Department of Applied Mathematics Center for Interdisciplinary Scientific Computation Illinois Institute of Technology hickernell@iit.edu mypages.iit.edu/~hickernell Happy Birthday and Congratulations to Ian! Thanks to the organizers, the GAIL team, NSF-DMS-1522687 and NSF-DMS-1638521 (SAMSI) Ian Sloan’s 80th Birthday Conference, June 18–19, 2018
Stop Summary References The Advantages of Integration Lattices They fill space better than IID They provide superior approximations to the high dimensional integrals when the coordinates decrease in importance We have ways to choose the sample size adaptively 2/19
Stop Summary References Integration in One Dimension In calculus we learn that the rectangle rule satisfies 1 0 f(x) dx − 1 n n−1 i=0 f i n √ 2 √ 3n f 2 We might have missed that 1 0 f(x) dx − 1 n n−1 i=0 f i n 2 ζ(2r) (2πn)r f(r) 2 , provided f, f , . . . , f(r−1) are continuous and periodic What can be done for [0,1)d f(x) dx for d = 2, 6, 12, 100, 1000? 3/19
Stop Summary References The Sloan-Kachoyan Unification Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d Grid (Strength d OA) C = 1 2 Id Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). 9/19
Stop Summary References The Sloan-Kachoyan Unification Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d Grid (Strength d OA) C = 1 2 Id Strength 3 Near OA C = 1 4 1 0 0 1 2 3 0 1 0 2 3 1 0 0 1 3 1 2 Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). 9/19
Stop Summary References The Sloan-Kachoyan Unification Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d Grid (Strength d OA) C = 1 2 Id Strength 3 Near OA C = 1 4 1 0 0 1 2 3 0 1 0 2 3 1 0 0 1 3 1 2 Strength 2 Near OA C = 1 8 1 0 1 3 2 3 0 1 1 2 3 1 Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). 9/19
Stop Summary References The Sloan-Kachoyan Unification Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d Grid (Strength d OA) C = 1 2 Id Strength 3 Near OA C = 1 4 1 0 0 1 2 3 0 1 0 2 3 1 0 0 1 3 1 2 Strength 2 Near OA C = 1 8 1 0 1 3 2 3 0 1 1 2 3 1 Good Lattice Point Set C = 1 64 31 11 23 3 19 13 Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). 9/19
Stop Summary References The Sloan-Kachoyan Unification Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d Grid (Strength d OA) C = 1 2 Id Strength 3 Near OA C = 1 4 1 0 0 1 2 3 0 1 0 2 3 1 0 0 1 3 1 2 Strength 2 Near OA C = 1 8 1 0 1 3 2 3 0 1 1 2 3 1 Good Lattice Point Set C = 1 64 31 11 23 3 19 13 How do we decide which kind of lattice is best? Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). 9/19
Stop Summary References An Ian Story My first interaction with Ian was when he acted as editor for my first quasi-Monte Carlo manuscript . Ian is always quite encouraging, H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19
Stop Summary References An Ian Story My first interaction with Ian was when he acted as editor for my first quasi-Monte Carlo manuscript . Ian is always quite encouraging, and his feedback included page after page of encourage- ment, H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19
Stop Summary References An Ian Story My first interaction with Ian was when he acted as editor for my first quasi-Monte Carlo manuscript . Ian is always quite encouraging, and his feedback included page after page of encourage- ment, which led to a much better paper. Thank you, Ian. H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19
Stop Summary References Sloan-Woźniakowski Discovery of Coordinate Weights [0,1]d f(x) − 1 n n−1 i=0 f(xi) −1 + 1 n n−1 i=0 d j=1 1 + γ2 j 2 B2(xij) ∂uf 2 j∈u γj u 2 Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? J. Complexity 14, 1–33 (1998). 14/19
Stop Summary References Sloan-Woźniakowski Discovery of Coordinate Weights [0,1]d f(x) − 1 n n−1 i=0 f(xi) −1 + 1 n n−1 i=0 d j=1 1 + γ2 j 2 B2(xij) ∂uf 2 j∈u γj u 2 An idea that launched 1000 papers Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? J. Complexity 14, 1–33 (1998). 14/19
Stop Summary References A Few of Those 1000 Papers Thank you, Ian, for our fruitful collaboration. H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1 Approximation over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. & Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19
Stop Summary References A Few of Those 1000 Papers Thank you, Ian, for our fruitful collaboration. Ian’s friend: “Ian collaborates with everybody!” H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1 Approximation over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. & Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19
Stop Summary References A Few of Those 1000 Papers Thank you, Ian, for our fruitful collaboration. Ian’s friend: “Ian collaborates with everybody!” Ian’s UNSW colleague: “Ian does the work of ten.” H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1 Approximation over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. & Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19
Stop Summary References In Practice, How Large Should n Be? [0,1]d f(x) − 1 n n−1 i=0 f(xi) −1 + 1 n n−1 i=0 d j=1 1 + γ2 j 2 B2(xij) ∂uf 2 j∈u γj u 2 ε H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for Cubature. in Contemporary Computational Mathematics — a celebration of the 80th birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. & Woźniakowski, H.) (Springer-Verlag, 2018), 597–619. doi:10.1007/978-3-319-72456-0, Jagadeeswaran, R. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. 16/19
Stop Summary References In Practice, How Large Should n Be? [0,1]d f(x) − 1 n n−1 i=0 f(xi) −1 + 1 n n−1 i=0 d j=1 1 + γ2 j 2 B2(xij) ∂uf 2 j∈u γj u 2 ε Look at the discrete Fourier coefficients (O(n log n) additional cost) and infer the decay rate of true coefficients assuming that the integrand lies in a cone C of nice functions. H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for Cubature. in Contemporary Computational Mathematics — a celebration of the 80th birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. & Woźniakowski, H.) (Springer-Verlag, 2018), 597–619. doi:10.1007/978-3-319-72456-0, Jagadeeswaran, R. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. 16/19
Stop Summary References The Advantages of Integration Lattices They fill space better than IID They provide superior approximations to the high dimensional integrals when the coordinates decrease in importance We have ways to choose the sample size adaptively 18/19
Stop Summary References Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987). Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994). H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? J. Complexity 14, 1–33 (1998). H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004). 19/19
Stop Summary References H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71. H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911 (2004). H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1 Approximation over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005). H., F. J., Sloan, I. H. & Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for Cubature. in Contemporary Computational Mathematics — a celebration of the 80th 19/19
Stop Summary References birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. & Woźniakowski, H.) (Springer-Verlag, 2018), 597–619. doi:10.1007/978-3-319-72456-0. Jagadeeswaran, R. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. Choi, S.-C. T. et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software. 2013–2017. http://gailgithub.github.io/GAIL_Dev/. (ed Niederreiter, H.) Monte Carlo and Quasi-Monte Carlo Methods 2002. (Springer-Verlag, Berlin, 2004). 19/19