메뉴 건너뛰기




Volumn 12, Issue 5, 2012, Pages 573-623

Greedy Approximation of High-Dimensional Ornstein-Uhlenbeck Operators

Author keywords

Fokker Planck equation; Greedy algorithm; Nonlinear approximation

Indexed keywords

FINITELY EXTENSIBLE NONLINEAR ELASTIC POTENTIALS; GREEDY ALGORITHMS; HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS; KINETIC POLYMER MODELS; MATHEMATICAL ANALYSIS; NONLINEAR APPROXIMATION; NONLINEAR APPROXIMATION THEORY; ORTHOGONAL GREEDY ALGORITHMS;

EID: 84866540803     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-012-9122-z     Document Type: Article
Times cited : (32)

References (46)
  • 1
    • 0003796630 scopus 로고    scopus 로고
    • 2nd edn., Pure and Applied Mathematics, Amsterdam: Elsevier/Academic Press
    • R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2nd edn. Pure and Applied Mathematics, vol. 140 (Elsevier/Academic Press, Amsterdam, 2003).
    • (2003) Sobolev Spaces , vol.140
    • Adams, R.A.1    Fournier, J.J.F.2
  • 2
    • 33751260933 scopus 로고    scopus 로고
    • A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids
    • doi:10.1016/j.jnnfm.2006.07.007
    • A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J. Non-Newton. Fluid Mech. 139(3), 153-176 (2006). doi: 10. 1016/j. jnnfm. 2006. 07. 007. http://www. sciencedirect. com/science/article/pii/S0377025706001662.
    • (2006) J. Non-Newton. Fluid Mech. , vol.139 , Issue.3 , pp. 153-176
    • Ammar, A.1    Mokdad, B.2    Chinesta, F.3    Keunings, R.4
  • 3
    • 34249706773 scopus 로고    scopus 로고
    • A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part II: Transient simulation using space-time separated representations
    • doi:10.1016/j.jnnfm.2007.03.009
    • A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part II: Transient simulation using space-time separated representations, J. Non-Newton. Fluid Mech. 144(2-3), 98-121 (2007). doi: 10. 1016/j. jnnfm. 2007. 03. 009. http://www. sciencedirect. com/science/article/pii/S0377025707000821.
    • (2007) J. Non-Newton. Fluid Mech. , vol.144 , Issue.2-3 , pp. 98-121
    • Ammar, A.1    Mokdad, B.2    Chinesta, F.3    Keunings, R.4
  • 4
    • 77954641379 scopus 로고    scopus 로고
    • Non incremental strategies based on separated representations: applications in computational rheology
    • A. Ammar, M. Normandin, F. Daim, D. González, E. Cueto, F. Chinesta, Non incremental strategies based on separated representations: applications in computational rheology, Commun. Math. Sci. 8(3), 671-695 (2010). http://projecteuclid. org/getRecord?id=euclid. cms/1282747135.
    • (2010) Commun. Math. Sci. , vol.8 , Issue.3 , pp. 671-695
    • Ammar, A.1    Normandin, M.2    Daim, F.3    González, D.4    Cueto, E.5    Chinesta, F.6
  • 5
    • 39049139418 scopus 로고    scopus 로고
    • Existence of global weak solutions to some regularized kinetic models for dilute polymers
    • (electronic), doi:10.1137/060666810
    • J. W. Barrett, E. Süli, Existence of global weak solutions to some regularized kinetic models for dilute polymers, Multiscale Model. Simul. 6(2), 506-546 (2007). (electronic). doi: 10. 1137/060666810.
    • (2007) Multiscale Model. Simul. , vol.6 , Issue.2 , pp. 506-546
    • Barrett, J.W.1    Süli, E.2
  • 6
    • 44349177436 scopus 로고    scopus 로고
    • Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off
    • doi:10.1142/S0218202508002917
    • J. W. Barrett, E. Süli, Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off, Math. Models Methods Appl. Sci. 18(6), 935-971 (2008). doi: 10. 1142/S0218202508002917.
