Skip to main content

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

Abstract

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers and contrasts the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. I. Babus̆ka, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45 (2007), 1005–1034.

    MathSciNet  Article  MATH  Google Scholar 

  2. I. Babus̆ka, R. Tempone, and G.E. Zouraris, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal. 42 (2004), 800–825.

    MathSciNet  Article  MATH  Google Scholar 

  3. J. Baldeaux and J. Dick, QMC rules of arbitrary high order: reproducing kernel Hilbert space approach, Constr. Approx. 30 (2009), 495–527.

    MathSciNet  Article  MATH  Google Scholar 

  4. J. Baldeaux, J. Dick, J. Greslehner, and F. Pillichshammer, Construction algorithms for higher order polynomial lattice rules, J. Complexity 27 (2011), 281–299.

    MathSciNet  Article  MATH  Google Scholar 

  5. J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens, and F. Pillichshammer, Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules, Numer. Algorithms 59 (2012), 403–431.

    MathSciNet  Article  MATH  Google Scholar 

  6. A. Barth, Ch. Schwab, and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math. 119 (2011), 123–161.

    MathSciNet  Article  MATH  Google Scholar 

  7. V. I. Bogachev, Gaussian Measures, AMS Monographs Vol. 62, American Mathematical Society, R.I., USA, 1998.

    Book  Google Scholar 

  8. H. Bungartz and M. Griebel, Sparse grids, Acta Numer. 13 (2004), 147–269.

    MathSciNet  Article  MATH  Google Scholar 

  9. R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1 (1997), 27–46.

    Article  Google Scholar 

  10. J. Charrier, Strong and weak error estimates for elliptic partial differential equations with random coefficients, SIAM J. Numer. Anal. 50 (2012), 216–246.

    MathSciNet  Article  MATH  Google Scholar 

  11. J. Charrier, R. Scheichl, and A. L. Teckentrup, Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods, SIAM J. Numer. Anal. 51 (2013), 322–352.

    MathSciNet  Article  MATH  Google Scholar 

  12. K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup, Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients, Computing and Visualization in Science Science 14 (2011), 3–15.

    MathSciNet  Article  MATH  Google Scholar 

  13. A. Cohen, A. Chkifa, and Ch. Schwab, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, Journ. Math. Pures et Appliquees 103 (2015), 400–428.

    MathSciNet  Article  MATH  Google Scholar 

  14. A. Cohen and R. DeVore, Approximation of high-dimensional parametric PDEs, Acta Numer. 24 (2015), 1–159.

    MathSciNet  Article  MATH  Google Scholar 

  15. A. Cohen, R. DeVore, and Ch. Schwab, Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic sPDEs, Found. Comp. Math. 10 (2010), 615–646.

    MathSciNet  Article  MATH  Google Scholar 

  16. N. Collier, A.-L. Haji-Ali, F. Nobile, E. von Schwerin, and R. Tempone, A continuation multilevel Monte Carlo algorithm, BIT, 55 (2015), 399–432.

    MathSciNet  Article  MATH  Google Scholar 

  17. R. Cools, F. Y. Kuo, and D. Nuyens, Constructing embedded lattice rules for multivariate integration, SIAM J. Sci. Comput. 28 (2006), 2162–2188.

    MathSciNet  Article  MATH  Google Scholar 

  18. G. Dagan, Solute transport in heterogeneous porous formations, J. Fluid Mech. 145 (1984), 151–177.

    Article  MATH  Google Scholar 

  19. J. Dick, On the convergence rate of the component-by-component construction of good lattice rules, J. Complexity 20 (2004), 493–522.

    MathSciNet  Article  MATH  Google Scholar 

  20. J. Dick, Explicit constructions of Quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions, SIAM J. Numer. Anal. 45 (2007), 2141–2176.

    MathSciNet  Article  MATH  Google Scholar 

  21. J. Dick, Walsh spaces containing smooth functions and Quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008), 1519–1553.

    MathSciNet  Article  MATH  Google Scholar 

  22. J. Dick, The decay of the Walsh coefficients of smooth functions, Bull. Aust. Math. Soc. 80 (2009), 430–453.

    MathSciNet  Article  MATH  Google Scholar 

  23. J. Dick, F. Y. Kuo, Q. T. Le Gia, D. Nuyens, and Ch. Schwab, Higher order QMC Galerkin discretization for parametric operator equations, SIAM J. Numer. Anal. 52 (2014), 2676–2702.

    MathSciNet  Article  MATH  Google Scholar 

  24. J. Dick, F. Y. Kuo, Q. T. Le Gia, and Ch. Schwab, Fast QMC matrix-vector multiplication, SIAM J. Sci. Comput. 37 (2015), A1436–A1450.

    MathSciNet  Article  MATH  Google Scholar 

  25. J. Dick, F. Y. Kuo, Q. T. Le Gia, and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J. Numer. Anal. 54 (2016), 2541–2568.

  26. J. Dick, F .Y. Kuo, F. Pillichshammer, and I. H. Sloan, Construction algorithms for polynomial lattice rules for multivariate integration, Math. Comp. 74 (2005), 1895–1921.

    MathSciNet  Article  MATH  Google Scholar 

  27. J. Dick, F. Y. Kuo, and I. H. Sloan, High-dimensional integration: the Quasi-Monte Carlo way, Acta Numer. 22 (2013), 133–288.

    MathSciNet  Article  MATH  Google Scholar 

  28. J. Dick, Q. T. Le Gia, and Ch. Schwab, Higher order Quasi-Monte Carlo integration for holomorphic, parametric operator equations, in review.

  29. J. Dick, D. Nuyens, and F. Pillichshammer, Lattice rules for nonperiodic smooth integrands, Numer. Math. 126 (2014), 259–291.

    MathSciNet  Article  MATH  Google Scholar 

  30. J. Dick and F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces, J. Complexity 21 (2005), 149–195.

    MathSciNet  Article  MATH  Google Scholar 

  31. J. Dick and F. Pillichshammer, Digital Nets and Sequences, Cambridge University Press, 2010.

  32. J. Dick, F. Pillichshammer, and B. J. Waterhouse, The construction of good extensible rank-\(1\) lattices, Math. Comp. 77 (2008), 2345–2374.

    MathSciNet  Article  MATH  Google Scholar 

  33. J. Dick, I. H. Sloan, X. Wang, and H. Woźniakowski, Liberating the weights, J. Complexity 20 (2004), 593–623.

    MathSciNet  Article  MATH  Google Scholar 

  34. C. R. Dietrich and G. H. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. 18, 1088–1107 (1997).

    MathSciNet  Article  MATH  Google Scholar 

  35. R. Freeze, A stochastic conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media, Water Resour. Res. 11 (1975), 725–741.

    Article  Google Scholar 

  36. M. Ganesh and S. C. Hawkins, A high performance computing and sensitivity analysis algorithm for stochastic many-particle wave scattering, SIAM J. Sci. Comput. 37 (2015), A1475–A1503.

    MathSciNet  Article  MATH  Google Scholar 

  37. R. N. Gantner and Ch. Schwab, Computational Higher-Order QMC integration, Research Report 2014-24, Seminar for Applied Mathematics, ETH Zürich.

  38. R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements, Dover, 1991.

  39. M. B. Giles, Improved multilevel Monte Carlo convergence using the Milstein scheme, Monte Carlo and Quasi-Monte Carlo methods 2006, Springer, 2007, pp 343–358.

  40. M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 256 (2008), 981–986.

    MathSciNet  MATH  Google Scholar 

  41. M. B. Giles, Multilevel Monte Carlo methods, Acta Numer. 24 (2015), 259–328.

    MathSciNet  Article  MATH  Google Scholar 

  42. M. B. Giles and B. J. Waterhouse, Multilevel quasi-Monte Carlo path simulation. Radon Series Comp. Appl. Math. 8 (2009), 1–18.

    MathSciNet  MATH  Google Scholar 

  43. T. Goda, Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces, J. Comput. Appl. Math. 285 (2015), 279–294.

    MathSciNet  Article  MATH  Google Scholar 

  44. T. Goda and J. Dick, Construction of interlaced scrambled polynomial lattice rules of arbitrary high order, Found. Comput. Math. 15 (2015), 1245–1278.

    MathSciNet  Article  MATH  Google Scholar 

  45. T. Goda, K. Suzuki, and T. Yoshiki, Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration, J. Complexity 33 (2016), 30–54.

    MathSciNet  Article  MATH  Google Scholar 

  46. I. G. Graham, F. Y. Kuo, J. Nichols, R. Scheichl, Ch. Schwab, and I. H. Sloan, QMC FE methods for PDEs with log-normal random coefficients, Numer. Math. 131 (2015), 329–368.

    MathSciNet  Article  MATH  Google Scholar 

  47. I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, J. Comput. Phys. 230 (2011), 3668–3694.

    MathSciNet  Article  MATH  Google Scholar 

  48. I. G. Graham, F. Y. Kuo, D. Nuyens, R. Scheichl, and I. H. Sloan, Analysis of QMC methods with circulant embedding for elliptic PDEs with lognormal coefficients, in preparation.

  49. M. Gunzburger, C. Webster, and G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numer. 23 (2014), 521–650.

    MathSciNet  Article  Google Scholar 

  50. A.L. Haji-Ali, F. Nobile, and R. Tempone, Multi-index Monte Carlo: when sparsity meets sampling, Numer. Math. 132 (2016), 767–806.

    MathSciNet  Article  MATH  Google Scholar 

  51. H. Harbrecht, M. Peters, and M. Siebenmorgen, On multilevel quadrature for elliptic stochastic partial differential equations, in Sparse Grids and Applications, Lecture Notes in Computational Science and Engineering, Volume 88, 2013, pp.161–179.

    Article  Google Scholar 

  52. H. Harbrecht, M. Peters, and M. Siebenmorgen, Multilevel accelerated quadrature for PDEs with log-normal distributed random coefficient. Preprint 2013-18, Math. Institut, Universität Basel, 2013.

  53. H. Harbrecht, M. Peters, and M. Siebenmorgen, On the quasi-Monte Carlo method with Halton points for elliptic PDEs with log-normal diffusion, Math. Comp., appeared online.

  54. S. Heinrich, Monte Carlo complexity of global solution of integral equations, J. Complexity 14 (1998), 151–175.

    MathSciNet  Article  MATH  Google Scholar 

  55. S. Heinrich, Multilevel Monte Carlo methods, Lecture notes in Compu. Sci. Vol. 2179, Springer, 2001, pp. 3624–3651.

  56. F. J. Hickernell, Obtaining \(O(N^{-2+\epsilon })\) convergence for lattice quadrature rules, in Monte Carlo and Quasi-Monte Carlo Methods 2000 (K. T. Fang, F. J. Hickernell, and H. Niederreiter, eds.), Springer, Berlin, 2002, pp. 274–289.

