Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Owhadi, Houman; Zhang, Lei; Berlyand, Leonid

doi:10.1051/m2an/2013118

Owhadi, Houman ; Zhang, Lei ; Berlyand, Leonid

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 2, pp. 517-552.

Résumé

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L^∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L² norm of the source terms; its (pre-)computation involves minimizing 𝒪(H^-d) quadratic (cell) problems on (super-)localized sub-domains of size 𝒪(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator -div(a∇·) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (𝒪(H) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

MR Zbl

DOI : 10.1051/m2an/2013118

Classification : 41A15, 34E13, 35B27
Mots clés : homogenization, polyharmonic splines, localization

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     author = {Owhadi, Houman and Zhang, Lei and Berlyand, Leonid},
     title = {Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {517--552},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {2},
     year = {2014},
     doi = {10.1051/m2an/2013118},
     mrnumber = {3177856},
     zbl = {1296.41007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013118/}
}

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Owhadi, Houman; Zhang, Lei; Berlyand, Leonid. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 2, pp. 517-552. doi : 10.1051/m2an/2013118. http://www.numdam.org/articles/10.1051/m2an/2013118/

Bibliographie
Cité par

[1] A. Abdulle and M.J. Grote, Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul. 9 (2011) 766-792. | MR | Zbl

[2] A. Abdulle and Ch. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Model. Simul. 3 (2004) 195-220. | MR | Zbl

[3] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4 (2005) 790-812. | MR | Zbl

[4] T. Arbogast and K.J. Boyd. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150-1171. | MR | Zbl

[5] T. Arbogast, C.-S. Huang and S.-M. Yang, Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements. Math. Models Methods Appl. Sci. 17 (2007) 1279-1305. | MR | Zbl

[6] S.N. Armstrong and P.E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pures Appl. 97 (2012) 460-504. | MR | Zbl

[7] M. Atteia, Fonctions spline et noyaux reproduisants d'Aronszajn-Bergman. Rev. Française Informat. Recherche Opérationnelle 4 (1970) 31-43. | Numdam | MR | Zbl

[8] I. Babuška, G. Caloz and J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945-981. | MR | Zbl

[9] I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9 (2011) 373-406. | MR | Zbl

[10] I. Babuška and J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510-536. | MR | Zbl

[11] I. Babuška and J.E. Osborn, Can a finite element method perform arbitrarily badly? Math. Comput. 69 (2000) 443-462. | MR | Zbl

[12] G. Bal and W. Jing, Corrector theory for MSFEM and HMM in random media. Multiscale Model. Simul. 9 (2011) 1549-1587. | MR | Zbl

[13] G. Ben Arous and H. Owhadi, Multiscale homogenization with bounded ratios and anomalous slow diffusion. Comm. Pure Appl. Math. 56 (2003) 80-113. | MR | Zbl

[14] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structure. North Holland, Amsterdam (1978). | MR | Zbl

[15] L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast. Arch. Rational Mech. Anal. 198 (2010) 677-721. | MR | Zbl

[16] X. Blanc, C. Le Bris and P.-L. Lions, Une variante de la théorie de l'homogénéisation stochastique des opérateurs elliptiques. C. R. Math. Acad. Sci. Paris 343 (2006) 717-724. | MR | Zbl

[17] X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices. J. Math. Pures Appl. 88 (2007) 34-63. | MR | Zbl

[18] A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal. 21 (1999) 303-315. | MR | Zbl

[19] L.V. Branets, S.S. Ghai, L.L. and X.-H. Wu, Challenges and technologies in reservoir modeling. Commun. Comput. Phys. (2009) 6 1-23. | MR

[20] R.A. Brownlee, Error estimates for interpolation of rough and smooth functions using radial basis functions. Ph.D. thesis. University of Leicester (2004).