    • (2008) Math. Models Methods Appl. Sci. , vol.18 , Issue.6 , pp. 935-971
    • Barrett, J.W.1    Süli, E.2
  • 7
    • 71049139773 scopus 로고    scopus 로고
    • Numerical approximation of corotational dumbbell models for dilute polymers
    • doi:10.1093/imanum/drn022
    • J. W. Barrett, E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers, IMA J. Numer. Anal. 29(4), 937-959 (2009). doi: 10. 1093/imanum/drn022.
    • (2009) IMA J. Numer. Anal. , vol.29 , Issue.4 , pp. 937-959
    • Barrett, J.W.1    Süli, E.2
  • 8
    • 79960351791 scopus 로고    scopus 로고
    • Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: finitely extensible nonlinear bead-spring chains
    • doi:10.1142/S0218202511005313
    • J. W. Barrett, E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: finitely extensible nonlinear bead-spring chains, Math. Models Methods Appl. Sci. 21(6), 1211-1289 (2011). doi: 10. 1142/S0218202511005313.
    • (2011) Math. Models Methods Appl. Sci. , vol.21 , Issue.6 , pp. 1211-1289
    • Barrett, J.W.1    Süli, E.2
  • 9
    • 79952566134 scopus 로고    scopus 로고
    • Finite element approximation of kinetic dilute polymer models with microscopic cut-off
    • doi:10.1051/m2an/2010030
    • J. W. Barrett, E. Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off, Math. Model. Numer. Anal. 45(1), 39-89 (2011). doi: 10. 1051/m2an/2010030.
    • (2011) Math. Model. Numer. Anal. , vol.45 , Issue.1 , pp. 39-89
    • Barrett, J.W.1    Süli, E.2
  • 11
    • 0002610108 scopus 로고
    • The principal term of the spectral asymptotics for "non-smooth" elliptic problems
    • Translated in Funct. Anal. Appl.4, 265-275 (1970)
    • M.Š. Birman, M. Z. Solomjak, The principal term of the spectral asymptotics for "non-smooth" elliptic problems, Funkc. Anal. Ego Prilož. 4(4), 1-13 (1970). Translated in Funct. Anal. Appl. 4, 265-275 (1970).
    • (1970) Funkc. Anal. Ego Prilož. , vol.4 , Issue.4 , pp. 1-13
    • Birman, M.Š.1    Solomjak, M.Z.2
  • 12
    • 0001768965 scopus 로고
    • Spectral asymptotics of nonsmooth elliptic operators. I
    • Translated in Trans. Mosc. Math. Soc.27, 1-5 (1972)
    • M.Š. Birman, M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, Tr. Mosk. Mat. Obŝ. 27, 3-52 (1972). Translated in Trans. Mosc. Math. Soc. 27, 1-5 (1972).
    • (1972) Tr. Mosk. Mat. Obŝ. , vol.27 , pp. 3-52
    • Birman, M.Š.1    Solomjak, M.Z.2
  • 14
    • 0003075578 scopus 로고
    • On the spectral theory of elliptic differential operators. I
    • doi:10.1007/BF01343363
    • F. E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142, 22-130 (1961). doi: 10. 1007/BF01343363.
    • (1961) Math. Ann. , vol.142 , pp. 22-130
    • Browder, F.E.1
  • 15
    • 80255141549 scopus 로고    scopus 로고
    • Convergence of a greedy algorithm for high-dimensional convex nonlinear problems
    • doi:10.1142/S0218202511005799
    • E. Cancès, V. Ehrlacher, T. Lelièvre, Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, Math. Models Methods Appl. Sci. 21(12), 2433-2467 (2011). doi: 10. 1142/S0218202511005799.