    Chapter  Google Scholar 

  57. F. J. Hickernell, P. Kritzer, F. Y. Kuo, and D. Nuyens, Weighted compound integration rules with higher order convergence for all \(N\), Numer. Algorithms 59 (2012), 161–183.

    MathSciNet  Article  MATH  Google Scholar 

  58. V. H. Hoang and Ch. Schwab, Regularity and generalized polynomial chaos approximation of parametric and random second-order hyperbolic partial differential equations, Analysis and Applications (Singapore) 10 (2012), 295–326.

    MathSciNet  Article  MATH  Google Scholar 

  59. V. H. Hoang and Ch. Schwab, Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs, Analysis and Applications (Singapore) 11 (2013), 1350001, 50.

    MathSciNet  MATH  Google Scholar 

  60. V. Hoang, Ch. Schwab, and A. Stuart, Sparse MCMC gpc finite element methods for Bayesian inverse problems, Inverse Problems 29 (2013), 085010.

    MathSciNet  Article  MATH  Google Scholar 

  61. R. J. Hoeksema and P. K. Kitanidis, Analysis of the spatial structure of properties of selected aquifers, Water Resour. Res. 21 (1985), 536–572.

    Google Scholar 

  62. S. Joe, Construction of good rank-\(1\) lattice rules based on the weighted star discrepancy, in Monte Carlo and Quasi-Monte Carlo Methods 2004 (H. Niederreiter and D. Talay, eds.), Springer Verlag, pp. 181–196, 2006.

  63. S. Joe and F. Y. Kuo, Constructing Sobol\(^{\prime }\) sequences with better two-dimensional projections, SIAM J. Sci. Comput. 30 (2008), 2635–2654.

    MathSciNet  Article  MATH  Google Scholar 

  64. A. Kunoth and Ch. Schwab, Analytic regularity and GPC approximation for stochastic control problems constrained by linear parametric elliptic and parabolic PDEs, SIAM J. Control and Optimization 51 (2013), 2442–2471.

    MathSciNet  Article  MATH  Google Scholar 

  65. F. Y. Kuo, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, J. Complexity 19 (2003), 301–320.

    MathSciNet  Article  MATH  Google Scholar 

  66. F. Y. Kuo, R. Scheichl, Ch. Schwab, I. H. Sloan, and E. Ullmann, Multilevel quasi-Monte Carlo methods for lognormal diffusion problems. Math. Comp., to appear

  67. F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Quasi-Monte Carlo methods for high dimensional integration: the standard weighted-space setting and beyond, ANZIAM J. 53 (2011), 1–37.

    MathSciNet  Article  MATH  Google Scholar 

  68. F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient, SIAM J. Numer. Anal. 50 (2012), 3351–3374.

    MathSciNet  Article  MATH  Google Scholar 

  69. F. Y. Kuo, Ch. Schwab, and I. H. Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient, Found. Comput. Math. 15 (2015), 411–449.

    MathSciNet  Article  MATH  Google Scholar 

  70. F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, and B. J. Waterhouse, Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands, J. Complexity 26 (2010), 135–160.

    MathSciNet  Article  MATH  Google Scholar 

  71. Q. T. Le Gia, A QMC-spectral method for elliptic PDEs with random coefficients on the unit sphere, in Monte Carlo and Quasi-Monte Carlo Methods 2012 (J. Dick, F. Y. Kuo, G. W. Peters and I. H. Sloan, eds.), Springer Verlag, Heidelberg, 2013, pp. 491–508.

    Google Scholar 

  72. M. Loève. Probability Theory, Volume II. Springer-Verlag, New York, 4th edition, 1978.

    MATH  Google Scholar 

  73. Z. Lu and D. Zhang, A comparative study on quantifying uncertainty of flow in randomly heterogeneous media using Monte Carlo simulations, the conventional and KL-based moment-equation approaches, SIAM J. Sci. Comput. 26 (2004), 558–577.

    MathSciNet  Article  MATH  Google Scholar 

  74. R. L. Naff, D. F. Haley, and E. A. Sudicky, High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 1. Methodology and flow results, Water Resour. Res., 34 (1998), 663–677.

    Article  Google Scholar 

  75. R. L. Naff, D. F. Haley, and E. A. Sudicky, High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 2. Transport Results, Water Resour. Res., 34 (1998), 679–697.

    Article  Google Scholar 

  76. J. A. Nichols and F. Y. Kuo, Fast CBC construction of randomly shifted lattice rules achieving \(\cal {O}(N^{-1+\delta })\) in weighted spaces with POD weights, J. Complexity 30 (2014), 444–468.

    MathSciNet  Article  MATH  Google Scholar 

  77. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.

    Book  MATH  Google Scholar 

  78. V. Nistor and C. Schwab, High order Galerkin approximations for parametric second order elliptic partial differential equations, Math. Mod. Meth. Appl. Sci. 23 (2013), 1729–1760.

    MathSciNet  Article  MATH  Google Scholar 

  79. F. Nobile, R. Tempone, and C. G. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46 (2008), 2309–2345.

    MathSciNet  Article  MATH  Google Scholar 

  80. F. Nobile, R. Tempone, and C. G. Webster, An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal. 46 (2008), 2411–2442.

    MathSciNet  Article  MATH  Google Scholar 

  81. E. Novak and H. Woźniakowski, Tractability of Multivariate Problems, Volume I: Linear Information, European Mathematical Society, Zürich, 2008.

    Book  MATH  Google Scholar 

  82. E. Novak and H. Woźniakowski, Tractability of Multivariate Problems, II: Standard Information for Functionals, European Mathematical Society, Zürich, 2010.

    Book  MATH  Google Scholar 

  83. E. Novak and H. Woźniakowski, Tractability of Multivariate Problems, Volume III: Standard Information for Operators, European Mathematical Society, Zürich, 2012.

    Book  MATH  Google Scholar 

  84. D. Nuyens, The construction of good lattice rules and polynomial lattice rules, In: Uniform Distribution and Quasi-Monte Carlo Methods (P. Kritzer, H. Niederreiter, F. Pillichshammer, A. Winterhof, eds.), Radon Series on Computational and Applied Mathematics Vol. 15, De Gruyter, 2014, pp. 223–256.

  85. D. Nuyens and R. Cools, Fast algorithms for component-by-component construction of rank-\(1\) lattice rules in shift-invariant reproducing kernel Hilbert spaces, Math. Comp. 75 (2006), 903–920.

    MathSciNet  Article  MATH  Google Scholar 

  86. D. Nuyens and R. Cools, Fast component-by-component construction of rank-\(1\) lattice rules with a non-prime number of points, J. Complexity 22 (2006), 4–28.

    MathSciNet  Article  MATH  Google Scholar 

  87. D. Nuyens and R. Cools, Fast component-by-component construction, a reprise for different kernels, in Monte Carlo and quasi-Monte Carlo methods 2004 (H. Niederreiter and D. Talay, eds.), Springer, Berlin, 2006, pp. 373–387.

    Chapter  Google Scholar 

  88. S. H. Paskov and J. F. Traub, Faster valuation of financial derivatives, J. Portfolio Management 22 (1995), 113–120.

    Article  Google Scholar 

  89. P. Robbe, D. Nuyens, and S. Vandewalle, A practical multilevel quasi-Monte Carlo method for elliptic PDEs with random coefficients. In Master Thesis “Een parallelle multilevel Monte-Carlo-methode voor de simulatie van stochastische partiële differentiaalvergelijkingen” by P. Robbe, June 2015.

  90. Y. Rubin, Applied Stochastic Hydrogeology, Oxford University Press, New York, 2003.

    Google Scholar 

  91. C. Schillings and Ch. Schwab, Sparse, adaptive smolyak algorithms for Bayesian inverse problems, Inverse Problems 29 (2013), 065011.

    MathSciNet  Article  MATH  Google Scholar 

  92. C. Schillings and Ch. Schwab, Sparsity in Bayesian inversion of parametric operator equations, Inverse Problems 30 (2014), 065007.

    MathSciNet  Article  MATH  Google Scholar 

  93. Ch. Schwab, QMC Galerkin discretizations of parametric operator equations, in Monte Carlo and Quasi-Monte Carlo Methods 2012 (J. Dick, F. Y. Kuo, G. W. Peters and I. H. Sloan, eds.), Springer Verlag, Heidelberg, 2013, pp. 613–629.

    Chapter  Google Scholar 

  94. Ch. Schwab and C. J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stoch astic PDEs, Acta Numer. 20 (2011), 291–467.

    MathSciNet  Article  MATH  Google Scholar 

  95. Ch. Schwab and R. A. Todor, Karhunen-Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phy. 217 (2006), 100–122.

    MathSciNet  Article  MATH  Google Scholar 

  96. Ch. Schwab and R. A. Todor, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal. 27 (2007), 232–261.

    MathSciNet  MATH  Google Scholar 

  97. I. H. Sloan, What’s New in high-dimensional integration? – designing quasi-Monte Carlo for applications, In: Proceedings of the ICIAM, Beijing, China (L. Guo and Z. Ma eds), Higher Education Press, Beijing, 2015, pp. 365–386.

    Google Scholar 

  98. I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration, Oxford University Press, Oxford, 1994.

    MATH  Google Scholar 

  99. I. H. Sloan, F. Y. Kuo, and S. Joe, Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal. 40 (2002), 1650–1665.

    MathSciNet  Article  MATH  Google Scholar 

  100. I. H. Sloan and A, V. Reztsov, Component-by-component construction of good lattice rules, Math. Comp. 71 (2002), 263–273.

    MathSciNet  Article  MATH  Google Scholar 

  101. I. H. Sloan, X. Wang, and H. Woźniakowski, Finite-order weights imply tractability of multivariate integration, J. Complexity 20 (2004), 46–74.

    MathSciNet  Article  MATH  Google Scholar 

  102. I. H. Sloan and H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals?, J. Complexity 14 (1998), 1–33.

    MathSciNet  Article  MATH  Google Scholar 

  103. A. L. Teckentrup, P. Jantsch, C. G. Webster, and M. Gunzburger, A multilevel stochastic collocation method for partial differential equations with random input data, Preprint arXiv:1404.2647 (2014).

  104. A. L. Teckentrup, R. Scheichl, M. B. Giles, and E. Ullmann, Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficient, Numer. Math. 125 (2013), 569–600.

