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Richardson extrapolation and higher order QMC Takashi Goda School of Engineering, University of Tokyo MCM 2019 Partly joint with Josef Dick and Takehito Yoshiki Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 1 / 61 Outline QMC

  1. Explicit construction For s = 1, C 1 can be the identity matrix, which generates the famous digital (0 , 1)-sequence called van der Corput sequence . For s = 2, C 1 and C 2 can be     1 0 · · · 0 0 · · · 0 1 0 1 · · · 0 0 · · · 1 0         C 1 =  , C 2 =  , . . . . . . ... ...     . . . . . . . . . . . .   0 0 · · · 1 1 · · · 0 0 which generates a digital (0 , m , 2)-net, known as the Hammersley point set. Not extensible in m . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 15 / 61

  2. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  3. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  4. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  5. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  6. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  7. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  8. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  9. Explicit construction Figure: Hammersley point set for m = 6 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 16 / 61

  10. Explicit construction Let p 1 , p 2 , . . . ∈ F 2 [ x ] be a sequence of distinct primitive/irreducible polynomials over F 2 with e 1 ≤ e 2 ≤ · · · where e j = deg( p j ). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 17 / 61

  11. Explicit construction Let p 1 , p 2 , . . . ∈ F 2 [ x ] be a sequence of distinct primitive/irreducible polynomials over F 2 with e 1 ≤ e 2 ≤ · · · where e j = deg( p j ). For each j , C j = ( c ( j ) k , l ) is given by the coefficients of the following Laurent series: c ( j ) c ( j ) x e j − 1 1 , 1 1 , 2 p j ( x ) = + x 2 + · · · x . . . c ( j ) c ( j ) 1 e j , 1 e j , 2 p j ( x ) = + x 2 + · · · x c ( j ) c ( j ) x e j − 1 e j +1 , 1 e j +1 , 2 ( p j ( x )) 2 = + + · · · x 2 x . . . (Sobol’ 1967; Niederreiter 1988; Tezuka 1993) Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 17 / 61

  12. Explicit construction All matrices C j ∈ F N × N are upper triangular and generate a digital 2 ( t , s )-sequence with t = ( e 1 − 1) + · · · + ( e s − 1) . (I used a C implementation for this sequence in at least more than 58636 dimensions due to Tomohiko Hironaka.) Figure: 2D projections of the first 2 6 points of the Niederreiter sequence. Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 18 / 61

  13. Polynomial lattice point sets (Niederreiter, 1992) Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m Let q = ( q 1 , . . . , q s ) ∈ ( F 2 [ x ]) s with deg( q j ) < m . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61

  14. Polynomial lattice point sets (Niederreiter, 1992) Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m Let q = ( q 1 , . . . , q s ) ∈ ( F 2 [ x ]) s with deg( q j ) < m . For each j , C j is given by the square Hankel matrix   a ( j ) a ( j ) a ( j ) · · · m 1 2 a ( j ) a ( j ) a ( j )   . . .   2 3 m +1 ∈ F m × m C j =   . . . ... 2 . . .   . . .   a ( j ) a ( j ) a ( j ) . . . m m +1 2 m − 1 where a ( j ) 1 , a ( j ) 2 , . . . are the coefficients of the Laurent series p ( x ) = a ( j ) + a ( j ) q j ( x ) 1 2 x 2 + · · · . x Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61

  15. Polynomial lattice point sets (Niederreiter, 1992) Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m Let q = ( q 1 , . . . , q s ) ∈ ( F 2 [ x ]) s with deg( q j ) < m . For each j , C j is given by the square Hankel matrix   a ( j ) a ( j ) a ( j ) · · · m 1 2 a ( j ) a ( j ) a ( j )   . . .   2 3 m +1 ∈ F m × m C j =   . . . ... 2 . . .   . . .   a ( j ) a ( j ) a ( j ) . . . m m +1 2 m − 1 where a ( j ) 1 , a ( j ) 2 , . . . are the coefficients of the Laurent series p ( x ) = a ( j ) + a ( j ) q j ( x ) 1 2 x 2 + · · · . x The resulting digital net is called a polynomial lattice point set P ( p , q ). Usually the vector q is constructed by a (fast) computer search algorithm (Nuyens & Cools, 2006; Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61

  16. Polynomial lattice point sets (Niederreiter, 1992) Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m Let q = ( q 1 , . . . , q s ) ∈ ( F 2 [ x ]) s with deg( q j ) < m . For each j , C j is given by the square Hankel matrix   a ( j ) a ( j ) a ( j ) · · · m 1 2 a ( j ) a ( j ) a ( j )   . . .   2 3 m +1 ∈ F m × m C j =   . . . ... 2 . . .   . . .   a ( j ) a ( j ) a ( j ) . . . m m +1 2 m − 1 where a ( j ) 1 , a ( j ) 2 , . . . are the coefficients of the Laurent series p ( x ) = a ( j ) + a ( j ) q j ( x ) 1 2 x 2 + · · · . x The resulting digital net is called a polynomial lattice point set P ( p , q ). Usually the vector q is constructed by a (fast) computer search algorithm (Nuyens & Cools, 2006; P. Kritzer, this afternoon!). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 19 / 61