    • (2011) Math. Models Methods Appl. Sci. , vol.21 , Issue.12 , pp. 2433-2467
    • Cancès, E.1    Ehrlacher, V.2    Lelièvre, T.3
  • 16
    • 79955469685 scopus 로고    scopus 로고
    • An overview of the proper generalized decomposition with applications in computational rheology
    • doi:10.1016/j.jnnfm.2010.12.012
    • F. Chinesta, A. Ammar, A. Leygue, R. Keunings, An overview of the proper generalized decomposition with applications in computational rheology, J. Non-Newton. Fluid Mech. 166(11), 578-592 (2011). doi: 10. 1016/j. jnnfm. 2010. 12. 012. http://www. sciencedirect. com/science/article/pii/S0377025711000061.
    • (2011) J. Non-Newton. Fluid Mech. , vol.166 , Issue.11 , pp. 578-592
    • Chinesta, F.1    Ammar, A.2    Leygue, A.3    Keunings, R.4
  • 17
    • 28844445147 scopus 로고
    • The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems
    • doi:10.1137/1009105
    • C. Clark, The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems, SIAM Rev. 9, 627-646 (1967). doi: 10. 1137/1009105.
    • (1967) SIAM Rev. , vol.9 , pp. 627-646
    • Clark, C.1
  • 18
    • 0000186368 scopus 로고
    • A Padé approximant to the inverse Langevin function
    • doi:10.1007/BF00366640
    • A. Cohen, A Padé approximant to the inverse Langevin function, Rheol. Acta30(3), 270-273 (1991). doi: 10. 1007/BF00366640. http://www. springerlink. com/content/t240j86m14247501.
    • (1991) Rheol. Acta , vol.30 , Issue.3 , pp. 270-273
    • Cohen, A.1
  • 20
    • 0003882735 scopus 로고
    • Cambridge Studies in Advanced Mathematics10.1017/CBO9780511623721, Cambridge: Cambridge University Press
    • E. B. Davies, Spectral Theory and Differential Operators. Cambridge Studies in Advanced Mathematics, vol. 42 (Cambridge University Press, Cambridge, 1995). doi: 10. 1017/CBO9780511623721.
    • (1995) Spectral Theory and Differential Operators , vol.42
    • Davies, E.B.1
  • 21
    • 0039439070 scopus 로고    scopus 로고
    • Some remarks on greedy algorithms
    • doi:10.1007/BF02124742
    • R. A. DeVore, V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math. 5(2-3), 173-187 (1996). doi: 10. 1007/BF02124742.
    • (1996) Adv. Comput. Math. , vol.5 , Issue.2-3 , pp. 173-187
    • DeVore, R.A.1    Temlyakov, V.N.2
  • 25
    • 76249085920 scopus 로고    scopus 로고
    • A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model
    • doi:10.1051/m2an/2009034
    • D. J. Knezevic, E. Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model, Math. Model. Numer. Anal. 43(6), 1117-1156 (2009). doi: 10. 1051/m2an/2009034.
    • (2009) Math. Model. Numer. Anal. , vol.43 , Issue.6 , pp. 1117-1156
    • Knezevic, D.J.1    Süli, E.2
  • 26
    • 0003440536 scopus 로고
    • Translated from the Czech, New York: Wiley
    • A. Kufner, Weighted Sobolev Spaces (Wiley, New York, 1985). Translated from the Czech.
    • (1985) Weighted Sobolev Spaces
    • Kufner, A.1
  • 27
    • 0001651444 scopus 로고
    • How to define reasonably weighted Sobolev spaces
    • A. Kufner, B. Opic, How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carol. 25(3), 537-554 (1984).
    • (1984) Comment. Math. Univ. Carol. , vol.25 , Issue.3 , pp. 537-554
    • Kufner, A.1    Opic, B.2
  • 28
    • 34347098764 scopus 로고
    • Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe
    • doi:10.1007/BF01793684
    • W. Kuhn, F. Grün, Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe, Kolloid Z. 101(3), 248-271 (1942). doi: 10. 1007/BF01793684.