    MathSciNet  Article  MATH  Google Scholar 

  105. X. Wang, Strong tractability of multivariate integration using quasi-Monte Carlo algorithms, Math. Comp. 72 (2002), 823–838.

    MathSciNet  Article  MATH  Google Scholar 

  106. D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys. 187 (2003), 137–167.

    MathSciNet  Article  MATH  Google Scholar 

  107. T. Yoshiki, Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration. arXiv:1504.03175 (2015).

  108. D. Zhang, Stochastic Methods for Flow in Porous Media: Coping with Uncertainties, Academic Press, San Diego, 2002.

    Google Scholar 

Download references

Acknowledgments

We graciously acknowledge many insightful discussions and valuable comments from our collaborators Josef Dick, Mahadevan Ganesh, Thong Le Gia, Alexander Gilbert, Ivan Graham, Yoshihito Kazashi, James Nichols, Pieterjan Robbe, Robert Scheichl, Christoph Schwab and Ian Sloan. We especially thank Mahadevan Ganesh for suggesting an alternative proof strategy to improve some existing estimates. We are also grateful for the financial supports from the Australian Research Council (FT130100655 and DP150101770) and the KU Leuven research fund (OT:3E130287 and C3:3E150478).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Frances Y. Kuo.

Additional information

Communicated by Albert Cohen.

Appendix: Selected Proofs

Appendix: Selected Proofs

In this section we provide the proofs for Lemmas 6.16.8. For simplicity of presentation, in the proofs we will often omit the arguments \({\pmb {x}}\) and \({\pmb {y}}\) in our notation. We start by collecting some identities and estimates that we need for the proofs.

We will make repeated use of the Leibniz product rule

$$\begin{aligned} \partial ^{\pmb {\nu }}(AB) = \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {m}}} A)\, (\partial ^{{\pmb {\nu }}-{\pmb {m}}} B), \end{aligned}$$
(9.1)

and the identity

$$\begin{aligned} \nabla \cdot (A\, \nabla B) = A\, \varDelta B + \nabla A \cdot \nabla B. \end{aligned}$$
(9.2)

We also need the combinatorial identity

$$\begin{aligned} \sum _{\mathop {\scriptstyle {|{\pmb {m}}|=i}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} = {\textstyle {\left( {\begin{array}{c}|{\pmb {\nu }}|\\ i\end{array}}\right) }}, \end{aligned}$$
(9.3)

which follows from a simple counting argument (i.e., consider the number of ways to select i distinct balls from some baskets containing a total number of \(|{\pmb {\nu }}|\) distinct balls). The identity (9.3) is used to establish the following identities

$$\begin{aligned} \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {m}}|!\,|{\pmb {\nu }}-{\pmb {m}}|!&= (|{\pmb {\nu }}|+1)!, \end{aligned}$$
(9.4)
$$\begin{aligned} \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} |{\pmb {m}}|! \,(|{\pmb {\nu }}-{\pmb {m}}|+1)!&= \frac{(|{\pmb {\nu }}|+2)!}{2}, \end{aligned}$$
(9.5)
$$\begin{aligned} \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \frac{(|{\pmb {m}}|+2)!}{2}\,\frac{(|{\pmb {\nu }}-{\pmb {m}}|+2)!}{2}&= \frac{(|{\pmb {\nu }}|+5)!}{120}. \end{aligned}$$
(9.6)

Additionally, we need the recursive estimates in the next two lemmas. The proofs can be found in [25] and [66], respectively.

Lemma 9.1

Given a sequence of non-negative numbers \({\pmb {b}}=(b_j)_{j\in {\mathbb {N}}}\), let \(({\mathbb {A}}_{\pmb {\nu }})_{{\pmb {\nu }}\in {\mathfrak {F}}}\) and \(({\mathbb {B}}_{\pmb {\nu }})_{{\pmb {\nu }}\in {\mathfrak {F}}}\) be non-negative numbers satisfying the inequality

$$\begin{aligned} {\mathbb {A}}_{\pmb {\nu }}\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,b_j\, {\mathbb {A}}_{{\pmb {\nu }}-{\pmb {e}}_j} + {\mathbb {B}}_{\pmb {\nu }}\quad \text{ for } \text{ any } {\pmb {\nu }}\in {\mathfrak {F}}\quad (\hbox {including }~{\pmb {\nu }}={\pmb {0}}). \end{aligned}$$

Then

$$\begin{aligned} {\mathbb {A}}_{\pmb {\nu }}\le \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {m}}|!\, {\pmb {b}}^{\pmb {m}}\,{\mathbb {B}}_{{\pmb {\nu }}-{\pmb {m}}} \quad \text{ for } \text{ all } {\pmb {\nu }}\in {\mathfrak {F}}. \end{aligned}$$

The result holds also when both inequalities are replaced by equalities.

Lemma 9.2

Given a sequence of non-negative numbers \({\pmb {\beta }}=(\beta _j)_{j\in {\mathbb {N}}}\), let \(({\mathbb {A}}_{\pmb {\nu }})_{{\pmb {\nu }}\in {\mathfrak {F}}}\) and \(({\mathbb {B}}_{\pmb {\nu }})_{{\pmb {\nu }}\in {\mathfrak {F}}}\) be non-negative numbers satisfying the inequality

$$\begin{aligned} {\mathbb {A}}_{\pmb {\nu }}\le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} {\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}} {\mathbb {A}}_{\pmb {m}}+ {\mathbb {B}}_{\pmb {\nu }}\quad \text{ for } \text{ any } {\pmb {\nu }}\in {\mathfrak {F}}\quad (\hbox {including } \ {\pmb {\nu }}={\pmb {0}}). \end{aligned}$$

Then

$$\begin{aligned} {\mathbb {A}}_{\pmb {\nu }}\le \sum _{{\pmb {k}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {k}}\end{array}}\right) }} \Lambda _{|{\pmb {k}}|}\, {\pmb {\beta }}^{\pmb {k}}\, {\mathbb {B}}_{{\pmb {\nu }}-{\pmb {k}}} \quad \text{ for } \text{ all } {\pmb {\nu }}\in {\mathfrak {F}}, \end{aligned}$$

where the sequence \((\Lambda _n)_{n\ge 0}\) is defined recursively by

$$\begin{aligned} \Lambda _0 := 1 \quad \text{ and }\quad \Lambda _n := \sum _{i=0}^{n-1} {\textstyle {\left( {\begin{array}{c}n\\ i\end{array}}\right) }} \Lambda _i \quad \text{ for } \text{ all } n\ge 1. \end{aligned}$$
(9.7)

The result holds also when both inequalities are replaced by equalities. Moreover, we have

$$\begin{aligned} \Lambda _n \le \frac{n!}{\alpha ^n} \quad \text{ for } \text{ all } n\ge 0. \end{aligned}$$
(9.8)

Proof of Lemma 6.1

This result was proved in [15]. Since the same proof strategy is used repeatedly in subsequent more complicated proofs, we include this relatively simple proof as a first illustration.

Let \(f\in V^*\) and \({\pmb {y}}\in U\). We prove this result by induction on \(|{\pmb {\nu }}|\). For \({\pmb {\nu }}={\pmb {0}}\), we take \(v = u(\cdot ,{\pmb {y}})\) in (3.5) to obtain

$$\begin{aligned} \int _D a({\pmb {x}},{\pmb {y}})\, |\nabla u({\pmb {x}},{\pmb {y}})|^2 \,{\mathrm {d}}{\pmb {x}}= \int _D f({\pmb {x}})\,u({\pmb {x}},{\pmb {y}})\,{\mathrm {d}}{\pmb {x}}, \end{aligned}$$

which leads to

$$\begin{aligned} a_{\min }\, \Vert u(\cdot ,{\pmb {y}})\Vert _V^2 \le \Vert f\Vert _{V^*} \Vert u(\cdot ,{\pmb {y}})\Vert _V \quad \implies \quad \Vert u(\cdot ,{\pmb {y}})\Vert _V \le \frac{\Vert f\Vert _{V^*}}{a_{\min }}, \end{aligned}$$

as required (see also (3.7)).

Given any multi-index \({\pmb {\nu }}\) with \(|{\pmb {\nu }}|\ge 1\), suppose that the result holds for any multi-index of order \(\le |{\pmb {\nu }}|-1\). Applying the mixed derivative operators \(\partial ^{\pmb {\nu }}\) to the variational formulation (3.5), recalling that f is independent of \({\pmb {y}}\), and using the Leibniz product rule (9.1), we obtain the identity (suppressing \({\pmb {x}}\) and \({\pmb {y}}\) in our notation)

$$\begin{aligned} \int _D \bigg ( \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {m}}} a)\, \nabla (\partial ^{{\pmb {\nu }}-{\pmb {m}}} u) \cdot \nabla z \bigg )\,{\mathrm {d}}{\pmb {x}}= 0 \qquad \text{ for } \text{ all }\quad z\in V. \end{aligned}$$

Observe that due to the linear dependence of \(a({\pmb {x}},{\pmb {y}})\) on the parameters \({\pmb {y}}\), the partial derivative \(\partial ^{{\pmb {m}}}\) of a with respect to \({\pmb {y}}\) satisfies

$$\begin{aligned} \partial ^{{\pmb {m}}} a({\pmb {x}},{\pmb {y}}) = {\left\{ \begin{array}{ll} a({\pmb {x}},{\pmb {y}}) &{} \text{ if } {\pmb {m}}= {\pmb {0}}, \\ \psi _j({\pmb {x}}) &{} \text{ if } {\pmb {m}}= {\pmb {e}}_j, \\ 0 &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$
(9.9)

Taking \(z = \partial ^{\pmb {\nu }}u(\cdot ,{\pmb {y}})\) and separating out the \({\pmb {m}}= {\pmb {0}}\) term, we obtain

$$\begin{aligned} \int _D a\, |\nabla (\partial ^{\pmb {\nu }}u)|^2 \,{\mathrm {d}}{\pmb {x}}= - \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \int _D \psi _j\,\nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j}u) \cdot \nabla (\partial ^{{\pmb {\nu }}} u)\,{\mathrm {d}}{\pmb {x}}, \end{aligned}$$

which yields

$$\begin{aligned} a_{\min }\, \Vert \nabla (\partial ^{\pmb {\nu }}u) \Vert _{L^2}^2&\le \sum _{j\ge 1}\nu _j\,\Vert \psi _j\Vert _{L^\infty } \Vert \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2}\, \Vert \nabla (\partial ^{{\pmb {\nu }}} u)\Vert _{L^2}, \end{aligned}$$

and therefore

$$\begin{aligned} \Vert \nabla (\partial ^{\pmb {\nu }}u) \Vert _{L^2} \le \sum _{j\ge 1}\nu _j\, b_j\, \Vert \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2}, \end{aligned}$$

where we used the definition of \(b_j\) in (2.3). The induction hypothesis then gives

$$\begin{aligned} \Vert \nabla (\partial ^{\pmb {\nu }}u) \Vert _{L^2}&\le \sum _{j\ge 1}\nu _j\, b_j\, |{\pmb {\nu }}-{\pmb {e}}_j|!\, {\pmb {b}}^{{\pmb {\nu }}-{\pmb {e}}_j}\,\frac{\Vert f\Vert _{V^*}}{a_{\min }} = |{\pmb {\nu }}|!\,{\pmb {b}}^{\pmb {\nu }}\,\frac{\Vert f\Vert _{V^*}}{a_{\min }}. \end{aligned}$$

This completes the proof. \(\square \)

Proof of Lemma 6.2

This result corresponds to [69, Theorem 6]. Here we take a more direct route with the proof and the bound depends on the sequence \(\overline{{\pmb {b}}}\) which is simpler than the sequence in [69] (there the sequence depends on an additional parameter \(\kappa \in (0,1]\) and other constants), at the expense of increasing the factorial factor from \(|{\pmb {\nu }}|!\) to \((|{\pmb {\nu }}|+1)!\). For simplicity we consider here the case \(f\in L^2(D)\), but the proof can be generalized to the case \(f\in H^{-1+t}(D)\) for \(t\in [0,1]\) as in [69].

