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Published online by Cambridge University Press:  10 October 2019

Houman Owhadi
Affiliation:
California Institute of Technology
Clint Scovel
Affiliation:
California Institute of Technology
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Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
From a Game Theoretic Approach to Numerical Approximation and Algorithm Design
, pp. 444 - 459
Publisher: Cambridge University Press
Print publication year: 2019

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References

[1] Abdulle, A. and Grote, M. J.. Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul., 9(2):766792, 2011.Google Scholar
[2] Abdulle, A. and Schwab, C.. Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Model. Simul., 3(1):195220 (electronic), 2004/05.CrossRefGoogle Scholar
[3] Abdulle, A., Weinan, E., Engquist, B., and Vanden-Eijnden, E.. The heterogeneous multiscale method. Acta Numerica, 21:187, 2012.CrossRefGoogle Scholar
[4] Adams, R. A. and Fournier, J. J. F.. Sobolev Spaces, volume 140. Academic Press, 2003.Google Scholar
[5] Agmon, S.. The coerciveness problem for integro-differential forms. J. Analyse Math., 6:183223, 1958.CrossRefGoogle Scholar
[6] Albiac, F. and Kalton, N. J.. Topics in Banach Space Theory, volume 233. Springer Science & Business Media, 2006.Google Scholar
[7] Aliprantis, C. D. and Border, K. C.. Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin, third edition, 2006.Google Scholar
[8] Allaire, G.. Homogenization and two-scale convergence. SIAM J. Math. Anal., 23:14821518, 1992.Google Scholar
[9] Allaire, G.. Two-scale convergence: a new method in periodic homogenization. nonlinear partial differential equations and their applications. In Collège de France Seminar Vol. XII (Paris, 1991–1993), volume 302, 1–14. Pitman Res. Notes Math. Ser., 1994.Google Scholar
[10] Allaire, G. and Brizzi, R.. A multiscale finite element method for numerical homogenization. Multiscale Model. Simul., 4(3):790812 (electronic), 2005.CrossRefGoogle Scholar
[11] Alpert, B., Beylkin, G., Coifman, R., and Rokhlin, V.. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 14(1):159184, 1993.Google Scholar
[12] Ancona, A.. Some results and examples about the behavior of harmonic functions and Greens functions with respect to second order elliptic operators. Nagoya Math. J., 165:123158, 2002.CrossRefGoogle Scholar
[13] Anderson, W. N., Jr. and Trapp, G. E.. Shorted operators. II. SIAM J. Appl. Math., (1):6071, 1975.Google Scholar
[14] Arbogast, T. and Boyd, K. J.. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal., 44(3):11501171 (electronic), 2006.CrossRefGoogle Scholar
[15] Arbogast, T., Huang, C.-S., and Yang, S.-M.. Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements. Math. Models Methods Appl. Sci., 17(8):12791305, 2007.CrossRefGoogle Scholar
[16] Aronszajn, N.. Theory of reproducing kernels. Transactions of the American Mathematical Society. 68(3) 337404, 1950.Google Scholar
[17] Ash, R. B.. Real Analysis and Probability. Academic Press, 1972. Probability and Mathematical Statistics, No. 11.Google Scholar
[18] Averbuch, A., Beylkin, G., Coifman, R., Fischer, P., and Israeli, M.. Adaptive solution of multidimensional PDEs via tensor product wavelet decomposition. Int. J. Pure Appl. Math., 44(1):75115, 2008.Google Scholar
[19] Averbuch, A., Beylkin, G., Coifman, R., and Israeli, M.. Multiscale inversion of elliptic operators. In Zeevi, Y. and Coifman, R., editors, Signal and Image Representation in Combined Spaces, volume 7 of Wavelet Anal. Appl., 341–359. Academic Press, San Diego, CA, 1998.Google Scholar
[20] Babuška, I., Caloz, G., and Osborn, J. E.. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal., 31(4):945981, 1994.CrossRefGoogle Scholar
[21] Babuška, I. and Lipton, R.. Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul., 9:373406, 2011.CrossRefGoogle Scholar
[22] Babuška, I. and Osborn, J. E.. Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal., 20(3):510536, 1983.CrossRefGoogle Scholar
[23] Babuška, I. and Osborn, J. E.. Can a finite element method perform arbitrarily badly? Math. Comp., 69(230):443462, 2000.Google Scholar
[24] Backus, G. E.. Bayesian inference in geomagnetism. Geophys. J., 92(1):125142, 1988.CrossRefGoogle Scholar
[25] Backus, G. E.. Trimming and procrastination as inversion techniques. Phys. Earth Planet. Inter., 98(3):101142, 1996.Google Scholar
[26] Bacry, E., Mallat, S., and Papanicolaou, G.. A wavelet based space-time adaptive numerical method for partial differential equations. RAIRO Modél. Math. Anal. Numér., 26(7):793834, 1992.CrossRefGoogle Scholar
[27] Bacry, E., Mallat, S., and Papanicolaou, G.. A wavelet space-time adaptive scheme for partial differential equations. In Meyer, Y. and Roques, S., editors, Progress in Wavelet Analysis and Applications (Toulouse, 1992), 677682. Frontières, Gif-sur-Yvette, 1993.Google Scholar
[28] Bakhvalov, N. S.. On the approximate evaluation of multiple integrals. Vestnik MGU, Ser. Math. Mech. Astron. Pbuys. Chem., 4:318, 1959. In Russian.Google Scholar
[29] Bakhvalov, N. S.. On the approximate calculation of multiple integrals. J. Complexity, 31(4):502516, 2015.CrossRefGoogle Scholar
[30] Bal, G. and Jing, W.. Corrector theory for MSFEM and HMM in random media. Multiscale Model. Simul., 9(4):15491587, 2011.CrossRefGoogle Scholar
[31] Bank, R. E., Dupont, T. F., and Yserentant, H.. The hierarchical basis multigrid method. Numer. Math., 52(4):427458, 1988.Google Scholar
[32] Barinka, A., Barsch, T., Charton, P., Cohen, A., Dahlke, S., Dahmen, W., and Urban, K.. Adaptive wavelet schemes for elliptic problems—implementation and numerical experiments. SIAM Journal on Scientific Computing, 23(3):910939, 2001.CrossRefGoogle Scholar
[33] Bebendorf, M.. Hierarchical Matrices, volume 63 of Lect. Notes in Computational Science and Engineering. Springer, 2008.Google Scholar
[34] Bebendorf, M.. Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients. Math. Comp., 74(251):11791199 (electronic), 2005.CrossRefGoogle Scholar
[35] Bebendorf, M.. Low-rank approximation of elliptic boundary value problems with high-contrast coefficients. SIAM J. Math. Anal., 48(2):932949, 2016.Google Scholar
[36] Ben Arous, G. and Owhadi, H.. Multiscale homogenization with bounded ratios and anomalous slow diffusion. Comm. Pure Appl. Math., 56(1):80113, 2003.Google Scholar
[37] Bensoussan, A., Lions, J. L., and Papanicolaou, G.. Asymptotic Analysis for Periodic Structure. North-Holland, 1978.Google Scholar
[38] Berlyand, L. and Owhadi, H.. Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast. Arch. Ration. Mech. Anal., 198(2):677721, 2010.Google Scholar
[39] Bertoluzza, S., Maday, Y., and Ravel, J.-C.. A dynamically adaptive wavelet method for solving partial differential equations. Comput. Methods Appl. Mech. Eng., 116(1–4):293299, 1994.Google Scholar
