Generalizing the Tolerance Function for Guaranteed Algorithms

Transcription

1 Generalizing the Tolerance Function for Guaranteed Algorithms Lluís Antoni Jiménez Rugama Joint work with Professor Fred J. Hickernell Room 120, Bldg E1, Department of Applied Mathematics April 11, 2015 Hybrid Error CASSC / 18

2 Outline Introduction Assumptions Problem Introduction New Definition of Tolerance Tolerance Function New Biased Algorithm Algorithm Redefinition Results Upper Bound on the Complexity of r I n Complexity Lemma Conclusions lluisantoni@gmail.com Hybrid Error CASSC / 18

3 Outline Introduction Assumptions Problem Introduction New Definition of Tolerance Tolerance Function New Biased Algorithm Algorithm Redefinition Results Upper Bound on the Complexity of r I n Complexity Lemma Conclusions lluisantoni@gmail.com Hybrid Error CASSC / 18

4 Assumptions What Configuration Do We Assume? Hybrid Error CASSC / 18

5 Assumptions What Configuration Do We Assume? Ipfq P R, solution to be approximated. lluisantoni@gmail.com Hybrid Error CASSC / 18

6 Assumptions What Configuration Do We Assume? Ipfq P R, solution to be approximated. p In pfq P R, approximating algorithm based on functionals using n data points. lluisantoni@gmail.com Hybrid Error CASSC / 18

7 Assumptions What Configuration Do We Assume? Ipfq P R, solution to be approximated. p In pfq P R, approximating algorithm based on functionals using n data points. pε n pfq P R`, decreasing absolute error bound of the algorithm above for a particular set of functions C, i.e. I pi n pfq ď pε n pfq lluisantoni@gmail.com Hybrid Error CASSC / 18

8 Assumptions What Configuration Do We Assume? Ipfq P R, solution to be approximated. p In pfq P R, approximating algorithm based on functionals using n data points. pε n pfq P R`, decreasing absolute error bound of the algorithm above for a particular set of functions C, i.e. I pi n pfq ď pε n pfq tol P R`, the maximum tolerance we are willing to accept. In other words, we want I p In pfq ď tol lluisantoni@gmail.com Hybrid Error CASSC / 18

9 Assumptions What Configuration Do We Assume? Ipfq P R, solution to be approximated. p In pfq P R, approximating algorithm based on functionals using n data points. pε n pfq P R`, decreasing absolute error bound of the algorithm above for a particular set of functions C, i.e. I pi n pfq ď pε n pfq tol P R`, the maximum tolerance we are willing to accept. In other words, we want I p In pfq ď tol Thus, we increase n until pε n pfq ď tol. lluisantoni@gmail.com Hybrid Error CASSC / 18

10 Assumptions Example cublattice g and cubsobol g algorithms in GAIL (free Matlab library that you can find on Ipfq : ş r0,1q d fpxqdx. p In pfq : 1 n ř n 1 i 0 fpx iq, where tx i u rank-1 lattices or Sobol points. pε n pfq defined according to (Jiménez Rugama and Hickernell, 2014) and (Hickernell and Jiménez Rugama, 2014). lluisantoni@gmail.com Hybrid Error CASSC / 18

11 Problem Introduction Problem to Solve Can we define a new generalized hybrid error tolerance whose inputs are ε a and ε r? tol pε a, ε r I q lluisantoni@gmail.com Hybrid Error CASSC / 18

12 Problem Introduction Problem to Solve For example: Can we define a new generalized hybrid error tolerance whose inputs are ε a and ε r? I pi n pfq ď ε r I. tol pε a, ε r I q lluisantoni@gmail.com Hybrid Error CASSC / 18

13 Outline Introduction Assumptions Problem Introduction New Definition of Tolerance Tolerance Function New Biased Algorithm Algorithm Redefinition Results Upper Bound on the Complexity of r I n Complexity Lemma Conclusions lluisantoni@gmail.com Hybrid Error CASSC / 18

14 Tolerance Function Generalization Consider tolpa, bq P R to have the following properties: Lipschitz-1 in b. Non-decreasing in both arguments, i.e. tolpa, bq ď tolpa 1, b 1 q for a ď a 1 and b ď b 1. lluisantoni@gmail.com Hybrid Error CASSC / 18