Let \(f\in L^2(D)\) and \({\pmb {y}}\in U\). For \({\pmb {\nu }}={\pmb {0}}\), we apply the identity (9.2) to the strong formulation (2.1) to obtain (formally, at this stage, since we do not yet know that \(\varDelta u(\cdot ,{\pmb {y}})\in L^2(D)\))

$$\begin{aligned} - a({\pmb {x}},{\pmb {y}})\, \varDelta u({\pmb {x}},{\pmb {y}}) \ = \nabla a({\pmb {x}},{\pmb {y}})\cdot \nabla u({\pmb {x}},{\pmb {y}}) + f({\pmb {x}}), \end{aligned}$$

which leads to

$$\begin{aligned} a_{\min }\, \Vert \varDelta u(\cdot ,{\pmb {y}})\Vert _{L^2} \le \Vert \nabla a(\cdot ,{\pmb {y}})\Vert _{L^\infty }\,\Vert \nabla u(\cdot ,{\pmb {y}})\Vert _{L^2} + \Vert f\Vert _{L^2}. \end{aligned}$$

Combining this with (3.7) gives

$$\begin{aligned} \Vert \varDelta u(\cdot ,{\pmb {y}})\Vert _{L^2} \le \frac{\sup _{{\pmb {z}}\in U}\Vert \nabla a(\cdot ,{\pmb {z}})\Vert _{L^\infty }}{a_{\min }}\,\frac{\Vert f\Vert _{V^*}}{a_{\min }} + \frac{\Vert f\Vert _{L^2}}{a_{\min }} \le C\,\Vert f\Vert _{L^2}, \end{aligned}$$
(9.10)

where we could take

$$\begin{aligned} C := C_\mathrm{emb}\left( \frac{\sup _{{\pmb {z}}\in U}\Vert \nabla a(\cdot ,{\pmb {z}})\Vert _{L^\infty }}{a_{\min }^2} + \frac{1}{a_{\min }}\right) , \quad \text{ with }\quad C_\mathrm{emb} := \sup _{f\in L^2(D)} \frac{\Vert f\Vert _{V^*}}{\Vert f\Vert _{L^2}}. \end{aligned}$$

Thus, the result holds for \({\pmb {\nu }}= {\pmb {0}}\) (see also (3.8)).

For \({\pmb {\nu }}\ne {\pmb {0}}\), we apply \(\partial ^{\pmb {\nu }}\) to the strong formulation (2.1) and use the Leibniz product rule (9.1) to obtain (suppressing \({\pmb {x}}\) and \({\pmb {y}}\))

$$\begin{aligned} \nabla \cdot \bigg (\sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {m}}} a)\, \nabla (\partial ^{{\pmb {\nu }}-{\pmb {m}}} u) \bigg ) = 0. \end{aligned}$$

Using again (9.9) and separating out the \({\pmb {m}}= {\pmb {0}}\) term yield the following identity

$$\begin{aligned} \nabla \cdot (a\nabla (\partial ^{\pmb {\nu }}u)) = - \nabla \cdot \left( \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,\psi _j({\pmb {x}})\, \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u) \right) . \end{aligned}$$

Applying the identity (9.2) to both sides yields (formally)

$$\begin{aligned} a\,\varDelta (\partial ^{\pmb {\nu }}u) + \nabla a \cdot \nabla (\partial ^{\pmb {\nu }}u) = - \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \left( \psi _j\, \varDelta (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u) \, + \, \nabla \psi _j\cdot \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u) \right) . \end{aligned}$$

In turn, we obtain

$$\begin{aligned} a_{\min } \,\Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2}&\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \bigg (\Vert \psi _j\Vert _{L^\infty }\, \Vert \varDelta (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2} + \Vert \nabla \psi _j\Vert _{L^\infty } \, \Vert \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2} \bigg ) \\&\qquad + \Vert \nabla a\Vert _{L^\infty } \,\Vert \nabla (\partial ^{\pmb {\nu }}u)\Vert _{L^2}, \end{aligned}$$

which leads to

$$\begin{aligned} \underbrace{\Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {\nu }}}&\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,b_j\, \underbrace{\Vert \varDelta (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2}}_{{\mathbb {A}}_{{\pmb {\nu }}-{\pmb {e}}_j}} \,+\, B_{\pmb {\nu }}, \end{aligned}$$

where we used the definition of \(b_j\) in (2.3), and

$$\begin{aligned} B_{\pmb {\nu }}:= \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \frac{\Vert \nabla \psi _j\Vert _{L^\infty }}{a_{\min }} \, \Vert \nabla (\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} u)\Vert _{L^2} + \frac{\Vert \nabla a\Vert _{L^\infty }}{a_{\min }} \,\Vert \nabla (\partial ^{\pmb {\nu }}u)\Vert _{L^2}. \end{aligned}$$

Note that this formulation of \(B_{\pmb {\nu }}\) cannot be used as \({\mathbb {B}}_{\pmb {\nu }}\) in Lemma 9.1 because the base step \({\mathbb {A}}_{\pmb {0}}\le B_{\pmb {0}}\) does not hold. From Lemma 6.1 we can estimate

$$\begin{aligned} B_{\pmb {\nu }}&\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \frac{\Vert \nabla \psi _j\Vert _{L^\infty }}{a_{\min }} \, |{\pmb {\nu }}-{\pmb {e}}_j|!\, {\pmb {b}}^{{\pmb {\nu }}-{\pmb {e}}_j}\, \frac{\Vert f\Vert _{V^*}}{a_{\min }} + \frac{\sup _{{\pmb {z}}\in U} \Vert \nabla a(\cdot ,{\pmb {z}})\Vert _{L^\infty }}{a_{\min }} \,|{\pmb {\nu }}|!\, {\pmb {b}}^{{\pmb {\nu }}}\, \frac{\Vert f\Vert _{V^*}}{a_{\min }} \\&\le C\,|{\pmb {\nu }}|!\, \overline{{\pmb {b}}}^{{\pmb {\nu }}}\, \Vert f\Vert _{L^2(D)} \,=:\, {\mathbb {B}}_{\pmb {\nu }}, \end{aligned}$$

where we used the definition of \(\overline{b}_j\ge b_j\) in (2.4). This definition of \({\mathbb {B}}_{\pmb {\nu }}\) ensures that the base step \({\mathbb {A}}_{\pmb {0}}\le {\mathbb {B}}_{\pmb {0}}\) does hold; see (9.10). Now we apply Lemma 9.1 to conclude that

$$\begin{aligned} \Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2}&\le \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {m}}|!\, {\pmb {b}}^{\pmb {m}}\, C\,|{\pmb {\nu }}-{\pmb {m}}|!\, \overline{{\pmb {b}}}^{{\pmb {\nu }}-{\pmb {m}}}\, \Vert f\Vert _{L^2} \le C\,(|{\pmb {\nu }}|+1)!\, \overline{{\pmb {b}}}^{{\pmb {\nu }}}\, \Vert f\Vert _{L^2}, \end{aligned}$$

where we used the identity (9.4). This completes the proof. \(\square \)

Proof of Lemma 6.3

This result appeared as a technical step in the proof of [69, Theorem 7], but only first derivatives were considered there, i.e., \(\nu _j\le 1\) for all j. Here we consider general derivatives, and we make use of Lemma 6.2 instead of [69, Theorem 6] so that the sequence \(\overline{{\pmb {b}}}\) is different, the factorial factor is larger, and we restrict to \(f\in L^2(D)\).

Let \(f\in L^2(D)\), \({\pmb {y}}\in U\) and \({\pmb {\nu }}\in {\mathfrak {F}}\). Galerkin orthogonality for the FE method yields

$$\begin{aligned} {\mathscr {A}}({\pmb {y}};u(\cdot ,{\pmb {y}})-u_h(\cdot ,{\pmb {y}}), z_h)\,= \,0 \qquad \text {for all} \quad z_h \in V_h, \end{aligned}$$
(9.11)

Let \({\mathscr {I}}:V\rightarrow V\) denote the identity operator and let \({\mathscr {P}}_h = {\mathscr {P}}_h({\pmb {y}}) :V\rightarrow V_h\) denote the parametric FE projection onto \(V_h\) which is defined, for arbitrary \(w\in V\), by

$$\begin{aligned} {\mathscr {A}}({\pmb {y}}; {\mathscr {P}}_h({\pmb {y}}) w - w, z_h) = 0 \qquad \text {for all} \quad z_h\in V_h . \end{aligned}$$
(9.12)

In particular, we have \(u_h = {\mathscr {P}}_h u \in V_h\) and

$$\begin{aligned} {\mathscr {P}}_h^2({\pmb {y}}) \,\equiv \, {\mathscr {P}}_h({\pmb {y}}) \quad \text{ on }\quad V_h . \end{aligned}$$
(9.13)

Moreover, since \(\partial ^{\pmb {\nu }}u_h \in V_h\) for every \({\pmb {\nu }}\in {\mathfrak {F}}\), it follows from (9.13) that

$$\begin{aligned} ({\mathscr {I}}- {\mathscr {P}}_h({\pmb {y}}))(\partial ^{\pmb {\nu }}u_h(\cdot ,{\pmb {y}})) \,\equiv \, 0. \end{aligned}$$
(9.14)

Thus,

$$\begin{aligned} \Vert \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2}&= \Vert \nabla {\mathscr {P}}_h \partial ^{\pmb {\nu }}(u - u_{h}) + \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{\pmb {\nu }}(u - u_{h}) \Vert _{L^2} \nonumber \\&\le \Vert \nabla {\mathscr {P}}_h \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2} \,+\, \Vert \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{\pmb {\nu }}u \Vert _{L^2} . \end{aligned}$$
(9.15)

We stress here that, since the parametric FE projection \({\mathscr {P}}_h({\pmb {y}})\) depends on \({\pmb {y}}\), in general we have \(\partial ^{\pmb {\nu }}(u(\cdot ,{\pmb {y}}) - u_{h}(\cdot ,{\pmb {y}})) \ne ({\mathscr {I}}- {\mathscr {P}}_h({\pmb {y}})) (\partial ^{\pmb {\nu }}u(\cdot ,{\pmb {y}}))\); this is why we need the estimate (9.15).

