Data-sparse approximation to a class of operator-valued functions

Gavrilyuk, Ivan; Hackbusch, Wolfgang; Khoromskij, Boris

doi:10.1090/S0025-5718-04-01703-X

Data-sparse approximation to a class of operator-valued functions
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by Ivan P. Gavrilyuk, Wolfgang Hackbusch and Boris N. Khoromskij PDF

Math. Comp. 74 (2005), 681-708 Request permission

Abstract:

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator $\mathcal {L}$.

References

K. I. Babenko, Osnovy chislennogo analiza, “Nauka”, Moscow, 1986 (Russian). MR 889669
R. Bartels and G.W. Stewart: Algorithm 432: Solution of the matrix equation $AX+XB=C$. Comm. ACM 15 (1972), 820-826.
Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR 0122820
Ivan P. Gavrilyuk, Strongly $P$-positive operators and explicit representations of the solutions of initial value problems for second-order differential equations in Banach space, J. Math. Anal. Appl. 236 (1999), no. 2, 327–349. MR 1704587, DOI 10.1006/jmaa.1999.6430
Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, $\scr H$-matrix approximation for the operator exponential with applications, Numer. Math. 92 (2002), no. 1, 83–111. MR 1917366, DOI 10.1007/s002110100360
I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, $\scr H$-matrix approximation for elliptic solution operators in cylinder domains, East-West J. Numer. Math. 9 (2001), no. 1, 25–58. MR 1839197
I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij: Data-sparse approximation to operator-valued functions of elliptic operators. Math. Comp., 73 (2004) 1297-1324.
I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij: Data-sparse approximation to a hierarchy of operator-valued functions. In: Proceedings of the 18th GAMM-Seminar Leipzig 2002, 31-52 (ISBN 3-00-009258-7, http://www.mis.mpg.de/conferences/gamm/2002).
I.P. Gavrilyuk and V.L. Makarov: Exponentially convergent parallel discretization methods for the first order evolution equations, Appl. Math. Inform. 5 (2000), 47-69, 79 (2001).
L. Grasedyck and W. Hackbusch: Construction and arithmetics of $\mathcal {H}$ -matrices. Computing 70 (2003), 295-334.
L. Grasedyck, W. Hackbusch and B.N. Khoromskij: Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing 70 (2003), 121-165.
W. Hackbusch, A sparse matrix arithmetic based on $\scr H$-matrices. I. Introduction to $\scr H$-matrices, Computing 62 (1999), no. 2, 89–108. MR 1694265, DOI 10.1007/s006070050015
W. Hackbusch and B. N. Khoromskij, A sparse $\scr H$-matrix arithmetic. II. Application to multi-dimensional problems, Computing 64 (2000), no. 1, 21–47. MR 1755846, DOI 10.1007/PL00021408
W. Hackbusch and B. N. Khoromskij, A sparse $\scr H$-matrix arithmetic: general complexity estimates, J. Comput. Appl. Math. 125 (2000), no. 1-2, 479–501. Numerical analysis 2000, Vol. VI, Ordinary differential equations and integral equations. MR 1803209, DOI 10.1016/S0377-0427(00)00486-6
Wolfgang Hackbusch and Boris N. Khoromskij, Towards $\scr H$-matrix approximation of linear complexity, Problems and methods in mathematical physics (Chemnitz, 1999) Oper. Theory Adv. Appl., vol. 121, Birkhäuser, Basel, 2001, pp. 194–220. MR 1847212
W. Hackbusch and B.N. Khoromskij: Hierarchical Kronecker tensor-product approximation to a class of nonlocal operators in high dimensions. Preprint MPI MIS, No. 16, Leipzig 2004; Computing (to appear).
Marlis Hochbruck and Christian Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 34 (1997), no. 5, 1911–1925. MR 1472203, DOI 10.1137/S0036142995280572
Yu. L. Daletskiĭ and M. G. Kreĭn, Ustoĭchivost′resheniĭ differentsial′nykh uravneniĭ v banakhovom prostranstve, Nonlinear Analysis and its Applications Series, Izdat. “Nauka”, Moscow, 1970 (Russian). MR 0352638
L. A. Fialkow, Spectral properties of elementary operators, Acta Sci. Math. (Szeged) 46 (1983), no. 1-4, 269–282. MR 739043
Lawrence A. Fialkow, Spectral properties of elementary operators. II, Trans. Amer. Math. Soc. 290 (1985), no. 1, 415–429. MR 787973, DOI 10.1090/S0002-9947-1985-0787973-9
Zoran Gajic and Muhammad Tahir Javed Qureshi, Lyapunov matrix equation in system stability and control, Mathematics in Science and Engineering, vol. 195, Academic Press, Inc., San Diego, CA, 1995. MR 1343974
G. H. Golub, S. Nash, and C. Van Loan, A Hessenberg-Schur method for the problem $AX+XB=C$, IEEE Trans. Automat. Control 24 (1979), no. 6, 909–913. MR 566448, DOI 10.1109/TAC.1979.1102170
Marlis Hochbruck and Gerhard Starke, Preconditioned Krylov subspace methods for Lyapunov matrix equations, SIAM J. Matrix Anal. Appl. 16 (1995), no. 1, 156–171. MR 1311424, DOI 10.1137/S0895479892239238
Peter Lancaster, Explicit solutions of linear matrix equations, SIAM Rev. 12 (1970), 544–566. MR 279115, DOI 10.1137/1012104
Irena Lasiecka and Roberto Triggiani, Algebraic Riccati equations arising in boundary/point control: a review of theoretical and numerical results. I. Continuous case, Perspectives in control theory (Sielpia, 1988) Progr. Systems Control Theory, vol. 2, Birkhäuser Boston, Boston, MA, 1990, pp. 175–210. MR 1046880
A. Lu and E.L. Wachspress: Solution of Lyapunov equations by ADI iteration. Comp. Math. Appl. 21 (1991), 43-58.
Gunter Lumer and Marvin Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41. MR 104167, DOI 10.1090/S0002-9939-1959-0104167-0
Dongwoo Sheen, Ian H. Sloan, and Vidar Thomée, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comp. 69 (2000), no. 229, 177–195. MR 1648403, DOI 10.1090/S0025-5718-99-01098-4
Frank Stenger, Numerical methods based on sinc and analytic functions, Springer Series in Computational Mathematics, vol. 20, Springer-Verlag, New York, 1993. MR 1226236, DOI 10.1007/978-1-4612-2706-9
Frank Stenger, Collocating convolutions, Math. Comp. 64 (1995), no. 209, 211–235. MR 1270624, DOI 10.1090/S0025-5718-1995-1270624-7
K. L. Bowers and J. Lund (eds.), Computation and control. III, Progress in Systems and Control Theory, vol. 15, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1247462, DOI 10.1007/978-1-4612-0321-6
E.L. Wachspress: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1 (1988) 87-90.