Numerical homogenization via approximation of the solution operator

Lecture Notes in Computational Science and Engineering

Volumn 82, Issue , 2012, Pages 187-216

  Gillman, Adrianna ^a   Young, Patrick ^a   Martinsson, Per Gunnar ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

PARTIAL DIFFERENTIAL EQUATIONS; POROUS MATERIALS;

BOUNDARY INTEGRAL OPERATORS; DISCRETE PROBLEMS; ELLIPTIC PARTIAL DIFFERENTIAL EQUATION; FLOWS IN POROUS MEDIA; GOVERNING DIFFERENTIAL EQUATIONS; HETEROGENEOUS MEDIA; NUMERICAL HOMOGENIZATION; ORIGINAL DIFFERENTIAL EQUATIONS;

NUMERICAL ANALYSIS;

EID: 84884378035     PISSN: 14397358     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-21943-6_9     Document Type: Conference Paper

References (35)

1. Ulf Andersson, Björn Engquist, Gunnar Ledfelt, and Olof Runborg, A contribution to wavelet-based subgrid modeling, Appl. Comput. Harmon. Anal. 7 (1999), no. 2, 151–164. MR MR1711012 (2000f:65130)
2. N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Kluwer Academic Publishers Group, Dordrecht, 1989, Mathematical problems in the mechanics of composite materials, Translated from the Russian by D. Leĭtes. MR 92d:73002
3. J. Barnes and P. Hut, A hierarchical O.nlogn/ force-calculation algorithm, Nature 324 (1986), no. 4.
4. G. Beylkin, R. Coifman, and V. Rokhlin, Wavelets in numerical analysis, Wavelets and their applications, Jones and Bartlett, Boston, MA, 1992, pp. 181–210. MR 93j:65215
5. 2 -matrix arithmetics in linear complexity, Computing 77 (2006), no. 1, 1–28. MR MR2207953 (2006k:65111)
6. 2 –matrices, Tech. Report 85/2007, Max Planck Institute, 2007.
7. 2 -matrices by hierarchical compression, Tech. Report 92/2007, Max Planck Institute, 2007.
8. J. Bremer and V. Rokhlin, Efficient discretization of Laplace boundary integral equations on polygonal domains, J. Comput. Phys. 229 (2010), no. 7, 2507–2525. MR MR2586199
9. M. E. Brewster and G. Beylkin, A multiresolution strategy for numerical homogenization, Appl. Comput. Harmon. Anal. 2 (1995), no. 4, 327–349. MR 96h:65156
10. S. Chandrasekaran and M. Gu, Fast and stable algorithms for banded plus semiseparable systems of linear equations, SIAM J. Matrix Anal. Appl. 25 (2003), no. 2, 373–384 (electronic). MR MR2047424 (2005f:65039)
11. S. Chandrasekaran, M. Gu, X.S. Li, and J. Xia, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 17 (2010), 953–976.
12. Superfast multifrontal method for structured linear systems of equations, SIAM J. Matrix Anal. Appl. 31 (2009), 1382–1411.
13. S. Chandrasekaran, M. Gu, and W. Lyons, A fast adaptive solver for hierarchically semiseparable representations, Calcolo 42 (2005), no. 3-4, 171–185. MR MR2191196 (2006i:65038)
14. Doina Cioranescu and Jeannine Saint Jean Paulin, Homogenization of reticulated structures, Applied Mathematical Sciences, vol. 136, Springer-Verlag, New York, 1999. MR MR1676922 (2000d:74064)
15. Patrick Dewilde and Shivkumar Chandrasekaran, A hierarchical semi-separable Moore-Penrose equation solver, Wavelets, multiscale systems and hypercomplex analysis, Oper. Theory Adv. Appl., vol. 167, Birkhäuser, Basel, 2006, pp. 69–85. MR MR2240291 (2007c:65039)
16. Mihai Dorobantu and Björn Engquist, Wavelet-based numerical homogenization, SIAM J. Numer. Anal. 35 (1998), no. 2, 540–559 (electronic). MR 99a:65183
17. Yalchin Efendiev and Thomas Y. Hou, Multiscale finite element methods, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4, Springer, New York, 2009, Theory and applications. MR MR2477579
18. Björn Engquist and Olof Runborg, Wavelet-based numerical homogenization with applications, Multiscale and multiresolution methods, Lect. Notes Comput. Sci. Eng., vol. 20, Springer, Berlin, 2002, pp. 97–148. MR MR1928565 (2003k:65180)
19. Björn Engquist and Olof Runborg Wavelet-based numerical homogenization, Highly oscillatory problems, London Math. Soc. Lecture Note Ser., vol. 366, Cambridge Univ. Press, Cambridge, 2009, pp. 98–126. MR MR2562507
20. Jacob Fish and Amir Wagiman, Multiscale finite element method for a locally nonperiodic heterogeneous medium, Computational Mechanics 12 (1993), 164–180, 10.1007/BF00371991.
21. A. George, Nested dissection of a regular finite element mesh, SIAM J. on Numerical Analysis 10 (1973), 345–363.
22. A. Gillman and P.G. Martinsson, Fast and accurate numerical methods for solving elliptic difference equations defined on lattices, J. Comput. Phys. 220(24) (2010), 9026–9041.
23. H -LU preconditioning, Numer. Math. 112 (2009), no. 4, 565–600. MR MR2507619 (2010e:65200)
24. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348.
25. Leslie Greengard and Vladimir Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta numerica, 1997, Acta Numer., vol. 6, Cambridge Univ. Press, Cambridge, 1997, pp. 229–269.
26. W. Hackbusch, The panel clustering technique for the boundary element method (invited contribution), Boundary elements IX, Vol. 1 (Stuttgart, 1987), Comput. Mech., Southampton, 1987, pp. 463–474. MR MR965331 (89i:76011)
27. 2 -matrices, Lectures on Applied Mathematics, Springer Berlin, 2002, pp. 9–29.
28. Wolfgang Hackbusch, A sparse matrix arithmetic based on H-matrices; Part I: Introduction to H-matrices, Computing 62 (1999), 89–108.
29. P.G. Martinsson, A fast direct solver for a class of elliptic partial differential equations, J. Sci. Comput. 38 (2009), no. 3, 316–330. MR MR2475654 (2010c:65041)
30. P.G. Martinsson and V. Rokhlin, A fast direct solver for boundary integral equations in two dimensions, J. Comput. Phys. 205 (2005), no. 1, 1–23.
31. An accelerated kernel independent fast multipole method in one dimension, SIAM Journal of Scientific Computing 29 (2007), no. 3, 1160–11178.
32. 2 N/ operations, IEE Proc. Microw. Antennas Propag. 143 (1996), no. 4, 277–283.
33. Zhifeng Sheng, Patrick Dewilde, and Shivkumar Chandrasekaran, Algorithms to solve hierarchically semi-separable systems, System theory, the Schur algorithm and multidimensional analysis, Oper. Theory Adv. Appl., vol. 176, Birkhäuser, Basel, 2007, pp. 255–294. MR MR2342902
34. Page Starr and Vladimir Rokhlin, On the numerical solution of two-point boundary value problems. II, Comm. Pure Appl. Math. 47 (1994), no. 8, 1117–1159. MR MR1288634 (95j:65090)
35. Jinchao Xu and Ludmil Zikatanov, On an energy minimizing basis for algebraic multigrid methods, Computing and Visualization in Science 7 (2004), 121–127, 10.1007/s00791-004-0147-y.