[40] Beylkin, G.. On multiresolution methods in numerical analysis. Doc. Math., Extra, 3:481490, 1998.Google Scholar
[41] Beylkin, G., Coifman, R., and Rokhlin, V.. Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math., 44(2):141183, 1991.Google Scholar
[42] Beylkin, G. and Coult, N.. A multiresolution strategy for reduction of elliptic PDEs and eigenvalue problems. Appl. Comput. Harmon. Anal., 5(2):129155, 1998.Google Scholar
[43] Bezhaev, A. and Vasilenko, V. A.. Variational Theory of Splines. Springer, 2001.CrossRefGoogle Scholar
[44] Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., and Wojtaszczyk, P.. Data assimilation in reduced modeling. SIAM/ASA J. Uncertain. Quant., 5(1):129, 2017.Google Scholar
[45] Bochev, P. B. and Scovel, C.. On quadratic invariants and symplectic structure. BIT Numer. Math., 34(3):337345, 1994.Google Scholar
[46] Bogachev, V. I.. Gaussian Measures, Vol. 62. American Mathematical Soc., 1998.CrossRefGoogle Scholar
[47] Bogachev, V. I.. Measure Theory, volume I. Springer-Verlag, 2007.Google Scholar
[48] Boom, P. D. and Zingg, D. W.. High-order implicit time-marching methods based on generalized summation-by-parts operators. SIAM J. Sci. Comput., 37(6):A2682– A2709, 2015.CrossRefGoogle Scholar
[49] Bourgeat, A. and Piatnitski, A.. Approximations of effective coefficients in stochastic homogenization. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 40:153165, 2004.Google Scholar
[50] Boyd, S. and Vandenberghe, L.. Convex Optimization. Cambridge University Press, 2004.Google Scholar
[51] Brandt, A.. Multi-level adaptive technique (MLAT) for fast numerical solutions to boundary value problems. In Bartlemann, M. et al., editors, Lect. Notes in Physics 1882–89. Springer, 1973.Google Scholar
[52] Branets, L. V., Ghai, S. S., L., L., and Wu, X.-H.. Challenges and technologies in reservoir modeling. Commun. Comput. Phys., 6(1):123, 2009.CrossRefGoogle Scholar
[53] Brenner, S. and Scott, R.. The Mathematical Theory of Finite Element Methods, volume 15. Springer Science & Business Media, 2007.Google Scholar
[54] Brewster, M. E. and Beylkin, G.. A multiresolution strategy for numerical homogenization. Appl. Comput. Harmon. Anal., 2(4):327349, 1995.CrossRefGoogle Scholar
[55] Brezis, H.. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media, 2010.Google Scholar
[56] Briol, F.-X., Oates, C. J., Girolami, M., Osborne, M. A., and Sejdinovic, D.. Probabilistic integration: a role for statisticians in numerical analysis? arXiv:1512.00933, 2015.Google Scholar
[57] Brown, D. L., Gedicke, J., and Peterseim, D.. Numerical homogenization of heterogeneous fractional Laplacians. Multiscale Model. Simul., 16(3):13051332, 2018.CrossRefGoogle Scholar
[58] Budninsky, M., Owhadi, H., and Desbrun, M.. Operator-adapted wavelets for finite-element differential forms. J. Comput. Phys., 388(July): 144177, 2019.CrossRefGoogle Scholar
[59] Bunke, O.. Minimax linear, ridge and shrunken estimators for linear parameters. Mathematische Operationsforschung und Statistik, 6(5):697701, 1975.CrossRefGoogle Scholar
[60] Caffarelli, L. A. and Souganidis, P. E.. A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math., 61(1):117, 2008.CrossRefGoogle Scholar
[61] Cameron, R. H. and Martin, W. T.. An expression for the solution of a class of nonlinear integral equations. Am. J. Math., 66(2):281298, 1944.CrossRefGoogle Scholar
[62] Cameron, R. H. and Martin, W. T.. Transformations of Weiner integrals under translations. Ann. of Math. (2), 45:386396, 1944.Google Scholar
[63] Cameron, R. H. and Martin, W. T.. Transformations of Wiener integrals under a general class of linear transformations. Transactions of the American Mathematical Society, 58(2):184219, 1945.Google Scholar
[64] Carnicer, J. M., Dahmen, W., and Peña, J. M.. Local decomposition of refinable spaces and wavelets. Appl. Comput. Harmon. Anal., 3(2):127153, 1996.Google Scholar
[65] Chiavassa, G. and Liandrat, J.. A fully adaptive wavelet algorithm for parabolic partial differential equations. Appl. Numer. Math., 36(2–3):333358, 2001.Google Scholar
[66] Chkrebtii, O. A., Campbell, D. A., Calderhead, B., and Girolami, M. A.. Bayesian solution uncertainty quantification for differential equations. Bayesian Analysis, 11(4):12391267, 2016.Google Scholar
[67] Chow, E. and Vassilevski, P. S.. Multilevel block factorizations in generalized hierarchical bases. Numer. Linear Algebra Appl., 10(1–2):105127, 2003.Google Scholar
[68] Chu, C.-C., Graham, I. G., and Hou, T. Y.. A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comp., 79:19151955, 2010.CrossRefGoogle Scholar
[69] Ciarlet, P. G.. The Finite Element Method for Elliptic Problems, volume 4 of Studies in Mathematics and Its Applications. North-Holland, 1978.Google Scholar
[70] Cioranescu, D. and Donato, P.. Introduction to Homogenization. Oxford University Press, 2000.Google Scholar
[71] Ph. Clément. Approximation by finite element functions using local regularization. Revue Française d’Automatique, Informatique, Recherche Opérationnelle. Analyse Numérique, 9(2):7784, 1975.CrossRefGoogle Scholar
[72] Cockayne, J., Oates, C., Sullivan, T., and Girolami, M.. Bayesian probabilistic numerical methods. 2017. arXiv:1702.03673.Google Scholar
[73] Cockayne, J., Oates, C. J., Sullivan, T., and Girolami, M.. Probabilistic meshless methods for Bayesian inverse problems, 2016. arXiv:1605.07811.Google Scholar
[74] Cohen, A.. Adaptive methods for PDEs: wavelets or mesh refinement? In Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 607620. Higher Ed. Press, 2002.Google Scholar
[75] Cohen, A., Dahmen, W., and DeVore, R.. Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp., 70(233):2775, 2001.CrossRefGoogle Scholar
[76] Cohen, A., Dahmen, W., and DeVore, R.. Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math., 2(3):203245, 2002.CrossRefGoogle Scholar
[77] Cohen, A., Dahmen, W., and DeVore, R.. Adaptive wavelet techniques in numerical simulation. Encyclopedia of Computational Mechanics. Wiley, 2004.Google Scholar
[78] Cohen, A., Daubechies, I., and Feauveau, J.-C.. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math., 45(5):485560, 1992.CrossRefGoogle Scholar
[79] Cohen, A. and Masson, R.. Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition. Numer. Math., 86(2): 193238, 2000.CrossRefGoogle Scholar
[80] Cohen, M. B., Kyng, R., Miller, G. L., et al. Solving SDD linear systems in nearly m log1/2 n time. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 343352. ACM, 2014.Google Scholar
[81] Cohn, D. L.. Measure Theory. Birkhäuser, 1980.Google Scholar
[82] Coifman, R. R., Meyer, Y., and Wickerhauser, V.. Wavelet analysis and signal processing. In Ruskai, M. B. et al., editors, Wavelets and Their Applications, 153178. Jones and Bartlett, 1992.Google Scholar