15 Tolerance Function Generalization Consider tolpa, bq P R to have the following properties: Lipschitz-1 in b. Non-decreasing in both arguments, i.e. tolpa, bq ď tolpa 1, b 1 q for a ď a 1 and b ď b 1. Examples for a ε a and b ε r I : tolpa, bq maxpa, bq tolpa, bq θa ` p1 θqb with θ P r0, 1s. lluisantoni@gmail.com Hybrid Error CASSC / 18

16 Algorithm Redefinition New Algorithm r I n Define, with n, : 1 2 ri n : p I n ` n, tol ε a, ε r In p pε n tol ε a, ε r In p ı ` pε n lluisantoni@gmail.com Hybrid Error CASSC / 18

17 Results Main lemma Lemma 1 If pε n ď n,` then I r In ď tolpεa, ε r I q. lluisantoni@gmail.com Hybrid Error CASSC / 18

18 Results Main lemma Lemma 1 If pε n ď n,` then I r In ď tolpεa, ε r I q. Note: Before, we increased n until pε n ď tol to have, I pi n ď tol Now, we increase n until pε n ď n,` to have, I ri n ď tolpεa, ε r I q lluisantoni@gmail.com Hybrid Error CASSC / 18

19 Outline Introduction Assumptions Problem Introduction New Definition of Tolerance Tolerance Function New Biased Algorithm Algorithm Redefinition Results Upper Bound on the Complexity of r I n Complexity Lemma Conclusions lluisantoni@gmail.com Hybrid Error CASSC / 18

20 Complexity Lemma Lemma for Complexity Lemma 2 If then, pε n ď tolpε a, ε r I q 1 ` ε r pε n ď n,` Note: Thus, using lemma (1), I ri n ď tolpεa, ε r I q. lluisantoni@gmail.com Hybrid Error CASSC / 18

21 Complexity Lemma Upper Bound Define Mpε, C p q such that, n ě Mpε, C p q ùñ pε n ď ε where C p is the set of parameters specifying the functions under which the guarantees hold. If M M tolpεa,εr I q 1`ε r, C p, n ě M Mdef ùñ pε n ď tol pε a, ε r I q 1 ` ε r lem(2) ùñ pε n ď n,` lem(1) ùñ I ri n ď tolpεa, ε r I q lluisantoni@gmail.com Hybrid Error CASSC / 18

22 Complexity Lemma The Quasi-Monte Carlo Example For the previous example, where we evaluate the function at some points and compute the Fast Fourier/Walsh Transform, cost rin, C ď cm logpm q ` $pfqpm q lluisantoni@gmail.com Hybrid Error CASSC / 18

23 Complexity Lemma Pricing the Geometric Asian Call Figure : 500 replications pricing the Geometric Asian Call option for dimensions 1, 2, 4, 8, 16, 32 and 64 chosen randomly. The tolerance function was max `10 2, 10 2 I. lluisantoni@gmail.com Hybrid Error CASSC / 18

24 Outline Introduction Assumptions Problem Introduction New Definition of Tolerance Tolerance Function New Biased Algorithm Algorithm Redefinition Results Upper Bound on the Complexity of r I n Complexity Lemma Conclusions lluisantoni@gmail.com Hybrid Error CASSC / 18

25 Future work Lower bound on the computational complexity for r I n. Fitting the cone of functions into a more familiar space (Korobov). Working on Multi-Level quasi Monte Carlo guaranteed hybrid error for infinite dimensional integration. For instance: Asian option pricing in continuous time. Implementing and adding this code to GAIL ( lluisantoni@gmail.com Hybrid Error CASSC / 18

26 References References I Dick, J., F. Kuo, and I. H. Sloan High dimensional integration the Quasi-Monte Carlo way, Acta Numer., Hickernell, F. J. and Ll. A. Jiménez Rugama Reliable adaptive cubature using digital sequences. submitted for publication, arxiv: [math.na]. Hickernell, F. J. and H. Niederreiter The existence of good extensible rank-1 lattices, J. Complexity 19, Jiménez Rugama, Ll. A. and F. J. Hickernell Adaptive multidimensional integration based on rank-1 lattices. submitted for publication, arxiv: Liu, K. I., J. Dick, and F. J. Hickernell A multivariate fast discrete Walsh transform with an application to function interpolation, Math. Comp. 78, Niederreiter, H. and F. Pillichshammer Construction algorithms for good extensible lattice rules, Constr. Approx. 30, Sloan, I. H Lattice methods for multiple integration, J. Comput. Appl. Math. 12 & 13, lluisantoni@gmail.com Hybrid Error CASSC / 18