  17. Outline QMC and digital nets/sequences Classical QMC Higher order QMC Richardson extrapolation and QMC Application 1: Truncation of higher order nets and sequences Application 2: Extrapolated polynomial lattice rules Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 20 / 61

  18. What if f is smooth In some applications, such as PDEs with random coefficients, f can be smooth. A proper design of QMC point sets enables higher order convergence of the integration error than O (1 / N ) as expected from the KH inequality. So far, QMC point sets achieving higher order convergence for non-periodic smooth functions are 1 Higher order digital nets/sequences (Dick, 2008; ...), and 2 Tent-transformed lattice point sets (Hickernell, 2002; ...). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 21 / 61

  19. What if f is smooth In some applications, such as PDEs with random coefficients, f can be smooth. A proper design of QMC point sets enables higher order convergence of the integration error than O (1 / N ) as expected from the KH inequality. So far, QMC point sets achieving higher order convergence for non-periodic smooth functions are 1 Higher order digital nets/sequences (Dick, 2008; ...), and 2 Tent-transformed lattice point sets (Hickernell, 2002; ...). In this talk, I will focus on higher order digital nets/sequences . The contents of this section are mostly developed by Dick (2008). Please refer to a recent review by G. & Suzuki (arXiv:1903.12353). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 21 / 61

  20. Quality measure: high order t -value Let α ∈ N . Let t be an integer such that, for any choice 1 ≤ d j , v j < · · · < d j , 1 ≤ α m , 0 ≤ v j ≤ α m , 1 ≤ j ≤ s , with min( v j ,α ) s ∑ ∑ d j , i = α m − t , j =1 i =1 the d 1 , v 1 , . . . , d 1 , 1 -th rows of C 1 . . . the d s , v s , . . . , d s , 1 -th rows of C s are linearly independent over F 2 . P is called an order α digital ( t , m , s ) -net . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 22 / 61

  21. Quality measure: high order t -value Order α digital ( t , m , s )-nets hold equi-distribution properties: union of dyadic elementary boxes contains the fair number of points (shown later visually). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 23 / 61

  22. Quality measure: high order t -value Order α digital ( t , m , s )-nets hold equi-distribution properties: union of dyadic elementary boxes contains the fair number of points (shown later visually). Let t be an integer such that, for any α m ≥ t , the first 2 m points of S are an order α digital ( t , m , s )-net. S is called an order α digital ( t , s ) -sequence . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 23 / 61

  23. Explicit construction Define D α : [0 , 1) α → [0 , 1) by  x 1 = (0 .ξ (1) 1 ξ (1) 2 ξ (1) . . . ) 2  3   x 2 = (0 .ξ (2) 1 ξ (2) 2 ξ (2)  . . . ) 2  3 �→ (0 . ξ (1) 1 ξ (2) . . . ξ ( α ) ξ (1) 2 ξ (2) . . . ξ ( α ) . . . ) 2 . . 1 1 2 2 . .    � �� � � �� �  x α = (0 .ξ ( α ) ξ ( α ) ξ ( α )  α α . . . ) 2 1 2 3 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61

  24. Explicit construction Define D α : [0 , 1) α → [0 , 1) by  x 1 = (0 .ξ (1) 1 ξ (1) 2 ξ (1) . . . ) 2  3   x 2 = (0 .ξ (2) 1 ξ (2) 2 ξ (2)  . . . ) 2  3 �→ (0 . ξ (1) 1 ξ (2) . . . ξ ( α ) ξ (1) 2 ξ (2) . . . ξ ( α ) . . . ) 2 . . 1 1 2 2 . .    � �� � � �� �  x α = (0 .ξ ( α ) ξ ( α ) ξ ( α )  α α . . . ) 2 1 2 3 For x ∈ [0 , 1) α s , let D α ( x ) = ( D α ( x 1 , . . . , x α ) , D α ( x α +1 , . . . , x 2 α ) , . . . , D α ( x α ( s − 1)+1 , . . . , x α s )) ∈ [0 , 1) s . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61

  25. Explicit construction Define D α : [0 , 1) α → [0 , 1) by  x 1 = (0 .ξ (1) 1 ξ (1) 2 ξ (1) . . . ) 2  3   x 2 = (0 .ξ (2) 1 ξ (2) 2 ξ (2)  . . . ) 2  3 �→ (0 . ξ (1) 1 ξ (2) . . . ξ ( α ) ξ (1) 2 ξ (2) . . . ξ ( α ) . . . ) 2 . . 1 1 2 2 . .    � �� � � �� �  x α = (0 .ξ ( α ) ξ ( α ) ξ ( α )  α α . . . ) 2 1 2 3 For x ∈ [0 , 1) α s , let D α ( x ) = ( D α ( x 1 , . . . , x α ) , D α ( x α +1 , . . . , x 2 α ) , . . . , D α ( x α ( s − 1)+1 , . . . , x α s )) ∈ [0 , 1) s . For a digital ( t , m , α s )-net P , we write D α ( P ) = {D α ( x ) | x ∈ P } . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 24 / 61