[83] Conrad, P. R., Girolami, M., Särkä, S., Stuart, A., and Zygalakis, K.. Probability measures for numerical solutions of differential equations. Statistics and Computing, arXiv:1512.00933, 1–18, 2016.Google Scholar
[84] Conway, J. B.. A Course in Functional Analysis. Springer-Verlag, 1985.Google Scholar
[85] Dahlke, S. and Weinreich, I.. Wavelet-Galerkin methods: an adapted biorthogonal wavelet basis. Constr. Approx., 9(2–3):237262, 1993.Google Scholar
[86] Dahlke, S. and Weinreich, I.. Wavelet bases adapted to pseudodifferential operators. Appl. Comput. Harmon. Anal., 1(3):267283, 1994.Google Scholar
[87] Dahmen, W., Harbrecht, H., and Schneider, R.. Compression techniques for boundary integral equations – asymptotically optimal complexity estimates. SIAM J. Numer. Anal., 43(6):22512271, 2006.Google Scholar
[88] Dahmen, W. and Kunoth, A.. Adaptive wavelet methods for linear-quadratic elliptic control problems: convergence rates. SIAM J. Control Optim., 43(5):16401675, 2005.Google Scholar
[89] Daubechies, I.. The wavelet transform, time-frequency localization and signal analysis. IEEE T. Inform. Theory, 36(5):9611005, 1990.Google Scholar
[90] Daubechies, I.. Ten Lectures on Wavelets. SIAM, 1992.CrossRefGoogle Scholar
[91] De la Madrid, R.. The role of the rigged Hilbert space in quantum mechanics. European Journal of Physics, 26(2):287, 2005.Google Scholar
[92] Dekel, S. and Leviatan, D.. The Bramble–Hilbert lemma for convex domains. SIAM J. Math. Anal., 35(5):12031212, 2004.Google Scholar
[93] Demko, S., Moss, W. F., and Smith, P. W.. Decay rates for inverses of band matrices. Math. Comp., 43(168):491499, 1984.Google Scholar
[94] Devroye, L., Györfi, L., and Lugosi, G.. A Probabilistic Theory of Pattern Recognition, volume 31. Springer Science & Business Media, 2013.Google Scholar
[95] Diaconis, P.. Bayesian numerical analysis. In Gupta, S. S. and Berger, J. O., editors, Statistical Decision Theory and Related Topics, IV, Vol. 1 (West Lafayette, Ind., 1986), 163175. Springer, 1988.Google Scholar
[96] Donoho, D. L.. Statistical estimation and optimal recovery. Ann. Stat., 22(1): 238270, 1994.Google Scholar
[97] Donoho, D. L.. De-noising by soft-thresholding. IEEE T. Inform. Theory, 41(3): 613627, 1995.Google Scholar
[98] Donoho, D. L. and Johnstone, I. M.. Minimax estimation via wavelet shrinkage. Ann. Stat., 26(3):879921, 1998.Google Scholar
[99] Doob, J. L.. Measure Theory, volume 143. Springer Science & Business Media, 2012.Google Scholar
[100] Dorobantu, M. and Engquist, B.. Wavelet-based numerical homogenization. SIAM J. Numer. Anal., 35(2):540559 (electronic), 1998.CrossRefGoogle Scholar
[101] Driscoll, M. F.. The reproducing kernel Hilbert space structure of the sample paths of a Gaussian process. Probability Theory and Related Fields, 26(4):309316, 1973.Google Scholar
[102] Driscoll, M. F.. The signal-noise problem: a solution for the case that signal and noise are Gaussian and independent. J. Appl. Prob., 12:183187, 1975.CrossRefGoogle Scholar
[103] Du, J.. Screening effect, geostatistical. Wiley StatsRef: Statistics Reference Online, 2013.Google Scholar
[104] Duchon, J.. Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. RAIRO Analyse Numerique, 10(R-3):512, 1976.Google Scholar
[105] Duchon, J.. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Schempp, W. and Zeller, K., K., editors, Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), volume 571 of Lect. Notes in Math., 85100. Springer, 1977.CrossRefGoogle Scholar
[106] Duchon, J.. Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-splines. RAIRO Anal. Numér., 12(4):325334, vi, 1978.CrossRefGoogle Scholar
[107] Dupont, T. and Scott, R.. Polynomial approximation of functions in Sobolev spaces. Math. Comput., 34(150):441463, 1980.Google Scholar
[108] , W. E B. Engquist The heterogeneous multiscale methods. Commun. Math. Sci., 1(1):87132, 2003.Google Scholar
[109] T. Li, W. E, and Lu, J.. Localized bases of eigensubspaces and operator compression. PNAS, 107(4):12731278, 2010.Google Scholar
[110] Efendiev, Y., Galvis, J., and Hou, T. Y.. Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys., 251:116135, 2013.Google Scholar
[111] Efendiev, Y., Galvis, J., and Vassilevski, P. S.. Spectral element agglomerate algebraic multigrid methods for elliptic problems with high-contrast coefficients. In Huang, Y., Kornhuber, R., Widlund, O., and Xu, J., editors, Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lect. Notes Comput. Sci. Eng., 407–414. Springer, 2011.Google Scholar
[112] Efendiev, Y., Ginting, V., Hou, T., and Ewing, R.. Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys., 220(1):155174, 2006.CrossRefGoogle Scholar
[113] Efendiev, Y. and Hou, T.. Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math., 57(5–7):577596, 2007.CrossRefGoogle Scholar
[114] Efendiev, Y. and Hou, T. Y.. Multiscale Finite Element Methods: Theory and Applications, volume 4. Springer Science & Business Media, 2009.Google Scholar
[115] Ellam, L., Zabaras, N., and Girolami, M.. A Bayesian approach to multiscale inverse problems with on-the-fly scale determination. J. Comput. Phys., 326:115140, 2016.CrossRefGoogle Scholar
[116] Engquist, B., Holst, H., and Runborg, O.. Multi-scale methods for wave propagation in heterogeneous media. Commun. Math. Sci., 9(1):3356, 2011.CrossRefGoogle Scholar
[117] Engquist, B. and Luo, E.. Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients. SIAM J. Numer. Anal., 34(6):22542273, 1997.Google Scholar
[118] Engquist, B., Osher, S., and Zhong, S.. Fast wavelet based algorithms for linear evolution equations. SIAM J. Comput., 15(4):755775, 1994.Google Scholar
[119] Engquist, B. and Runborg, O.. Wavelet-based numerical homogenization with applications. In Barth, T. J., Chan, T., and Haimes, R., editors, Multiscale and Multiresolution Methods, volume 20 of Lect. Notes Comput. Sci. Eng., 97148. Springer, 2002.Google Scholar
[120] Engquist, B. and Runborg, O.. Wavelet-based numerical homogenization. In Engquist, B., Fokas, A., Hairer, E., and Iserles, A., editors, Highly Qscillatory Problems, volume 366 of London Math. Soc. Lect. Note Ser., 98126. Cambridge University Press, 2009.CrossRefGoogle Scholar
[121] Engquist, B. and Souganidis, P. E.. Asymptotic and numerical homogenization. Acta Numerica, 17:147190, 2008.CrossRefGoogle Scholar
[122] Evans, S. N. and Stark, P. B.. Inverse problems as statistics. Inverse Problems, 18(4):R55, 2002.CrossRefGoogle Scholar
[123] Fan, Y.. Schur complements and its applications to symmetric nonnegative and Z-matrices. Linear Algebra Appl., 353(1–3):289307, 2002.CrossRefGoogle Scholar
[124] Farge, M., Schneider, K., and Kevlahan, N.. Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis. Phys. Fluids, 11(8):21872201, 1999.Google Scholar