Now, applying \(\partial ^{\pmb {\nu }}\) to (9.11) and recalling (9.9), we get for all \(z_h\in V_h\),

$$\begin{aligned} \int _D a\, \nabla \partial ^{{\pmb {\nu }}} (u-u_h)\cdot \nabla z_h \,{\mathrm {d}}{\pmb {x}}= - \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \int _D \psi _j \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j}(u-u_h)\cdot \nabla z_h \,{\mathrm {d}}{\pmb {x}}. \end{aligned}$$
(9.16)

Choosing \(z_h = {\mathscr {P}}_h \partial ^{\pmb {\nu }}(u-u_h)\) and using the definition (9.12) of \({\mathscr {P}}_h\), the left-hand side of (9.16) is equal to \(\int _D a\, |\nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)|^2 \,{\mathrm {d}}{\pmb {x}}\). Using the Cauchy–Schwarz inequality, we then obtain

$$\begin{aligned}&a_{\min }\, \Vert \nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)\Vert _{L^2}^2 \\&\quad \le \, \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \Vert \psi _j\Vert _{L^\infty } \Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j}(u-u_h)\Vert _{L^2}\, \Vert \nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)\Vert _{L^2}. \end{aligned}$$

Canceling one common factor from both sides, we arrive at

$$\begin{aligned} \Vert \nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)\Vert _{L^2}&\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,b_j\, \Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j}(u-u_h)\Vert _{L^2}, \end{aligned}$$
(9.17)

where we used the definition of \(b_j\) in (2.3). Substituting (9.17) into (9.15), we then obtain

$$\begin{aligned} \underbrace{\Vert \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {\nu }}}&\le \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,b_j\, \underbrace{\Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j}(u-u_h)\Vert _{L^2}}_{{\mathbb {A}}_{{\pmb {\nu }}-{\pmb {e}}_j}} \,+\, \underbrace{\Vert \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{\pmb {\nu }}u\Vert _{L^2}}_{{\mathbb {B}}_{\pmb {\nu }}} . \end{aligned}$$

Noting that \({\mathbb {A}}_{\pmb {0}}= {\mathbb {B}}_{\pmb {0}}\), we now apply Lemma 9.1 to obtain

$$\begin{aligned} \Vert \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2}&\le \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}|!\, {\pmb {b}}^{\pmb {\nu }}\, \Vert \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{{\pmb {\nu }}-{\pmb {m}}} u\Vert _{L^2}. \end{aligned}$$

Next we use the FE estimate (3.9) that for all \({\pmb {y}}\in U\) and \(w\in H^2(D)\), we have \(\Vert \nabla ({\mathscr {I}}- {\mathscr {P}}_h) w \Vert _{L^2} \,\lesssim \, h\, \Vert \varDelta w\Vert _{L^2}\). Hence, from Lemma 6.2 we obtain

$$\begin{aligned} \Vert \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2}&\,\lesssim \, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}|!\, {\pmb {b}}^{\pmb {\nu }}\, h\, \Vert \varDelta (\partial ^{{\pmb {\nu }}-{\pmb {m}}} u)\Vert _{L^2} \\&\,\lesssim \, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}|!\, {\pmb {b}}^{\pmb {\nu }}\, h\, (|{\pmb {\nu }}-{\pmb {m}}|+1)!\, \overline{{\pmb {b}}}^{{\pmb {\nu }}-{\pmb {m}}}\, \Vert f\Vert _{L^2} \\&\,\lesssim \, h\, \frac{(|{\pmb {\nu }}|+2)!}{2}\,\overline{{\pmb {b}}}^{\pmb {\nu }}\, \Vert f\Vert _{L^2}, \end{aligned}$$

where we used the identity (9.5). This completes the proof.

Proof of Lemma 6.4

This result generalizes [69, Theorem 7] from first derivatives to general derivatives. The proof is based on a duality argument since G is a bounded linear functional. It makes use of Lemma 6.3, and therefore, the sequence \(\overline{{\pmb {b}}}\) is different, the factorial factor is larger, and we restrict to \(f,G\in L^2(D)\) here.

Let \(f,G\in L^2(D)\) and \({\pmb {y}}\in U\). We define \(v^G(\cdot ,{\pmb {y}})\in V\) and \(v_h^G(\cdot ,{\pmb {y}})\in V_h\) via the adjoint problems

$$\begin{aligned} {\mathscr {A}}({\pmb {y}}; w, v^G(\cdot ,{\pmb {y}})) = G( w ) \quad&\text {for all}\quad w\in V, \end{aligned}$$
(9.18)
$$\begin{aligned} {\mathscr {A}}({\pmb {y}}; w_h, v_h^G(\cdot ,{\pmb {y}})) = G ( w_h ) \;\;\;\;\,\quad&\text {for all}\quad w_h\in V_h. \end{aligned}$$
(9.19)

Due to Galerkin orthogonality (9.11) for the original problem, by choosing the test function \(w = u(\cdot ,{\pmb {y}}) - u_h(\cdot ,{\pmb {y}})\) in (9.18), we obtain

$$\begin{aligned} G(u(\cdot ,{\pmb {y}})-u_h(\cdot ,{\pmb {y}})) = {\mathscr {A}}({\pmb {y}}; u(\cdot ,{\pmb {y}})-u_h(\cdot ,{\pmb {y}}), v^G(\cdot ,{\pmb {y}}) - v_h^G(\cdot ,{\pmb {y}})). \end{aligned}$$
(9.20)

From the Leibniz product rule (9.1) and (9.9) we have for \({\pmb {\nu }}\in {\mathfrak {F}}\)

$$\begin{aligned} \partial ^{\pmb {\nu }}G(u - u_{h})&= \int _D \partial ^{\pmb {\nu }}\big (a \,\nabla (u - u_{h}) \cdot \nabla \big (v^G - v^G_h \big ) \big )\, {\mathrm {d}}{\pmb {x}}\\&= \int _D \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {m}}} a)\, \partial ^{{\pmb {\nu }}-{\pmb {m}}} \big ( \nabla (u - u_{h}) \cdot \nabla \big (v^G - v^G_h \big ) \big )\, {\mathrm {d}}{\pmb {x}}\\&= \int _D a\, \partial ^{{\pmb {\nu }}} \big ( \nabla (u - u_{h}) \cdot \nabla \big (v^G - v^G_h \big ) \big )\, {\mathrm {d}}{\pmb {x}}\\&\quad + \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \int _D \psi _j\,\partial ^{{\pmb {\nu }}-{\pmb {e}}_j} \big ( \nabla (u - u_{h}) \cdot \nabla \big (v^G - v^G_h \big ) \big )\, {\mathrm {d}}{\pmb {x}}\\&= \int _D a\, \sum _{{\pmb {k}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {k}}\end{array}}\right) }} \nabla \partial ^{\pmb {k}}(u- u_{h}) \cdot \nabla \partial ^{{\pmb {\nu }}-{\pmb {k}}} \big (v^G - v^G_h \big )\, {\mathrm {d}}{\pmb {x}}\\&\quad + \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \int _D \psi _j\,\sum _{{\pmb {k}}\le {\pmb {\nu }}-{\pmb {e}}_j} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}-{\pmb {e}}_j\\ {\pmb {k}}\end{array}}\right) }} \nabla \partial ^{\pmb {k}}(u- u_{h}) \cdot \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j-{\pmb {k}}} \big (v^G - v^G_h \big )\, {\mathrm {d}}{\pmb {x}}. \end{aligned}$$

The Cauchy–Schwarz inequality then yields

$$\begin{aligned}&|\partial ^{\pmb {\nu }}G(u - u_{h})| \le a_{\max } \sum _{{\pmb {k}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {k}}\end{array}}\right) }} \Vert \nabla \partial ^{\pmb {k}}(u- u_{h})\Vert _{L^2} \, \Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {k}}} \big (v^G - v^G_h \big )\Vert _{L^2} \nonumber \\&\quad + \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\, \Vert \psi _j\Vert _{L^\infty }\,\sum _{{\pmb {k}}\le {\pmb {\nu }}-{\pmb {e}}_j} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}{-}{\pmb {e}}_j\\ {\pmb {k}}\end{array}}\right) }} \Vert \nabla \partial ^{\pmb {k}}(u{-} u_{h})\Vert _{L^2}\Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {e}}_j{-}{\pmb {k}}} \big (v^G {-} v^G_h \big )\Vert _{L^2}. \end{aligned}$$
(9.21)

We see from (the proof of) Lemma 6.3 that

$$\begin{aligned} \Vert \nabla \partial ^{\pmb {k}}(u - u_{h})\Vert _{L^2}&\,\lesssim \, h\, \frac{(|{\pmb {k}}|+2)!}{2}\,\overline{{\pmb {b}}}^{\pmb {k}}\, \Vert f\Vert _{L^2}. \end{aligned}$$
(9.22)

Since the bilinear form \({\mathscr {A}}({\pmb {y}};\cdot ,\cdot )\) is symmetric and since the representer g for the linear functional G is in \(L^2(D)\), all the results hold verbatim also for the adjoint problem (9.18) and for its FE discretisation (9.19). Hence, as in (9.22), we obtain

$$\begin{aligned} \Vert \nabla \partial ^{{\pmb {\nu }}-{\pmb {k}}} (v^G - v^G_{h})\Vert _{L^2}&\,\lesssim \, h \, \frac{(|{\pmb {\nu }}-{\pmb {k}}|+2)!}{2}\, \overline{{\pmb {b}}}^{{\pmb {\nu }}-{\pmb {k}}}\, \Vert G\Vert _{L^2}. \end{aligned}$$
(9.23)