[21] L.A. Caffarelli and P.E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math. 61 (2008) 1-17. | MR | Zbl

[22] C.-C. Chu, I.G. Graham and T.Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915-1955. | MR | Zbl

[23] M. Desbrun, R. Donaldson and H. Owhadi. Modeling across scales: Discrete geometric structures in homogenization and inverse homogenization. Reviews of Nonlinear Dynamics and Complexity. Special issue on Multiscale Analysis and Nonlinear Dynamics (2012). | MR

[24] J. Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. RAIRO Anal. Numer. 10 (1976) 5-12. | Numdam | MR

[25] J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, in Constructive theory of functions of several variables, Proc. of Conf., Math. Res. Inst., Oberwolfach, 1976, in vol. 571. of Lect. Notes Math. Springer, Berlin (1977) 85-100. | MR | Zbl

[26] J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les Dm-splines. RAIRO Anal. Numér. (1978) 12 325-334. | Numdam | MR | Zbl

[27] W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87-132. | MR | Zbl

[28] Y. Efendiev, J. Galvis and X. Wu, Multiscale finite element and domain decomposition methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937-955.

[29] Y. Efendiev, V. Ginting, T. Hou and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 (2006) 155-174. | MR | Zbl

[30] Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math. 57 (2007) 577-596. | MR | Zbl

[31] Y. Efendiev and T.Y. Hou, Multiscale finite element methods, Theory and applications, in vol. 4, Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York (2009). | MR | Zbl

[32] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, vol. 28 of Classics in Appl. Math. Society for Industrial and Applied Mathematics (1987). | MR | Zbl

[33] B. Engquist and P.E. Souganidis, Asymptotic and numerical homogenization. Acta Numerica 17 (2008) 147-190. | MR | Zbl

[34] B. Engquist, H. Holst and O. Runborg, Multi-scale methods for wave propagation in heterogeneous media. Commun. Math. Sci. 9 (2011) 33-56. | MR | Zbl

[35] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, vol. 105. Ann. Math. Stud. Princeton University Press, Princeton, NJ (1983). | MR | Zbl

[36] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag (1983). | MR

[37] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'aera. Rendi Conti di Mat. 8 (1975) 277-294. | MR | Zbl

[38] A. Gloria, Analytical framework for the numerical homogenization of elliptic monotone operators and quasiconvex energies. SIAM MMS 5 (2006) 996-1043. | MR | Zbl

[39] A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci. 21 (2011) 1601-1630. | MR | Zbl

[40] A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 (2012) 1-28. | MR

[41] L. Grasedyck, I. Greff and S. Sauter, The al basis for the solution of elliptic problems in heterogeneous media. Multiscale Modeling and Simulation 10 (2012) 245-258. | MR | Zbl

[42] M. Grüter and K. Widman, The green function for uniformly elliptic equations. Manuscripta Math. 37 (1982) 303-342. | MR | Zbl

[43] R.L. Harder and R.N. Desmarais, Interpolation using surface splines. J. Aircr. 9 (1972) 189-191.

[44] T.Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913-943. | MR | Zbl

[45] T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl

[46] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1991). | MR | Zbl

[47] E. Kosygina, F. Rezakhanlou and S.R.S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59 (2006) 1489-1521. | MR | Zbl

[48] O. Kounchev and H. Render, Polyharmonic splines on grids Z× aZn and their limits. Math. Comput. 74 (2005) 1831-1841. | MR | Zbl

[49] S.M. Kozlov, The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188-202, 327. | MR | Zbl

[50] P.-L. Lions and P.E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56 (2003) 1501-1524. | MR | Zbl

[51] W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. Approx. Theory Appl. 4 (1988) 77-89. | MR | Zbl

[52] W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. II. Math. Comput. 54 (1990) 211-230. | MR | Zbl

[53] W.R. Madych and S.A. Nelson, Polyharmonic cardinal splines. J. Approx. Theory 60 (1990) 141-156. | MR | Zbl

[54] W.R. Madych and S.A. Nelson, Polyharmonic cardinal splines: a minimization property. J. Approx. Theory 63 (1990) 303-320. | MR | Zbl

[55] A. Malqvist and D. Peterseim, Localization of elliptic multiscale problems. Technical report arXiv:1110.0692 (2012). | MR | Zbl

[56] O.V. Matveev, Some methods for the reconstruction of functions of n variables defined on chaotic grids. Dokl. Akad. Nauk 326 (1992) 605-609. | MR | Zbl

[57] O.V. Matveev, Spline interpolation of functions of several variables and bases in Sobolev spaces. Trudy Mat. Inst. Steklov. 198 (1992) 125-152. | MR | Zbl