[125] Fedorenko, R. P.. A relaxation method of solution of elliptic difference equations. Ž. Vyčisl. Mat. i Mat. Fiz., 1:922927, 1961.Google Scholar
[126] Feischl, M. and Peterseim, D.. Sparse compression of expected solution operators. arXiv:1807.01741, 2018.Google Scholar
[127] Feshchenko, I. S.. On closeness of the sum of n subspaces of a Hilbert space. Ukrainian Mathematical Journal, 1–57, 2012.Google Scholar
[128] Fröhlich, J. and Schneider, K.. An adaptive wavelet Galerkin algorithm for one- and two-dimensional flame computations. European J. Mech. B Fluids, 13(4):439471, 1994.Google Scholar
[129] Furrer, R., Genton, M. G., and Nychka, D.. Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat, 15(3):502523, 2006.CrossRefGoogle Scholar
[130] Gal, S. and Micchelli, C. A.. Optimal sequential and non-sequential procedures for evaluating a functional. Appl. Anal., 10(2):105120, 1980.CrossRefGoogle Scholar
[131] Gallistl, D. and Peterseim, D.. Computation of local and quasi-local effective diffusion tensors in elliptic homogenization. Multiscale Modeling & Simulation, 15(4):15301552, 2017.CrossRefGoogle Scholar
[132] Gallistl, D. and Peterseim, D.. Numerical stochastic homogenization by quasilocal effective diffusion tensors. arXiv:1702.08858, 2017.Google Scholar
[133] Gantumur, T., Harbrecht, H., and Stevenson, R.. An optimal adaptive wavelet method without coarsening of the iterands. Math. Comput., 76(258):615629, 2007.CrossRefGoogle Scholar
[134] Gantumur, T. and Stevenson, R.. Computation of differential operators in wavelet coordinates. Math. Comput., 75(254):697709, 2006.Google Scholar
[135] Gantumur, T. and Stevenson, R. P.. Computation of singular integral operators in wavelet coordinates. Computing, 76(1):77107, 2006.Google Scholar
[136] Gazzola, F., Grunau, H.-C., and Sweers, G.. Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Springer Science & Business Media, 2010.Google Scholar
[137] Gel’fand, I. M. and Vilenkin, Y. N.. Generalized Functions, volume. 4. Academic Press, 1964.Google Scholar
[138] Gines, D., Beylkin, G., and Dunn, J.. LU factorization of non-standard forms and direct multiresolution solvers. Appl. Comput. Harmon. Anal., 5(2):156201, 1998.CrossRefGoogle Scholar
[139] De Giorgi, E.. Sulla convergenza di alcune successioni di integrali del tipo dell’aera. Rendi Conti di Mat., 8:277294, 1975.Google Scholar
[140] Gloria, A.. An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies. Multiscale Model. Simul., 5(3): 9961043, 2006.Google Scholar
[141] Gloria, A.. Reduction of the resonance error – Part 1: approximation of homogenized coefficients. Math. Models Methods Appl. Sci., 21(8):16011630, 2011.Google Scholar
[142] Gloria, A., Neukamm, S., and Otto, F.. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Inventiones Mathematicae, 199(2):455515, 2015.Google Scholar
[143] Gloria, A. and Otto, F.. An optimal error estimate in stochastic homogenization of discrete elliptic equations. Annals of Applied Probability, 22(1):128, 2012.CrossRefGoogle Scholar
[144] Golomb, M. and Weinberger, H.. Optimal approximation and error bounds. In Langer, R. E., editor, On Numerical Approximation: Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, April 21–23, 1958, 117. University of Wisconsin Press, 1959.Google Scholar
[145] Grasedyck, L., Greff, I., and Sauter, S.. The AL basis for the solution of elliptic problems in heterogeneous media. Multiscale Model. Simul., 10(1):245258, 2012.Google Scholar
[146] Greengard, L. and Rokhlin, V.. A fast algorithm for particle simulations. J. Comput. Phys., 73(2):325348, 1987.Google Scholar
[147] Griebel, M. and Oswald, P.. On the abstract theory of additive and multiplicative Schwarz algorithms. Numerische Mathematik, 70(2):163180, 1995.CrossRefGoogle Scholar
[148] Gross, L.. Abstract Wiener spaces. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, Part 1. Regents of the University of California, 1967.Google Scholar
[149] Hackbusch, W.. A fast iterative method for solving Poisson’s equation in a general region. In Numerical Treatment of Differential Equations (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1976), volume 631 of Lect. Notes in Math., 5162. Springer, 1978.CrossRefGoogle Scholar
[150] Hackbusch, W.. Multigrid Methods and Applications, volume 4 of Springer Series in Computational Mathematics. Springer-Verlag, 1985.Google Scholar
[151] Hackbusch, W., Grasedyck, L., and Börm, S.. An introduction to hierarchical matrices. In M. Krbec and J. Kuben, editors, Proceedings of EQUADIFF, 10 (Prague, 2001), volume 127:2, 229–241, 2002.CrossRefGoogle Scholar
[152] Hairer, E., Lubich, C., and Wanner, G.. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations., volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, second edition, 2006.Google Scholar
[153] Halko, N., Martinsson, P. G., and Tropp, J. A.. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., 53(2):217288, 2011.Google Scholar
[154] Halmos, P. R.. A Hilbert Space Problem Book, volume 19. Springer-Verlag, 1982.Google Scholar
[155] Harbrecht, H. and Schneider, R.. Wavelet Galerkin schemes for boundary integral equations – implementation and quadrature. SIAM J. Comput.,, 27(4):13471370, 2006.Google Scholar
[156] Harder, R. L. and Desmarais, R. N.. Interpolation using surface splines. J. Aircraft, 9:189191, 1972.Google Scholar
[157] Hennig, P.. Probabilistic interpretation of linear solvers. SIAM J. Optim., 25(1): 234260, 2015.CrossRefGoogle Scholar
[158] Hennig, P., Osborne, M. A., and Girolami, M.. Probabilistic numerics and uncertainty in computations. Proc. R. Soc. A., 471(2179):20150142, 2015.CrossRefGoogle ScholarPubMed
[159] Hestenes, M. R. and Stiefel, E.. Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards, 49:409–436 (1953), 1952.Google Scholar
[160] Ho, K. L. and Ying, L.. Hierarchical interpolative factorization for elliptic operators: differential equations. Comm. Pure Appl. Math., 69(8):14151451, 2016.Google Scholar
[161] Holmström, M. and Waldén, J.. Adaptive wavelet methods for hyperbolic PDEs. J. Sci. Comput., 13(1):1949, 1998.Google Scholar
[162] Horn, R. A. and Johnson, C. R.. Topics in Matrix Analysis. Cambridge University Press, 1991.CrossRefGoogle Scholar
[163] Horn, R.A. and Zhang, F.. Basic properties of the Schur complement. In Zhang, F., editor, The Schur Complement and Its Applications, volume 4, 17–46. Springer Science & Business Media, 2006.Google Scholar
[164] Hou, T. H. and Liu, P.. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficient. Discrete and Continuous Dynamical Systems, 36(8):44514476, 2016.CrossRefGoogle Scholar
[165] Hou, T. Y., Huang, D., Lam, K. C., and Zhang, P.. An adaptive fast solver for a general class of positive definite matrices via energy decomposition. Multiscale Modeling & Simulation, 16(2):615678, 2018.CrossRefGoogle Scholar