Substituting (9.22) and (9.23) into (9.21), and using \(\Vert \psi _j\Vert _{L^\infty } = a_{\min } b_j \le a_{\max } \overline{b}_j\), yields

$$\begin{aligned}&|\partial ^{\pmb {\nu }}G(u - u_{h})| \\&\quad \,\lesssim \, a_{\max }\,\Vert f\Vert _{L^2}\,\Vert G\Vert _{L^2}\, h^2\, \bigg ( \sum _{{\pmb {k}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {k}}\end{array}}\right) }}\, \frac{(|{\pmb {k}}|+2)!}{2}\,\overline{{\pmb {b}}}^{\pmb {k}}\, \frac{(|{\pmb {\nu }}-{\pmb {k}}|+2)!}{2}\,\overline{{\pmb {b}}}^{{\pmb {\nu }}-{\pmb {k}}} \\&\qquad + \sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,\overline{b}_j\, \sum _{{\pmb {k}}\le {\pmb {\nu }}-{\pmb {e}}_j} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}-{\pmb {e}}_j\\ {\pmb {k}}\end{array}}\right) }}\,\frac{(|{\pmb {k}}|+2)!}{2}\,\overline{{\pmb {b}}}^{\pmb {k}}\, \frac{(|{\pmb {\nu }}-{\pmb {e}}_j-{\pmb {k}}|+2)!}{2}\,\overline{{\pmb {b}}}^{{\pmb {\nu }}-{\pmb {e}}_j-{\pmb {k}}} \bigg ) \\&\quad \,\lesssim \, a_{\max }\,\Vert f\Vert _{L^2}\,\Vert G\Vert _{L^2}\, h^2\, \bigg (\frac{(|{\pmb {\nu }}|+5)!}{120} +\sum _{j\in {\mathrm {supp}}({\pmb {\nu }})} \nu _j\,\frac{(|{\pmb {\nu }}-{\pmb {e}}_j|+5)!}{120} \bigg )\overline{{\pmb {b}}}^{\pmb {\nu }}\\&\quad \,\lesssim \, a_{\max }\,\Vert f\Vert _{L^2}\,\Vert G\Vert _{L^2}\, h^2\, \frac{(|{\pmb {\nu }}|+5)!}{120}\, \overline{{\pmb {b}}}^{\pmb {\nu }}, \end{aligned}$$

where we used the identity (9.6). This completes the proof. \(\square \)

Proof of Lemma 6.5

This result is [46, Theorem 14]. We include the proof here since, unlike in the uniform case where we do induction directly for the quantity \(\Vert \nabla (\partial ^{{\pmb {\nu }}}u(\cdot ,{\pmb {y}}))\Vert _{L^2}\), here we need to work with \(\Vert a^{1/2}(\cdot ,{\pmb {y}})\,\nabla (\partial ^{{\pmb {\nu }}}u(\cdot ,{\pmb {y}}))\Vert _{L^2}\), and this technical step is needed for the subsequent proof.

Let \(f\in V^*\) and \({\pmb {y}}\in U_{\pmb {b}}\). We first prove by induction on \(|{\pmb {\nu }}|\) that

$$\begin{aligned} \Vert a^{1/2}(\cdot ,{\pmb {y}})\,\nabla (\partial ^{{\pmb {\nu }}}u(\cdot ,{\pmb {y}}))\Vert _{L^2} \le \Lambda _{|{\pmb {\nu }}|}\,{\pmb {\beta }}^{\pmb {\nu }}\, \frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}}, \end{aligned}$$
(9.24)

where the sequence \((\Lambda _n)_{n\ge 0}\) is defined recursively by (9.7) and satisfies (9.8).

We take \(v = u(\cdot ,{\pmb {y}})\) in the weak form (3.5) to obtain

$$\begin{aligned} \int _D a \,|\nabla u |^2\,{\mathrm {d}}{\pmb {x}}&\;\le \; \Vert f\Vert _{V^*}\,\Vert u(\cdot ,{\pmb {y}})\Vert _{V} \;\le \; \frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}} \left( \int _D a \, |\nabla u |^2 \,{\mathrm {d}}{\pmb {x}}\right) ^{1/2}, \end{aligned}$$

and then cancel the common factor from both sides to obtain (9.24) for the case \({\pmb {\nu }}= {\pmb {0}}\). Given any multi-index \({\pmb {\nu }}\) with \(|{\pmb {\nu }}| = n\ge 1\), we apply \(\partial ^{\pmb {\nu }}\) to (3.5) to obtain

$$\begin{aligned} \int _D \bigg ( \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a) \,\nabla (\partial ^{{\pmb {m}}} u) \cdot \nabla z\bigg ) {\mathrm {d}}{\pmb {x}}= 0 \qquad \text{ for } \text{ all }\quad z\in V\;. \end{aligned}$$

Taking \(z = \partial ^{\pmb {\nu }}u(\cdot ,{\pmb {y}})\), separating out the \({\pmb {m}}= {\pmb {\nu }}\) term, dividing and multiplying by a, and using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned}&\int _D a\, |\nabla (\partial ^{{\pmb {\nu }}} u) |^2 {\mathrm {d}}{\pmb {x}}= - \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \int _D (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a) \,\nabla (\partial ^{{\pmb {m}}} u) \cdot \nabla (\partial ^{{\pmb {\nu }}} u) \, {\mathrm {d}}{\pmb {x}}\\&\quad \;\le \; \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \bigg \Vert \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a (\cdot ,{\pmb {y}})}{a(\cdot ,{\pmb {y}})}\bigg \Vert _{L^\infty } \left( \int _D a |\nabla (\partial ^{{\pmb {m}}} u) |^2\,{\mathrm {d}}{\pmb {x}}\right) ^{1/2} \left( \int _D a |\nabla (\partial ^{{\pmb {\nu }}} u) |^2\,{\mathrm {d}}{\pmb {x}}\right) ^{1/2}. \end{aligned}$$

We observe from (2.6) that

$$\begin{aligned} \partial ^{{\pmb {\nu }}-{\pmb {m}}} a = a\prod _{j\ge 1} (\sqrt{\mu _j}\,\xi _j)^{\nu _j-m_j} \quad \text {for all} \quad {\pmb {\nu }}\ne {\pmb {m}}, \end{aligned}$$
(9.25)

and therefore

$$\begin{aligned} \bigg \Vert \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a (\cdot ,{\pmb {y}})}{a(\cdot ,{\pmb {y}})}\bigg \Vert _{L^\infty } = \bigg \Vert \prod _{j\ge 1} (\sqrt{\mu _j}\,\xi _j)^{\nu _j-m_j}\bigg \Vert _{L^\infty } \le {\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}}. \end{aligned}$$
(9.26)

Thus, we arrive at

$$\begin{aligned} \left( \int _D a |\nabla (\partial ^{{\pmb {\nu }}} u) |^2\,{\mathrm {d}}{\pmb {x}}\right) ^{1/2}&\le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, {\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}} \left( \int _D a |\nabla (\partial ^{{\pmb {m}}} u)|^2\,{\mathrm {d}}{\pmb {x}}\right) ^{1/2}. \end{aligned}$$

We now use the inductive hypothesis that (9.24) holds when \(\vert {\pmb {\nu }}\vert \le n-1\) in each of the terms on the right-hand side to obtain

$$\begin{aligned} \left( \int _D a |\nabla (\partial ^{{\pmb {\nu }}} u) |^2\,{\mathrm {d}}{\pmb {x}}\right) ^{1/2}&\le \sum _{i=0}^{n-1} \sum _{\mathop {\scriptstyle {|{\pmb {m}}|=i}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, {\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}}\, \Lambda _i\, {\pmb {\beta }}^{\pmb {m}}\, \frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}} \\&= \sum _{i=0}^{n-1} {\textstyle {\left( {\begin{array}{c}n\\ i\end{array}}\right) }} \Lambda _i\, {\pmb {\beta }}^{\pmb {\nu }}\frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}} = \Lambda _n\, {\pmb {\beta }}^{\pmb {\nu }}\frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}}, \end{aligned}$$

where we used the identity (9.3). This completes the induction proof of (9.24).

The desired bound in the lemma is obtained by applying (9.8) on the right-hand side of (9.24), and by noting that the left-hand side of (9.24) can be bounded from below by \(\sqrt{a_{\min }({\pmb {y}})}\,\Vert \partial ^{{\pmb {\nu }}} u(\cdot ,{\pmb {y}}) \Vert _V\). The case \({\pmb {\nu }}={\pmb {0}}\) corresponds to (3.12). This completes the proof. \(\square \)

Proof of Lemma 6.6

This result was proved in [66] based on an argument similar to the proof of Lemma 6.2 in the uniform case. The tricky point of the proof is in recognizing that for the recursion to work in the lognormal case we need to multiply the expression by \(a^{-1/2}(\cdot ,{\pmb {y}})\), which is not intuitive.

Let \(f\in L^2(D)\) and \({\pmb {y}}\in U_{\overline{{\pmb {\beta }}}}\). For any multi-index \({\pmb {\nu }}\ne {\pmb {0}}\), we apply \(\partial ^{\pmb {\nu }}\) to (2.1) to obtain (formally, at this stage)

$$\begin{aligned} \nabla \cdot \partial ^{\pmb {\nu }}(a\nabla u) = \nabla \cdot \left( \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a)\, \nabla (\partial ^{\pmb {m}}u) \right) = 0. \end{aligned}$$

Separating out the \({\pmb {m}}= {\pmb {\nu }}\) term yields the following identity

$$\begin{aligned} g_{\pmb {\nu }}:= \nabla \cdot (a\nabla (\partial ^{\pmb {\nu }}u))&= - \nabla \cdot \Bigg (\sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a)\, \nabla (\partial ^{\pmb {m}}u) \Bigg ) \\&= - \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \nabla \cdot \left( \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\, (a \nabla (\partial ^{\pmb {m}}u)) \right) \\&= - \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \left( \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\, g_{\pmb {m}}\, + \, \nabla \left( \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\right) \cdot (a \nabla (\partial ^{\pmb {m}}u)) \right) , \end{aligned}$$

where we used the identity (9.2). Due to Assumption (L2) we may multiply \(g_{\pmb {\nu }}\) by \(a^{-1/2}\) and obtain, for any \(|{\pmb {\nu }}| > 0\), the recursive bound

$$\begin{aligned} \Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2}&\le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \bigg (\bigg \Vert \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\bigg \Vert _{L^\infty }\, \Vert a^{-1/2} g_{\pmb {m}}\Vert _{L^2} \nonumber \\&\quad \quad \quad \quad \quad \quad + \; \bigg \Vert \nabla \bigg (\frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\bigg )\bigg \Vert _{L^\infty }\, \Vert a^{1/2} \nabla (\partial ^{\pmb {m}}u)\Vert _{L^2} \bigg ). \end{aligned}$$
(9.27)

By assumption, \(-g_{\pmb {0}}=f\in L^2(D)\), so that we obtain (by induction with respect to \(|{\pmb {\nu }}|\)) from (9.27) that \(a^{-1/2}(\cdot ,{\pmb {y}})\,g_{\pmb {\nu }}(\cdot ,{\pmb {y}})\in L^2(D)\), and hence from Assumption (L2) that \(g_{\pmb {\nu }}(\cdot ,{\pmb {y}})\in L^2(D)\) for every \({\pmb {\nu }}\in {\mathfrak {F}}\). The above formal identities therefore hold in \(L^2(D)\).