    • (1942) Kolloid Z. , vol.101 , Issue.3 , pp. 248-271
    • Kuhn, W.1    Grün, F.2
  • 29
    • 70449574498 scopus 로고    scopus 로고
    • Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations
    • doi:10.1007/s00365-009-9071-1
    • C. Le Bris, T. Lelièvre, Y. Maday, Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations, Constr. Approx. 30(3), 621-651 (2009). doi: 10. 1007/s00365-009-9071-1.
    • (2009) Constr. Approx. , vol.30 , Issue.3 , pp. 621-651
    • Le Bris, C.1    Lelièvre, T.2    Maday, Y.3
  • 30
    • 70449363611 scopus 로고    scopus 로고
    • On the solution of the Fokker-Planck equation using a high-order reduced basis approximation
    • doi:10.1016/j.cma.2009.09.028
    • G. M. Leonenko, T. N. Phillips, On the solution of the Fokker-Planck equation using a high-order reduced basis approximation, Comput. Methods Appl. Mech. Eng. 199(1-4), 158-168 (2009). doi: 10. 1016/j. cma. 2009. 09. 028.
    • (2009) Comput. Methods Appl. Mech. Eng. , vol.199 , Issue.1-4 , pp. 158-168
    • Leonenko, G.M.1    Phillips, T.N.2
  • 32
    • 55749083845 scopus 로고    scopus 로고
    • Well-posedness for the FENE dumbbell model of polymeric flows
    • doi:10.1002/cpa.20252
    • N. Masmoudi, Well-posedness for the FENE dumbbell model of polymeric flows, Commun. Pure Appl. Math. 61(12), 1685-1714 (2008). doi: 10. 1002/cpa. 20252.
    • (2008) Commun. Pure Appl. Math. , vol.61 , Issue.12 , pp. 1685-1714
    • Masmoudi, N.1
  • 33
    • 34548290926 scopus 로고    scopus 로고
    • A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations
    • doi:10.1016/j.cma.2007.05.016
    • A. Nouy, A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations, Comput. Methods Appl. Mech. Eng. 196(45-48), 4521-4537 (2007). doi: 10. 1016/j. cma. 2007. 05. 016.
    • (2007) Comput. Methods Appl. Mech. Eng. , vol.196 , Issue.45-48 , pp. 4521-4537
    • Nouy, A.1
  • 34
    • 53549096783 scopus 로고    scopus 로고
    • Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms
    • doi:10.1016/j.cma.2008.06.012
    • A. Nouy, Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms, Comput. Methods Appl. Mech. Eng. 197(51-52), 4718-4736 (2008). doi: 10. 1016/j. cma. 2008. 06. 012.
    • (2008) Comput. Methods Appl. Mech. Eng. , vol.197 , Issue.51-52 , pp. 4718-4736
    • Nouy, A.1
  • 35
    • 55549140120 scopus 로고    scopus 로고
    • Generalized spectral decomposition for stochastic nonlinear problems
    • doi:10.1016/j.jcp.2008.09.010
    • A. Nouy, O. P. Le Maître, Generalized spectral decomposition for stochastic nonlinear problems, J. Comput. Phys. 228(1), 202-235 (2009). doi: 10. 1016/j. jcp. 2008. 09. 010.
    • (2009) J. Comput. Phys. , vol.228 , Issue.1 , pp. 202-235
    • Nouy, A.1    Le Maître, O.P.2
  • 36
    • 0003997917 scopus 로고
    • Pitman Research Notes in Mathematics Series, Harlow: Longman Scientific & Technical
    • B. Opic, A. Kufner, Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series, vol. 219 (Longman Scientific & Technical, Harlow, 1990).