[166] Hou, T. Y. and Wu, X. H.. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134(1):169189, 1997.Google Scholar
[167] Hou, T. Y., Wu, X.-H., and Cai, Z.. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp., 68(227):913943, 1999.CrossRefGoogle Scholar
[168] Hou, T. Y. and Zhang, P.. Sparse operator compression of higher-order elliptic operators with rough coefficients. Research in the Mathematical Sciences, 4(1):24, 2017.Google Scholar
[169] Hughes, T. J. R., Feijóo, G. R., Mazzei, L., and Quincy, J.-B.. The variational multiscale methoda paradigm for computational mechanics. Comput Methods Appl Mech Eng., 166(1–2):324, 1998.CrossRefGoogle Scholar
[170] Jaffard, S.. Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(5):461476, 1990.CrossRefGoogle Scholar
[171] Janson, S.. Gaussian Hilbert Spaces, volume 129. Cambridge University Press, 1997.Google Scholar
[172] Jawerth, B. and Sweldens, W.. Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations. 259–273 of NASA. Langley Research Center, the Sixth Copper Mountain Conference on Multigrid Methods. NASA, 1993.Google Scholar
[173] Jikov, V. V., Kozlov, S. M., and Oleinik, O. A.. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, 1991.Google Scholar
[174] John, V.. Numerical methods for partial differential equations. www.wias-berlin.de/people/john/LEHRE/NUM_PDE_FUB/num_pde_fub.pdf, 2013.Google Scholar
[175] Kadane, J and Wasilkowski, G. Average case ϵ-complexity in computer science: a Bayesian view. In Bernardo, J. M., Degroot, M. H., Lindley, D. V., and Smith, A. F. M., editors, BAYESIAN. STATISTICS 2. Proceedings of the Second Valencia. International Meeting. September 6/10, 1983, 361374. North-Holland, 1985.Google Scholar
[176] Kallenberg, O.. Foundations of Modern Probability. Springer Science & Business Media, 2006.Google Scholar
[177] Kallianpur, G.. Zero-one laws for Gaussian processes. Transactions of the American Mathematical Society, 149(1):199211, 1970.Google Scholar
[178] Kallianpur, G.. Abstract Wiener processes and their reproducing kernel Hilbert spaces. Probability Theory and Related Fields, 17(2):113123, 1971.Google Scholar
[179] Kelner, J. A., Orecchia, L., Sidford, A., and Zhu, Z. A.. A simple, combinatorial algorithm for solving SDD systems in nearly-linear time. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 911920. ACM, 2013.CrossRefGoogle Scholar
[180] Kimeldorf, G. S. and Wahba, G.. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Statist., 41:495502, 1970.CrossRefGoogle Scholar
[181] Kohn, W.. Analytic properties of Bloch waves and Wannier functions. Phys. Rev., 115(4):809, 1959.Google Scholar
[182] Kornhuber, R., Peterseim, D., and Yserentant, H.. An analysis of a class of variational multiscale methods based on subspace decomposition. Math. Comp., 87(314): 27652774, 2018.Google Scholar
[183] Kornhuber, R. and Yserentant, H.. Numerical homogenization of elliptic multiscale problems by subspace decomposition. Multiscale Model. Simul., 14(3):10171036, 2016.CrossRefGoogle Scholar
[184] Koutis, I., Miller, G. L., and Peng, R.. Approaching optimality for solving SDD linear systems. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, 235244. IEEE, 2010.CrossRefGoogle Scholar
[185] Kozlov, S. M.. The averaging of random operators. Mat. Sb. (N.S.), 109(151)(2): 188202, 327, 1979.Google Scholar
[186] Kruskal, W.. When are Gauss–Markov and least squares estimators identical? A coordinate-free approach. Ann. Math. Statist., 39(1):7075, 1968.CrossRefGoogle Scholar
[187] Kuelbs, J.. Abstract Wiener spaces and applications to analysis. Pacific J. Math, 31(2):433450, 1969.CrossRefGoogle Scholar
[188] Kuelbs, J.. Expansions of vectors in a Banach space related to Gaussian measures. Proceedings of the American Mathematical Society, 27(2):364370, 1971.Google Scholar
[189] Kuelbs, J., Larkin, F. M., and Williamson, J. A.. Weak probability distributions on reproducing kernel Hilbert spaces. Rocky Mt. J Math., 2(3):369378, 1972.CrossRefGoogle Scholar
[190] Kuks, J. A. and Olman, W.. A minimax linear estimation of regression coefficients (ii). Iswestija Akademija Nauk Estonskoj SSR, 20:480482, 1971.Google Scholar
[191] Kuks, J. A. and Olman, W.. Minimax linear estimation of regression coefficients. Iswestija Akademija Nauk Estonskoj SSR, 21:6672, 1972.Google Scholar
[192] Kyng, R., Lee, Y. T., Peng, R., Sachdeva, S., and Spielman, D. A.. Sparsified Cholesky and multigrid solvers for connection Laplacians. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, 842850. ACM, 2016.Google Scholar
[193] Kyng, R. and Sachdeva, S.. Approximate Gaussian elimination for Laplacians-fast, sparse, and simple. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, 573582. IEEE, 2016.Google Scholar
[194] Kyng, R. and Zhang, P.. Hardness results for structured linear systems. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 684695. IEEE, 2017.Google Scholar
[195] Larkin, F. M.. Gaussian measure in Hilbert space and applications in numerical analysis. Rocky Mt. J Math., 2(3): 379422, 1972.CrossRefGoogle Scholar
[196] Läuter, H.. A minimax linear estimator for linear parameters under restrictions in form of inequalities. Mathematische Operationsforschung und Statistik, 6(5): 689695, 1975.CrossRefGoogle Scholar
[197] Lax, P. D.. Functional Analysis. Wiley-Interscience, 2002.Google Scholar
[198] Le Cam, L.. An extension of Wald’s theory of statistical decision functions. Ann. Math. Statist., 26:6981, 1955.Google Scholar
[199] Le Cam, L.. Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, 1986.CrossRefGoogle Scholar
[200] Lee, D.. Approximation of linear operators on a Wiener space. Rocky Mt. J Math., 16(4):641659, 1986.Google Scholar
[201] Lee, D. and Wasilkowski, G. W.. Approximation of linear functionals on a Banach space with a Gaussian measure. J. Complexity, 2(1):1243, 1986.Google Scholar
[202] Lehto, O.. Some remarks on the kernel function in Hilbert function space. Ann. Acad. Sci. Fenn. Ser. A I, 109:6, 1952.Google Scholar
[203] Li, K.-C.. Minimaxity of the method of regularization of stochastic processes. Ann. Stat., 10(3): 937942, 1982.Google Scholar
[204] Lounsbery, M., DeRose, T. D., and Warren, J.. Multiresolution analysis for surfaces of arbitrary topological type. ACM Transactions on Graphics (TOG), 16(1):3473, 1997.CrossRefGoogle Scholar
[205] Luenberger, D. G.. Optimization by Vector Space Methods. John Wiley & Sons, 1969.Google Scholar
[206] Lukić, M. and Beder, J.. Stochastic processes with sample paths in reproducing kernel Hilbert spaces. Transactions of the American Mathematical Society, 353(10): 39453969, 2001.CrossRefGoogle Scholar
[207] Mallat, S. G.. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674693, 1989.CrossRefGoogle Scholar