To complete the proof, it remains to bound the above \(L^2\) norm. Applying the product rule to (9.25) we obtain

$$\begin{aligned} \nabla \left( \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\right) = \sum _{k\ge 1} (\nu _k-m_k)(\sqrt{\mu _k}\,\xi _k)^{\nu _k-m_k-1} (\sqrt{\mu _k}\,\nabla \xi _k) \prod _{\mathop {\scriptstyle {j\ne k}}\limits ^{\scriptstyle {j\ge 1}}} (\sqrt{\mu _j}\,\xi _j)^{\nu _j-m_j}. \end{aligned}$$

Due to the definition of \(\overline{\beta }_j\) in (2.9), this implies, in a similar manner to (9.26), that

$$\begin{aligned} \bigg \Vert \nabla \bigg (\frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\bigg )\bigg \Vert _{L^\infty } \le |{\pmb {\nu }}-{\pmb {m}}| \,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}-{\pmb {m}}}. \end{aligned}$$
(9.28)

Substituting (9.26) and (9.28) into (9.27), we conclude that

$$\begin{aligned} \underbrace{\Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {\nu }}} \le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\,{\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}}\, \underbrace{\Vert a^{-1/2}g_{\pmb {m}}\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {m}}} \,+\, B_{\pmb {\nu }}, \end{aligned}$$

where

$$\begin{aligned} B_{\pmb {\nu }}\,:=&\, \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}-{\pmb {m}}| \,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}-{\pmb {m}}}\, \Vert a^{1/2} \nabla (\partial ^{\pmb {m}}u)\Vert _{L^2} \\ \,\le&\, \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}-{\pmb {m}}| \,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}-{\pmb {m}}}\, \Lambda _{|{\pmb {m}}|}\,{\pmb {\beta }}^{\pmb {m}}\, \frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}} \le \overline{\Lambda }_{|{\pmb {\nu }}|}\,\overline{{\pmb {\beta }}}^{\pmb {\nu }}\frac{\Vert f\Vert _{V^*}}{\sqrt{a_{\min }({\pmb {y}})}}, \end{aligned}$$

where we used (9.24) and again the identity (9.3) to write, with \(n = |{\pmb {\nu }}|\),

$$\begin{aligned} \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, |{\pmb {\nu }}-{\pmb {m}}|\, \Lambda _{|{\pmb {m}}|} = \sum _{i=0}^{n-1} \sum _{\mathop {\scriptstyle {|{\pmb {m}}|=i}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}\, (n-i)\, \Lambda _i = \sum _{i=0}^{n-1} {\textstyle {\left( {\begin{array}{c}n\\ i\end{array}}\right) }}\, (n-i)\, \Lambda _i \,=:\, \overline{\Lambda }_n. \end{aligned}$$

Since \({\mathbb {A}}_{\pmb {0}}= \Vert a^{-1/2} f\Vert _{L^2} \le \Vert f\Vert _{L^2}/\sqrt{a_{\min }({\pmb {y}})}\), we now define

$$\begin{aligned} {\mathbb {B}}_{\pmb {\nu }}:= C_\mathrm{emb}\, \overline{\Lambda }_{|{\pmb {\nu }}|}\,\overline{{\pmb {\beta }}}^{\pmb {\nu }}\frac{\Vert f\Vert _{L^2}}{\sqrt{a_{\min }({\pmb {y}})}}, \end{aligned}$$

so that \({\mathbb {A}}_{\pmb {0}}\le {\mathbb {B}}_{\pmb {0}}\) and \(B_{\pmb {\nu }}\le {\mathbb {B}}_{\pmb {\nu }}\) for all \({\pmb {\nu }}\). We may now apply Lemma 9.2 to obtain

$$\begin{aligned} \Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2}&\le \sum _{{\pmb {k}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {k}}\end{array}}\right) }}\Lambda _{|{\pmb {k}}|}\,{\pmb {\beta }}^{\pmb {k}}\, C_\mathrm{emb}\, \overline{\Lambda }_{|{\pmb {\nu }}-{\pmb {k}}|}\,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}-{\pmb {k}}} \frac{\Vert f\Vert _{L^2}}{\sqrt{a_{\min }({\pmb {y}})}}. \end{aligned}$$
(9.29)

Note the extra factor \(n-i\) in the definition of \(\overline{\Lambda }_n\) compared to \(\Lambda _n\) in (9.7) so that \(\Lambda _n\le \overline{\Lambda }_n\). Using the bound in (9.8) with \(\alpha \le \ln 2\), we have

$$\begin{aligned} \overline{\Lambda }_n \le \sum _{i=0}^{n-1} {\textstyle {\left( {\begin{array}{c}n\\ i\end{array}}\right) }} (n-i)\,\frac{i!}{\alpha ^i} = \frac{n!}{\alpha ^n}\,\alpha \,\sum _{i=0}^{n-1} \frac{\alpha ^{n-i-1}}{(n-i-1)!} = \frac{n!}{\alpha ^n}\,\alpha \,\sum _{k=0}^{n-1} \frac{\alpha ^k}{k!} \le \frac{n!}{\alpha ^n}\,\alpha \, e^\alpha \le \frac{n!}{\alpha ^n}, \end{aligned}$$

where the final step is valid provided that \(\alpha \,e^\alpha \le 1\). Thus, it suffices to choose \(\alpha \le 0.567\cdots \). For convenience we take \(\alpha = 0.5\) to bound (9.29). This together with the identity (9.4) gives

$$\begin{aligned} \Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2} \le C_\mathrm{emb}\, (|{\pmb {\nu }}|+1)!\,2^{|{\pmb {\nu }}|}\,\overline{{\pmb {\beta }}}^{\pmb {\nu }}\,\frac{\Vert f\Vert _{L^2}}{\sqrt{a_{\min }({\pmb {y}})}}. \end{aligned}$$
(9.30)

Since \(a^{-1/2}g_{\pmb {\nu }}= a^{-1/2}\nabla \cdot (a\nabla (\partial ^{\pmb {\nu }}u)) = a^{1/2}\varDelta (\partial ^{\pmb {\nu }}u) + a^{-1/2}\,(\nabla a\cdot \nabla (\partial ^{\pmb {\nu }}u))\) by applying (9.2), we have

$$\begin{aligned} \Vert a^{1/2}\varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2} \le \Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2} \,+\, \Vert a^{-1/2}\,(\nabla a\cdot \nabla (\partial ^{\pmb {\nu }}u))\Vert _{L^2}, \end{aligned}$$

which yields

$$\begin{aligned} \sqrt{a_{\min }({\pmb {y}})}\,\Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2} \le \Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2} \,+\, \frac{\Vert \nabla a(\cdot ,{\pmb {y}})\Vert _{L^\infty } }{a_{\min }({\pmb {y}})} \Vert a^{1/2}\nabla (\partial ^{\pmb {\nu }}u)\Vert _{L^2}, \end{aligned}$$

and in turn

$$\begin{aligned} \Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2} \le \frac{\Vert a^{-1/2}g_{\pmb {\nu }}\Vert _{L^2}}{\sqrt{a_{\min }({\pmb {y}})}} \,+\, \frac{\Vert \nabla a(\cdot ,{\pmb {y}})\Vert _{L^\infty } }{a_{\min }({\pmb {y}})}\, \frac{\Vert a^{1/2}\nabla (\partial ^{\pmb {\nu }}u)\Vert _{L^2}}{\sqrt{a_{\min }({\pmb {y}})}}. \end{aligned}$$
(9.31)

Substituting (9.30) and (9.24) into (9.31), and using \(\Lambda _{|{\pmb {\nu }}|}\le 2^{|{\pmb {\nu }}|}|{\pmb {\nu }}|!\) and \({\pmb {\beta }}^{\pmb {\nu }}\le \overline{{\pmb {\beta }}}^{\pmb {\nu }}\), we conclude that

$$\begin{aligned} \Vert \varDelta (\partial ^{\pmb {\nu }}u)\Vert _{L^2}&\le C_\mathrm{emb}\, \bigg ( \frac{1}{a_{\min }({\pmb {y}})} + \frac{\Vert \nabla a(\cdot ,{\pmb {y}})\Vert _{L^\infty }}{a_{\min }^2({\pmb {y}})}\bigg ) (|{\pmb {\nu }}|+1)!\,2^{|{\pmb {\nu }}|}\,\overline{{\pmb {\beta }}}^{\pmb {\nu }}\,\Vert f\Vert _{L^2}. \end{aligned}$$

This completes the proof. \(\square \)

Proof of Lemma 6.7

This result was proved in [66]. We include the proof here to provide a complete unified view of the proof techniques discussed in this survey.