  26. Another look at construction Let P be a digital ( t , m , α s )-net with C 1 , . . . , C α s ∈ F m × m . We write 2       c (1) c (2) c ( α ) 1 1 1 . . .  .   .   .  C 1 =  , C 2 =  , . . . , C α =  , . . . . . . .    c (1) c (2) c ( α ) m m m Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 25 / 61

  27. Another look at construction Let P be a digital ( t , m , α s )-net with C 1 , . . . , C α s ∈ F m × m . We write 2       c (1) c (2) c ( α ) 1 1 1 . . .  .   .   .  C 1 =  , C 2 =  , . . . , C α =  , . . . . . . .    c (1) c (2) c ( α ) m m m D α ( P ) is a digital net with D 1 , . . . , D s ∈ F α m × m where 2     c (1) c ( α +1) 1 1 c (2) c ( α +2)     1 .  1     .    . .     . .         c ( α ) c (2 α )     1 1 .     .  .   .  D 1 = , D 2 = , . . . . . .         c (1) c ( α +1)     m m     c (2) c ( α +2)         m m     . . . .     . .     c ( α ) c (2 α ) m m Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 25 / 61

  28. High order t -value For a digital ( t , m , α s )-net P , D α ( P ) is an order α digital ( t ′ , m , s )-net with { ⌊ s ( α − 1) ⌋} t ′ ≤ α min m , t + . 2 For a digital ( t , α s )-sequences S , D α ( S ) is an order α digital ( t ′ , s )-sequences with t ′ ≤ α t + s α ( α − 1) . 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 26 / 61

  29. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  30. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  31. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  32. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  33. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  34. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  35. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  36. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  37. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  38. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  39. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  40. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  41. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  42. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  43. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  44. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  45. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  46. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  47. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  48. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  49. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  50. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  51. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  52. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  53. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  54. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  55. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  56. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  57. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  58. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  59. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  60. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  61. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  62. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  63. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  64. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  65. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  66. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  67. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  68. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  69. Interlaced Sobol’ point sets with α = 2 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 27 / 61

  70. Interlaced polynomial lattice point sets Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m , and let q = ( q 1 , . . . , q α s ) ∈ ( F 2 [ x ]) α s with deg( q j ) < m . An interlaced polynomial lattice point set is just D α ( P ( p , q )) . Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 28 / 61

  71. Interlaced polynomial lattice point sets Let p ∈ F 2 [ x ] be irreducible with deg( p ) = m , and let q = ( q 1 , . . . , q α s ) ∈ ( F 2 [ x ]) α s with deg( q j ) < m . An interlaced polynomial lattice point set is just D α ( P ( p , q )) . The vector q can be constructed component by component. In the simplest case, the necessary construction cost is of O ( α sN log N ) with O ( N ) memory (G. & Dick, 2015; G., 2015). In applications to PDEs with random coefficients, the criterion sometimes becomes a bit complicated, requiring O ( α sN log N + α 2 s 2 N ) construction cost with O ( α sN ) memory (Dick, Kuo, Le Gia, Nuyens & Schwab, 2014). Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 28 / 61

  72. Outline QMC and digital nets/sequences Classical QMC Higher order QMC Richardson extrapolation and QMC Application 1: Truncation of higher order nets and sequences Application 2: Extrapolated polynomial lattice rules Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 29 / 61

  73. Walsh functions For k = ( . . . κ 1 κ 0 ) 2 ∈ N 0 , the k -th Walsh function is defined by wal k ( x ) = ( − 1) κ 0 ξ 1 + κ 1 ξ 2 + ··· , where x = (0 .ξ 1 ξ 2 . . . ) 2 ∈ [0 , 1), unique in the sense that infinitely many ξ i are equal to 0. Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 30 / 61

  74. Walsh functions For k = ( . . . κ 1 κ 0 ) 2 ∈ N 0 , the k -th Walsh function is defined by wal k ( x ) = ( − 1) κ 0 ξ 1 + κ 1 ξ 2 + ··· , where x = (0 .ξ 1 ξ 2 . . . ) 2 ∈ [0 , 1), unique in the sense that infinitely many ξ i are equal to 0. For s ≥ 1 and k = ( k 1 , . . . , k s ) ∈ N s 0 , the k -th Walsh function is defined by s ∏ wal k ( x ) = wal k j ( x j ) . j =1 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 30 / 61

  75. Walsh functions Figure: The k -th Walsh functions for k = 0 , 1 , 2 , 3 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 31 / 61

  76. Walsh functions Figure: The k -th Walsh functions for k = 4 , 5 , 6 , 7 Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 32 / 61

  77. Walsh functions Every Walsh function is a piecewise constant function. Takashi Goda (U. Tokyo) Extrapolation and HOQMC MCM 2019 33 / 61

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