[208] Målqvist, A. and Peterseim, D.. Localization of elliptic multiscale problems. Math. Comput., 83(290):25832603, 2014.Google Scholar
[209] Mandel, J., Brezina, M., and Vaněk, P.. Energy optimization of algebraic multigrid bases. Computing, 62(3):205228, 1999.CrossRefGoogle Scholar
[210] Mangasarian, O. L.. Nonlinear Programming. SIAM, 1994.Google Scholar
[211] Maniglia, S. and Rhandi, A.. Gaussian measures on separable Hilbert spaces and applications. Quaderni di Matematica, 2004(1), 2004.Google Scholar
[212] Martinsson, P.-G. and Rokhlin, V.. A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys., 205(1):123, 2005.CrossRefGoogle Scholar
[213] Marzari, N. and Vanderbilt, D.. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev., 56(20):12847, 1997.Google Scholar
[214] Melenk, J. M.. On n-widths for elliptic problems. J. Math. Anal. Appl., 247(1): 272289, 2000.CrossRefGoogle Scholar
[215] Meyer, Y.. Wavelets and Operators, volume 1. Cambridge University Press, 1995.Google Scholar
[216] Micchelli, C. A.. Optimal Estimation of Linear Functionals. IBM Thomas J. Watson Research Division, 1975.Google Scholar
[217] Micchelli, C. A.. Orthogonal projections are optimal algorithms. J. Approx. Theory, 40(2):101110, 1984.Google Scholar
[218] Micchelli, C. A. and Rivlin, T. J.. A survey of optimal recovery. In Micchelli, C. A. and Rivlin, T. J., editors, Optimal Estimation in Approximation Theory, 154. Springer, 1977.Google Scholar
[219] Micchelli, C. A. and Rivlin, T. J.. Lectures on Optimal Recovery. Springer, 1985.Google Scholar
[220] Milton, G. W.. The Theory of Composites, volume 6 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2002.Google Scholar
[221] Monasse, P. and Perrier, V.. Orthonormal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal., 29(4):10401065, 1998.Google Scholar
[222] Murat, F. and Tartar, L.. H-convergence. Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger, 1978.Google Scholar
[223] Nash, J.. Non-cooperative games. Ann. of Math. (2), 54:286295, 1951.CrossRefGoogle Scholar
[224] Nemirovsky, A. S.. Information-based complexity of linear operator equations. J. Complexity, 8(2):153175, 1992.Google Scholar
[225] Nguetseng, G.. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 21:608623, 1990.Google Scholar
[226] Nikolsky, S. M.. A Course Of Mathematical Analysis, volume 1. MIR Publishers, 1977.Google Scholar
[227] Nolen, J., Papanicolaou, G., and Pironneau, O.. A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul., 7(1):171196, 2008.Google Scholar
[228] Novak, E.. Quadrature formulas for convex classes of functions. In H. Brass and G. Hämmerlin, editors, Numerical Integration IV, 283296. Springer, 1993.CrossRefGoogle Scholar
[229] Novak, E. and Woźniakowski, H.. Tractability of Multivariate Problems. Vol. 1: Linear Information. Volume 6 of EMS Tracts in Mathematics. Eur. Math. Soc., Zürich, 2008.Google Scholar
[230] Oates, C. J., Cockayne, J., Aykroyd, R. G., and Girolami, M.. Bayesian probabilistic numerical methods in time-dependent state estimation for industrial hydrocyclone equipment. Journal of the American Statistical Association, (just-accepted):1–27, 2019.CrossRefGoogle Scholar
[231] O’Hagan, A.. Bayesian quadrature. University of Warwick, Dept. of Statistics Technical Report, 1985.Google Scholar
[232] O’Hagan, A.. Bayes–Hermite quadrature. J. Statist. Plann. Inference, 29(3):245260, 1991.Google Scholar
[233] O’Hagan, A.. Some Bayesian numerical analysis. In Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M., editors, Bayesian Statistics, 4 (Peñíscola, 1991), 345363. Oxford University Press, 1992.Google Scholar
[234] O’Sullivan, F.. A statistical perspective on ill-posed inverse problems. Statistical Science, 1(4):502518, 1986.Google Scholar
[235] Owhadi, H.. Anomalous slow diffusion from perpetual homogenization. Ann. Probab., 31(4):19351969, 2003.Google Scholar
[236] Owhadi, H.. Approximation of the effective conductivity of ergodic media by periodization. Probability Theory and Related Fields, 125(2):225258, 2003.Google Scholar
[237] Owhadi, H.. Averaging versus chaos in turbulent transport? Comm. Math. Phys., 247(3):553599, 2004.Google Scholar
[238] Owhadi, H.. Bayesian numerical homogenization. Multiscale Model. Simul., 13(3):812828, 2015.Google Scholar
[239] Owhadi, H.. Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games. SIAM Rev., 59(1):99149, 2017.CrossRefGoogle Scholar
[240] Owhadi, H. and Scovel, C.. Separability of reproducing kernel spaces. Proceedings of the American Mathematical Society, 145(5):21312138, 2017.Google Scholar
[241] Owhadi, H. and Scovel, C.. Towards Machine Wald. In Ghanem, R., Higdon, D., and Owhadi, H., editors, Handbook of Uncertainty Quantification, 157191. Springer International Publishing, 2017.Google Scholar
[242] Owhadi, H. and Scovel, C.. Universal scalable robust solvers from computational information games and fast eigenspace adapted multiresolution analysis. arXiv:1703.10761, 2017.Google Scholar
[243] Owhadi, H. and Scovel, C.. Conditioning Gaussian measure on Hilbert space. Journal of Mathematical and Statistical Analysis, 1(1):205, 2018.Google Scholar
[244] Owhadi, H. and Zhang, L.. Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal., 46(1):136, 2007.CrossRefGoogle Scholar
[245] Owhadi, H. and Zhang, L.. Metric-based upscaling. Comm. Pure Appl. Math., 60(5):675723, 2007.Google Scholar
[246] Owhadi, H. and Zhang, L.. Homogenization of the acoustic wave equation with a continuum of scales. Comput Methods Appl Mech Eng., 198(3–4):397406, 2008.CrossRefGoogle Scholar
[247] Owhadi, H. and Zhang, L.. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. SIAM Multiscale Model. Simul., 9:13731398, 2011.Google Scholar
[248] Owhadi, H. and Zhang, L.. Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients. J. Comput. Phys., 347:99128, 2017.Google Scholar
[249] Owhadi, H., Zhang, L., and Berlyand, L.. Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization. ESAIM Math. Model. Numer. Anal., 48(2):517552, 2014.CrossRefGoogle Scholar
[250] Ozoliņš, V., Lai, R., Caflisch, R., and Osher, S.. Compressed modes for variational problems in mathematics and physics. PNAS, 110(46):1836818373, 2013.Google Scholar
[251] Packel, E. W.. Linear problems (with extended range) have linear optimal algorithms. Aequationes Mathematicae, 31(1):1825, 1986.Google Scholar
[252] Packel, E. W.. The algorithm designer versus nature: a game-theoretic approach to information-based complexity. J. Complexity, 3(3):244257, 1987.Google Scholar
[253] Palasti, I. and Renyi, A.. On interpolation theory and the theory of games. MTA Mat. Kat. Int. Kozl, 1:529540, 1956.Google Scholar
[254] Papanicolaou, G. C. and Varadhan, S. R. S.. Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, 835–873. North-Holland, 1981.Google Scholar