[58] O.V. Matveev, Interpolation of functions on chaotic grids. Dokl. Akad. Nauk 339 (1994) 594-597. | MR | Zbl

[59] O.V. Matveev, Methods for the approximate recovery of functions defined on chaotic grids. Izv. Ross. Akad. Nauk Ser. Mat. 60 111-156, 1996. | MR | Zbl

[60] O.V. Matveev, On a method for the interpolation of functions on chaotic grids. Mat. Zametki 62 (1997) 404-417. | MR | Zbl

[61] J.M. Melenk, On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272-289. | MR | Zbl

[62] G.W. Milton, The theory of composites, vol. 6 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl

[63] P. Ming and X. Yue, Numerical methods for multiscale elliptic problems. J. Comput. Phys. 214 (2006) 421-445. | MR | Zbl

[64] R. Moser, Theory of partial differential equations. MA6000A. Lect. Notes (2012). Available on http://people.bath.ac.uk/rm257/MA6000A/notes.pdf.

[65] F. Murat and L. Tartar, H-convergence. Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger (1978).

[66] F.J. Narcowich, J.D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74 (2005) 743-763. | MR | Zbl

[67] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR | Zbl

[68] J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7 (2008) 171-196. | MR | Zbl

[69] H. Owhadi and L. Zhang, Metric-based upscaling. Comm. Pure Appl. Math. 60 (2007) 675-723. | MR | Zbl

[70] H. Owhadi and L. Zhang. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. SIAM Multiscale Model. Simul. 9 (2011) 1373-1398. arXiv:1011.0986. | MR | Zbl

[71] H. Owhadi, Anomalous slow diffusion from perpetual homogenization. Ann. Probab. 31 (2003) 1935-1969. | MR | Zbl

[72] H. Owhadi, Averaging versus chaos in turbulent transport? Comm. Math. Phys. 247 (2004) 553-599. | MR | Zbl

[73] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields, Vol. I, II (Esztergom (1979)), vol. 27. Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam (1981) 835-873. | MR | Zbl

[74] A. Pinkus, N-Widths in Approximation Theory. Springer-Verlag (1985). | MR | Zbl

[75] C. Rabut, B-splines Polyarmoniques Cardinales: Interpolation, Quasi-interpolation, filtrage. Thèse d'État. Université de Toulouse (1990).

[76] Ch. Rabut, Elementary m-harmonic cardinal B-splines. Numer. Algorithms 2 (1992) 39-61. | MR | Zbl

[77] Ch. Rabut, High level m-harmonic cardinal B-splines. Numer. Algorithms 2 (1992) 63-84. | MR | Zbl

[78] M. Rossini, Detecting discontinuities in two-dimensional signals sampled on a grid. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4 (2009) 203-215. | MR

[79] I.J. Schoenberg, Cardinal spline interpolation. Conference Board of the Mathematical Sciences Regional Conf. Ser. Appl. Math. No. 12. Society for Industrial and Applied Mathematics, Philadelphia, Pa., (1973). | MR | Zbl

[80] S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa 22 (1968) 571-597; errata, S. Spagnolo, Ann. Scuola Norm. Sup. Pisa 22 (1968) 673. | Numdam | MR | Zbl

[81] G. Stampacchia, Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl

[82] G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus. Séminaire Jean Leray No. 3 (1963-1964). Numdam (1964). | Zbl

[83] William Symes. Transfer of approximation and numerical homogenization of hyperbolic boundary value problems with a continuum of scales. TR12-20 Rice Tech Report (2012).

[84] J.L. Taylor, S. Kim and R.M. Brown, The green function for elliptic systems in two dimensions. arXiv:1205.1089 (2012). | MR | Zbl

[85] J. Vybiral, Widths of embeddings in function spaces. J. Complexity 24 (2008) 545-570. | MR | Zbl

[86] C.D. White and R.N. Horne, Computing absolute transmissibility in the presence of finescale heterogeneity. SPE Symposium on Reservoir Simulation 16011 (1987).

[87] X.H. Wu, Y. Efendiev and T.Y. Hou, Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst. Ser. B 2 (2002) 185-204. | MR | Zbl

[88] V.V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27 (1986) 167-180. | MR | Zbl

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