Nonlinear model reduction for porous media flow, in 2010 SIAM Annual Meeting, 2010. · Zbl 1217.65169 [39] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737–2764. · Zbl 1217.65169 [40] D. Sorensen, private communication, 2010. [41] D. C. Sorensen and M. Embree, A DEIM induced CUR factorization, CAAM Department Technical report TR14-04, Rice University, Houston, TX, 2014.
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