[255] Parzen, E.. Regression analysis of continuous parameter time series. In Proceedings of the Fourth Berkeley Symposion on Mathematical Statistics and Probability, volume 1, 469–489, 1961.Google Scholar
[256] Perdikaris, P., Venturi, D., and Karniadakis, G. E.. Multifidelity information fusion algorithms for high-dimensional systems and massive data sets. SIAM J. Comput., 38(4):B521B538, 2016.Google Scholar
[257] Poincaré, H.. Calcul des probabilités. Georges Carrés, Paris, 1896.Google Scholar
[258] Pollard, D.. Empirical Processes: Theory and Applications. NSF-CBMS Regional Conference Series Probability and Statistics. Institute of Mathematical Statistics and the American Statistical Association, 1990.Google Scholar
[259] Raissi, M., Perdikaris, P., and Karniadakis, G. E.. Inferring solutions of differential equations using noisy multi-fidelity data. J. Comput. Phys., 335:736746, 2017.Google Scholar
[260] Rao, C. R.. Estimation of parameters in a linear model. Ann. Stat., 4(6):10231037, 1976.Google Scholar
[261] Reed, M. and Simon, B.. Methods of Modern Mathematical Physics, volume 1. Academic Press, 1980.Google Scholar
[262] Ritter, K.. Approximation and optimization on the Wiener space. J. Complexity, 6(4):337364, 1990.CrossRefGoogle Scholar
[263] Ruge, J. W. and Stüben, K.. Algebraic multigrid. In McCormick, S. F., editor, Multigrid Methods, volume 3 of Frontiers Appl. Math., 73130. SIAM, 1987.Google Scholar
[264] Sacks, J. and Ylvisaker, D.. Linear estimation for approximately linear models. Ann. Stat., 6(5):11221137, 1978.Google Scholar
[265] Sard, A.. Best approximate integration formulas: best approximation formulas. Am. J. Math., 71(1):8091, 1949.CrossRefGoogle Scholar
[266] Sard, A.. Linear Approximation, volume 9. American Mathematical Society, 1963.Google Scholar
[267] Sard, A.. Optimal approximation. J. Funct., 1(2):222244, 1967.Google Scholar
[268] Schaefer, H. H.. Topological Vector Spaces. Springer, 1971.Google Scholar
[269] Schäfer, F.. Personal communication. 2017.Google Scholar
[270] Schäfer, F., Sullivan, T. J., and Owhadi, H.. Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity. arXiv:1706.02205, 2017.Google Scholar
[271] Schmitz, P. G. and Ying, L.. A fast direct solver for elliptic problems on general meshes in 2d. J. Comput. Phys., 231(4):13141338, 2012.Google Scholar
[272] Schober, M., Duvenaud, D. K., and Hennig, P.. Probabilistic ODE solvers with Runge–Kutta means. In Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., and Weinberger, K.Q., editors, Advances in Neural Information Processing Systems 27, 739–747. Curran Associates, Inc., 2014.Google Scholar
[273] Schoenberg, I. J.. Spline interpolation and best quadrature formulae. B. Am. Math. Soc., 70(1):143148, 1964.Google Scholar
[274] Schwab, C. and Stevenson, R.. Adaptive wavelet algorithms for elliptic PDE’s on product domains. Math. Comp., 77(261):7192, 2008.CrossRefGoogle Scholar
[275] Segal, I. E.. Tensor algebras over Hilbert spaces. I. Transactions of the American Mathematical Society, 81(1):106134, 1956.Google Scholar
[276] Segal, I. E.. Distributions in Hilbert space and canonical systems of operators. Transactions of the American Mathematical Society, 88(1):1241, 1958.Google Scholar
[277] Sendov, Bl.. Adapted multiresolution analysis and wavelets. In Leinder, L., editor, Functions, Series, Operators (Budapest, 1999), 2338. János Bolyai Math. Soc., 2002.Google Scholar
[278] Shaw, J. E. H.. A quasirandom approach to integration in Bayesian statistics. Ann. Statist., 16(2):895914, 1988.CrossRefGoogle Scholar
[279] Shewchuk, J. R.. An introduction to the conjugate gradient method without the agonizing pain. Technical report, Carnegie Mellon University, 1994.Google Scholar
[280] Sion, M.. On general minimax theorems. Pacific J. Math, 8(1):171176, 1958.Google Scholar
[281] Skilling, J.. Bayesian solution of ordinary differential equations. In Smith, C. R., Erickson, G. J., and Neudorfer, P. O., editors, Maximum Entropy and Bayesian Methods, 2337. Springer, 1992.CrossRefGoogle Scholar
[282] Smale, S.. On the efficiency of algorithms of analysis. Bulletin (New Series) of the American Mathematical Society, 13(2):87121, 1985.Google Scholar
[283] Spagnolo, S.. Convergence in energy for elliptic operators. In Hubbard, B., editor, Numerical Solutions of Partial Differential Equations III Synspade 1975, 468498. Academic Press 1976.Google Scholar
[284] Speckman, P.. Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Stat., 13(3):970983, 1985.CrossRefGoogle Scholar
[285] Spielman, D. A. and Teng, S.-H.. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, 8190. ACM, 2004.Google Scholar
[286] Spielman, D. A. and Teng, S.-H.. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J. Matrix Analysis Applications, 35(3):835885, 2014.Google Scholar
[287] Stampacchia, G.. Èquations elliptiques du second ordre à coefficients discontinus. Séminaire Jean Leray no 3 (1963–1964), 1–77, 1963–1964.Google Scholar
[288] Stein, M. L.. The screening effect in Kriging. Ann. Stat., 30(1):298323, 2002.CrossRefGoogle Scholar
[289] Stein, M. L.. 2010 Rietz lecture: when does the screening effect hold? Ann. Stat., 39(6):27952819, 2011.Google Scholar
[290] Steinwart, I. and Christmann, A.. Support Vector Machines. Springer Science & Business Media, 2008.Google Scholar
[291] Stevenson, R.. Adaptive wavelet methods for solving operator equations: an overview. In DeVore, R. A. and Kunoth, A., editors, Multiscale, Nonlinear and Adaptive Approximation, 543597. Springer, 2009.Google Scholar
[292] Strasser, H.. Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory, volume 7. Walter de Gruyter, 1985.Google Scholar
[293] Stuart, A. M.. Inverse problems: a Bayesian perspective. Acta Numerica, 19:451– 559, 2010.Google Scholar
[294] Stüben, K.. A review of algebraic multigrid. J Comput Appl Math., 128(1–2, 281–309), 2001.CrossRefGoogle Scholar
[295] Sudarshan, R.. Operator-Adapted Finite Element Wavelets: Theory and Applications to A Posteriori Error Estimation and Adaptive Computational modeling. ProQuest LLC, 2005. Thesis (Ph.D.), Massachusetts Institute of Technology.Google Scholar
[296] Sul’din, A. V.. Wiener measure and its applications to approximation methods. I. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, (6):145158, 1959.Google Scholar
[297] Sul’din, A. V. . Wiener measure and its applications to approximation methods. II. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, (5):165179, 1960.Google Scholar
[298] Sweldens, W.. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal., 29(2):511546, 1998.CrossRefGoogle Scholar
[299] Symes, W.. Transfer of approximation and numerical homogenization of hyperbolic boundary value problems with a continuum of scales. TR12–20 Rice Tech Report, 2012.Google Scholar
[300] Tarantola, A.. Inverse Problem Theory and Methods for Model Parameter Estimation, volume 89. SIAM, 2005.CrossRefGoogle Scholar