Let \(f\in L^2(D)\) and \({\pmb {y}}\in U_{\overline{{\pmb {\beta }}}}\). Following (9.12)–(9.15) in the uniform case, we can write in the lognormal case

$$\begin{aligned} \Vert a^{1/2} \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2} \le \Vert a^{1/2} \nabla {\mathscr {P}}_h \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2} \,+\, \Vert a^{1/2} \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{\pmb {\nu }}u \Vert _{L^2}. \end{aligned}$$
(9.32)

Now, applying \(\partial ^{\pmb {\nu }}\) to (9.11) and separating out the \({\pmb {m}}= {\pmb {\nu }}\) term, we get for all \(z_h\in V_h\) in the lognormal case that

$$\begin{aligned} \int _D a\, \nabla \partial ^{{\pmb {\nu }}} (u-u_h)\cdot \nabla z_h \,{\mathrm {d}}{\pmb {x}}= - \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \int _D (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a) \nabla \partial ^{{\pmb {m}}}(u-u_h)\cdot \nabla z_h \,{\mathrm {d}}{\pmb {x}}. \end{aligned}$$
(9.33)

Choosing \(z_h = {\mathscr {P}}_h \partial ^{\pmb {\nu }}(u-u_h)\) and using the definition (9.12) of \({\mathscr {P}}_h\), the left-hand side of (9.33) is equal to \(\int _D a\, |\nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)|^2 \,{\mathrm {d}}{\pmb {x}}\). Dividing and multiplying the right-hand side of (9.33) by a, and using the Cauchy–Schwarz inequality, we then obtain

$$\begin{aligned}&\int _D a\, |\nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)|^2 \,{\mathrm {d}}{\pmb {x}}\\&\quad \le \, \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \left\| \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a}\right\| _{L^\infty } \left( \int _D a\, |\nabla \partial ^{{\pmb {m}}}(u-u_h)|^2\,{\mathrm {d}}{\pmb {x}}\right) ^{\frac{1}{2}} \left( \int _D a\, |\nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)|^2\,{\mathrm {d}}{\pmb {x}}\right) ^{\frac{1}{2}}. \end{aligned}$$

Canceling one common factor from both sides and using (9.26), we arrive at

$$\begin{aligned} \Vert a^{1/2} \nabla {\mathscr {P}}_h \partial ^{{\pmb {\nu }}} (u-u_h)\Vert _{L^2}&\le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}{\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}}\, \Vert a^{1/2} \nabla \partial ^{{\pmb {m}}}(u-u_h)\Vert _{L^2}. \end{aligned}$$
(9.34)

Substituting (9.34) into (9.32), we then obtain

$$\begin{aligned}&\underbrace{\Vert a^{1/2} \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {\nu }}} \\&\le \sum _{\mathop {\scriptstyle {{\pmb {m}}\ne {\pmb {\nu }}}}\limits ^{\scriptstyle {{\pmb {m}}\le {\pmb {\nu }}}}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }}{\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}} \underbrace{\Vert a^{1/2} \nabla \partial ^{{\pmb {m}}}(u-u_h)\Vert _{L^2}}_{{\mathbb {A}}_{\pmb {m}}} \,+\, \underbrace{\Vert a^{1/2} \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{\pmb {\nu }}u \Vert _{L^2}}_{{\mathbb {B}}_{\pmb {\nu }}}. \end{aligned}$$

Note that we have \({\mathbb {A}}_{\pmb {0}}= {\mathbb {B}}_{\pmb {0}}\). Now applying Lemma 9.2 with \(\alpha =0.5\), together with (3.14), Lemma 6.6 and (9.5), we conclude that

$$\begin{aligned} \Vert a^{1/2} \nabla \partial ^{\pmb {\nu }}(u - u_{h})\Vert _{L^2} \,&\le \, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \Lambda _{|{\pmb {m}}|}\,{\pmb {\beta }}^{\pmb {m}}\, \Vert a^{1/2} \nabla ({\mathscr {I}}- {\mathscr {P}}_h) \partial ^{{\pmb {\nu }}-{\pmb {m}}} u \Vert _{L^2} \\&\,\lesssim \, h\, a_{\max }^{1/2}({\pmb {y}})\, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \Lambda _{|{\pmb {m}}|}\, {\pmb {\beta }}^{\pmb {m}}\, \Vert \varDelta (\partial ^{{\pmb {\nu }}-{\pmb {m}}} u)\Vert _{L^2} \\&\,\lesssim \, h\, T({\pmb {y}})\,a_{\max }^{1/2}({\pmb {y}})\, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \, |{\pmb {m}}|!\,2^{|{\pmb {m}}|}\, {\pmb {\beta }}^{\pmb {m}}\, (|{\pmb {\nu }}-{\pmb {m}}|+1)!\\&\quad \quad 2^{|{\pmb {\nu }}-{\pmb {m}}|}\,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}-{\pmb {m}}}\, \Vert f\Vert _{L^2} \\&\,\lesssim \, h T({\pmb {y}})\,\, a_{\max }^{1/2}({\pmb {y}})\, \frac{(|{\pmb {\nu }}|+2)!}{2}\,2^{|{\pmb {\nu }}|}\, \overline{{\pmb {\beta }}}^{{\pmb {\nu }}}\, \Vert f\Vert _{L^2} , \end{aligned}$$

where \(T({\pmb {y}})\) is defined in (3.13). This completes the proof. \(\square \)

Proof of Lemma 6.8

This result was proved in [66]. Again we include the proof here to provide a complete unified view of the proof techniques discussed in this survey.

Let \(f,G\in L^2(D)\) and \({\pmb {y}}\in U_{\overline{{\pmb {\beta }}}}\). Following (9.18)–(9.20) in the uniform case, and using the Leibniz product rule (9.1), we have for the lognormal case that

$$\begin{aligned} \partial ^{\pmb {\nu }}G(u - u_{h}) \,&=\, \int _D \partial ^{\pmb {\nu }}\left( a \,\nabla (u - u_{h}) \cdot \nabla \left( v^G - v^G_h \right) \right) \, {\mathrm {d}}{\pmb {x}}\\&= \int _D \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a)\, \partial ^{\pmb {m}}\left( \nabla (u - u_{h}) \cdot \nabla \left( v^G - v^G_h \right) \right) \, {\mathrm {d}}{\pmb {x}}\\&= \int _D \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} (\partial ^{{\pmb {\nu }}-{\pmb {m}}} a)\, \sum _{{\pmb {k}}\le {\pmb {m}}} {\textstyle {\left( {\begin{array}{c}{\pmb {m}}\\ {\pmb {k}}\end{array}}\right) }} \nabla \partial ^{\pmb {k}}(u- u_{h}) \cdot \nabla \partial ^{{\pmb {m}}-{\pmb {k}}} \left( v^G - v^G_h \right) \, {\mathrm {d}}{\pmb {x}}\\&= \int _D \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \frac{\partial ^{{\pmb {\nu }}-{\pmb {m}}} a}{a} \sum _{{\pmb {k}}\le {\pmb {m}}} {\textstyle {\left( {\begin{array}{c}{\pmb {m}}\\ {\pmb {k}}\end{array}}\right) }} \left( a^{1/2}\nabla \partial ^{\pmb {k}}(u - u_{h})\right) \nonumber \\&\quad \quad \cdot \left( a^{1/2}\nabla \partial ^{{\pmb {m}}-{\pmb {k}}} \left( v^G - v^G_h \right) \right) \, {\mathrm {d}}{\pmb {x}}. \end{aligned}$$

Using the Cauchy–Schwarz inequality and (9.26), we obtain

$$\begin{aligned}&|\partial ^{\pmb {\nu }}G(u - u_{h}) | \nonumber \\&\le \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} {\pmb {\beta }}^{{\pmb {\nu }}-{\pmb {m}}} \sum _{{\pmb {k}}\le {\pmb {m}}} {\textstyle {\left( {\begin{array}{c}{\pmb {m}}\\ {\pmb {k}}\end{array}}\right) }} \Vert a^{1/2} \nabla \partial ^{\pmb {k}}(u - u_{h})\Vert _{L^2}\, \Vert a^{1/2} \nabla \partial ^{{\pmb {m}}-{\pmb {k}}} \left( v^G - v^G_h \right) \Vert _{L^2}. \end{aligned}$$
(9.35)

We have from Lemma 6.7 that

$$\begin{aligned} \Vert a^{1/2} \nabla \partial ^{\pmb {k}}(u - u_{h})\Vert _{L^2} \,\lesssim \, h \, T({\pmb {y}})\,a_{\max }^{1/2}({\pmb {y}})\, \frac{(|{\pmb {k}}|+2)!}{2}\, 2^{|{\pmb {k}}|}\, \overline{{\pmb {\beta }}}^{{\pmb {k}}}\,\Vert f\Vert _{L^2}. \end{aligned}$$
(9.36)

Since the bilinear form \({\mathscr {A}}({\pmb {y}};\cdot ,\cdot )\) is symmetric and since the representer g for the linear functional G is in \(L^2\), all the results hold verbatim also for the adjoint problem (9.18) and for its FE discretisation (9.19). Hence, as in (9.36), we obtain

$$\begin{aligned} \Vert a^{1/2} \nabla \partial ^{{\pmb {m}}-{\pmb {k}}} (v^G - v^G_{h})\Vert _{L^2}&\lesssim h T({\pmb {y}})\,\,a_{\max }^{1/2}({\pmb {y}})\, \frac{(|{\pmb {m}}-{\pmb {k}}|+2)!}{2}\,2^{|{\pmb {m}}-{\pmb {k}}|}\,\overline{{\pmb {\beta }}}^{{\pmb {m}}-{\pmb {k}}}\,\Vert G\Vert _{L^2}. \end{aligned}$$
(9.37)

Substituting (9.36) and (9.37) into (9.35), and using the identity (9.6), we obtain

$$\begin{aligned} |\partial ^{\pmb {\nu }}G(u - u_{h}) |&\,\lesssim \, h^2\, T^2({\pmb {y}})\,a_{\max }({\pmb {y}})\, \sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} \frac{(|{\pmb {m}}|+5)!}{120}\,2^{|{\pmb {m}}|}\,\overline{{\pmb {\beta }}}^{{\pmb {\nu }}} \Vert f\Vert _{L^2} \,\Vert G\Vert _{L^2}. \end{aligned}$$

Using again (9.3), with \(n=|{\pmb {\nu }}|\) we have

$$\begin{aligned}&\sum _{{\pmb {m}}\le {\pmb {\nu }}} {\textstyle {\left( {\begin{array}{c}{\pmb {\nu }}\\ {\pmb {m}}\end{array}}\right) }} 2^{|{\pmb {m}}|}\,\frac{(|{\pmb {m}}|+5)!}{120} \\&\quad {=} \sum _{i=0}^{n} {\textstyle {\left( {\begin{array}{c}n\\ i\end{array}}\right) }} 2^{i}\,\frac{(i+5)!}{120} {=} n!\,\sum _{i=0}^{n} \frac{(i+1)(i{+}2)(i{+}3)(i+4)(i+5) 2^{i}}{120(n-i)!} \le \frac{(n+5)!}{120} 2^n e. \end{aligned}$$

This yields the required bound in the lemma. This completes the proof. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kuo, F.Y., Nuyens, D. Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation. Found Comput Math 16, 1631–1696 (2016). https://doi.org/10.1007/s10208-016-9329-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-016-9329-5

Keywords

  • Quasi-Monte Carlo methods
  • Infinite-dimensional integration
  • Partial differential equations with random coefficients
  • Uniform
  • Lognormal
  • Single-level
  • Multi-level
  • First order
  • Higher order
  • Deterministic
  • Randomized

Mathematics Subject Classification

  • 65D30
  • 65D32
  • 65N30