    • (1990) Hardy-Type Inequalities , vol.219
    • Opic, B.1    Kufner, A.2
  • 37
    • 0003687441 scopus 로고
    • 2nd edn., Functional analysis, New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers]
    • M. Reed, B. Simon, Methods of Modern Mathematical Physics. I, 2nd edn. (Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980). Functional analysis.
    • (1980) Methods of Modern Mathematical Physics. I
    • Reed, M.1    Simon, B.2
  • 38
    • 34250957105 scopus 로고
    • Zur Theorie der linearen und nichtlinearen Integralgleichungen
    • doi:10.1007/BF01449770
    • E. Schmidt, Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann. 63(4), 433-476 (1907). doi: 10. 1007/BF01449770. http://www. springerlink. com/content/k636412j41k8316w.
    • (1907) Math. Ann. , vol.63 , Issue.4 , pp. 433-476
    • Schmidt, E.1
  • 40
    • 84934812791 scopus 로고
    • The spectral asymptotic behavior of elliptic boundary value problems with weak degeneracy
    • Drogobych1973Functional Analysis and Its Applications(Russian), Moscow: Akad. Nauk SSSR Central. Èkonom.-Mat. Inst
    • G. M. Taščijan, The spectral asymptotic behavior of elliptic boundary value problems with weak degeneracy, in Proceedings of the Sixth Winter School on Mathematical Programming and Related Questions (Drogobych 1973), Functional Analysis and Its Applications, pp. 277-293 Akad. Nauk SSSR Central. Èkonom.-Mat. Inst., Moscow, 1975) (Russian).
    • (1975) Proceedings of the Sixth Winter School on Mathematical Programming and Related Questions , pp. 277-293
    • Taščijan, G.M.1
  • 41
    • 33748847228 scopus 로고
    • The classical formula of the asymptotic behavior of the spectrum of elliptic equations that are degenerate on the boundary of the domain
    • Translated in Math. Notes30(6), 937-942 (1981), doi:10.1007/BF01145775, 959
    • G. M. Taščijan, The classical formula of the asymptotic behavior of the spectrum of elliptic equations that are degenerate on the boundary of the domain, Mat. Zametki30(6), 871-880, 959 (1981). Translated in Math. Notes30(6), 937-942 (1981). doi: 10. 1007/BF01145775.
    • (1981) Mat. Zametki , vol.30 , Issue.6 , pp. 871-880
    • Taščijan, G.M.1
  • 42
    • 58149380535 scopus 로고    scopus 로고
    • Greedy approximation
    • doi:10.1017/S0962492906380014
    • V. N. Temlyakov, Greedy approximation, Acta Numer. 17, 235-409 (2008). doi: 10. 1017/S0962492906380014.
    • (2008) Acta Numer. , vol.17 , pp. 235-409
    • Temlyakov, V.N.1
  • 44
    • 84876507942 scopus 로고
    • Spectral asymptotic analysis for degenerate second order elliptic operators
    • Translated in Math. USSR-Izv.8(6), 1343-1371 (1974)
    • I. L. Vulis, M. Z. Solomjak, Spectral asymptotic analysis for degenerate second order elliptic operators, Izv. Akad. Nauk SSSR, Ser. Mat. 38, 1362-1392 (1974). http://mi. mathnet. ru/eng/izv2017. Translated in Math. USSR-Izv. 8(6), 1343-1371 (1974).
    • (1974) Izv. Akad. Nauk SSSR, Ser. Mat. , vol.38 , pp. 1362-1392
    • Vulis, I.L.1    Solomjak, M.Z.2
  • 45
    • 0347638498 scopus 로고
    • Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells
    • doi:10.1021/i160043a017
    • H. R. Warner, Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells, Ind. Eng. Chem. Fundam. 11(3), 379-387 (1972). doi: 10. 1021/i160043a017.
    • (1972) Ind. Eng. Chem. Fundam. , vol.11 , Issue.3 , pp. 379-387
    • Warner, H.R.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.