[301] Tartar, L.. Cours Peccot au Collège de France. Unpublished, 1977.Google Scholar
[302] Tartar, L.. Compensated compactness and applications to partial differential equations. In Knops, R. J., editor, Nonlinear Analysis and Mechanics: Herriot–Watt Symposium IV. 136212. Pitman Press, 1979.Google Scholar
[303] Tartar, L.. The General Theory of Homogenization: A Personalized Introduction, volume 7. Springer Science & Business Media, 2009.Google Scholar
[304] Tenorio, L.. Statistical regularization of inverse problems. SIAM Rev., 43(2):347366, 2001.Google Scholar
[305] Traub, J. F., Wasilkowski, G. W., and Woźniakowski, H.. Average case optimality for linear problems. Theoretical Computer Science, 29(1–2):125, 1984.Google Scholar
[306] Traub, J. F., Wasilkowski, G. W., and Woźniakowski, H.. Information-Based Complexity. Computer Science and Scientific Computing. Academic Press, Inc., 1988. With contributions by A. G. Werschulz and T. Boult.Google Scholar
[307] Traub, J. F. and Wozniakowski, H.. A General Theory of Optimal Algorithms. Academic Press, 1980.Google Scholar
[308] Traub, J. F. and Woźniakowski, H.. Information and computation. In Yovits, Y. C., editor, Advances in Computers, volume 23, 3592. Elsevier, 1984.Google Scholar
[309] Tyrtyshnikov, E.. Mosaic-skeleton approximations. Calcolo, 33(1):4757, 1996.Google Scholar
[310] Unser, M. and Tafti, P. D.. An Introduction to Sparse Stochastic Processes. Cambridge University Press, 2014.Google Scholar
[311] Vakhania, N., Tarieladze, V., and Chobanyan, S.. Probability Distributions on Banach Spaces, volume 14. Springer Science & Business Media, 1987.Google Scholar
[312] Van der Linde, A.. Splines from a Bayesian point of view. Test, 4(1):6381, 1995.Google Scholar
[313] Vasilyev, O. V. and Paolucci, S.. A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain. J. Comput. Phys., 125(2):498512, 1996.Google Scholar
[314] Vassilevski, P. S.. Multilevel preconditioning matrices and multigrid V-cycle methods. In Hackbusch, W., editor, Robust Multi-Grid Nethods (Kiel, 1988), volume 23 of Notes Numer. Fluid Mech., 200–208. Vieweg, 1989.Google Scholar
[315] Vassilevski, P. S.. On two ways of stabilizing the hierarchical basis multilevel methods. SIAM Rev., 39(1):1853, 1997.Google Scholar
[316] Vassilevski, P. S.. General constrained energy minimization interpolation mappings for AMG. SIAM J. Sci. Comput., 32(1):113, 2010.Google Scholar
[317] Vassilevski, P. S. and Wang, J.. Stabilizing the hierarchical basis by approximate wavelets. I. Theory. Numer. Linear Algebra Appl., 4(2):103126, 1997.3.0.CO;2-J>CrossRefGoogle Scholar
[318] Vassilevski, P. S. and Wang, J.. Stabilizing the hierarchical basis by approximate wavelets. II. Implementation and numerical results. SIAM J. Sci. Comput., 20(2):490514 (electronic), 1998.Google Scholar
[319] Verfürth, R.. A note on polynomial approximation in Sobolev spaces. ESAIM: Mathematical Modelling and Numerical Analysis, 33(4):715719, 1999.CrossRefGoogle Scholar
[320] von Neumann, J.. Zur Theorie der Gesellschaftsspiele. Math. Ann., 100(1):295320, 1928.Google Scholar
[321] von Neumann, J. and Morgenstern, O.. Theory of Games and Economic Behavior. Princeton University Press, 1944.Google Scholar
[322] Wald, A.. Statistical decision functions which minimize the maximum risk. Ann. of Math. (2), 46:265280, 1945.Google Scholar
[323] Wan, W. L., Chan, T. F., and Smith, B.. An energy-minimizing interpolation for robust multigrid methods. SIAM J. Sci. Comput., 21(4):16321649, 1999/2000.Google Scholar
[324] Wang, X.. Transfer-of-Approximation Approaches for Subgrid Modeling. Ph.D. thesis, Rice University, 2012.Google Scholar
[325] Wannier, G. H.. Dynamics of band electrons in electric and magnetic fields. Reviews of Modern Physics, 34(4):645, 1962.Google Scholar
[326] Wasilkowski, G. W.. Local average error. Columbia University Technical Report CUCS-70-83, 1983.Google Scholar
[327] Wasilkowski, G. W.. Optimal algorithms for linear problems with Gaussian measures. Rocky Mt. J Math., 16(4):727749, 1986.Google Scholar
[328] Wasilkowski, G. W.. Integration and approximation of multivariate functions: Average case complexity with isotropic Wiener measure. B. Am. Math. Soc., 28(2): 308314, 1993.CrossRefGoogle Scholar
[329] Wasilkowski, G. W. and Woźniakowski, H.. Can adaption help on the average? Numerische Mathematik, 44(2):169190, 1984.Google Scholar
[330] Wasilkowski, G. W. and Woźniakowski, H.. Average case optimal algorithms in Hilbert spaces. J. Approx. Theory, 47(1):1725, 1986.Google Scholar
[331] White, C. D. and Horne, R. N.. Computing absolute transmissibility in the presence of finescale heterogeneity. SPE Symposium on Reservoir Simulation, 16011. Society of Petroleum Engineers, 1987.Google Scholar
[332] Wolpert, D. H.. The lack of a priori distinctions between learning algorithms. Neural Computation, 8(7):13411390, 1996.Google Scholar
[333] Wolpert, D. H. and Macready, W. G.. No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, 1(1):6782, 1997.CrossRefGoogle Scholar
[334] Woźniakowski, H.. Probabilistic setting of information-based complexity. J. Complexity, 2(3):255269, 1986.Google Scholar
[335] Woźniakowski, H.. What is information-based complexity? In Essays on the Complexity of Continuous Problems, 8995. Eur. Math. Soc., Zürich, 2009.Google Scholar
[336] Xu, J.. Iterative methods by space decomposition and subspace correction. SIAM Rev., 34(4):581613, 1992.Google Scholar
[337] Xu, J. and Zhu, Y.. Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci., 18(1): 77105, 2008.Google Scholar
[338] Xu, J. and Zikatanov, L.. On an energy minimizing basis for algebraic multigrid methods. Comput. Vis. Sci., 7(3–4):121127, 2004.Google Scholar
[339] Yavneh, I.. Why multigrid methods are so efficient. Computing in Science and Eng., 8(6):1222, 2006.Google Scholar
[340] Yin, P. and Liandrat, J.. Coupling wavelets/vaguelets and smooth fictitious domain methods for elliptic problems: the univariate case. Comp. Appl. Math., 35(2): 351369, 2016.Google Scholar
[341] Ying, L., Biros, G., and Zorin, D.. A high-order 3d boundary integral equation solver for elliptic PDEs in smooth domains. J. Comput. Phys., 219(1):247275, 2006.CrossRefGoogle Scholar
[342] Yoo, R. and Owhadi, H.. De-noising by thresholding operator adapted wavelets. arXiv:1805.10736, to appear in Statistics and Computing, 2018.Google Scholar
[343] Yosida, K.. Functional Analysis. Springer-Verlag, 1980.Google Scholar
[344] Yserentant, H.. On the multilevel splitting of finite element spaces. Numer. Math., 49(4):379412, 1986.CrossRefGoogle Scholar
[345] Zhikov, V. V., Kozlov, S. M., Oleinik, O.A., and Ngoan, Kha T’en. Averaging and g-convergence of differential operators. Russian Math. Surveys, 34(5):69147, 1979.CrossRefGoogle Scholar