Benjamin Doerr Max-Planck-Institut für Informatik Saarbrücken Component-by-Component Construction of Low-Discrepancy Point Sets joint work with Michael Gnewuch (Kiel), Peter Kritzer (Salzburg), and Friedrich Pillichshammer (Linz)

Benjamin Doerr Reminder: Star Discrepancy  Definition: N –s ∈ N “ dimension ” (Austrian notation) – P = {p 0, p 1,..., p N-1 } multi-set of N points in [0,1) s –Discrepancy function: For x ∈ [0,1] s,  Δ(x,P) := λ([0,x)) – #{i : p i ∈ [0,x)} / N  “ (how many points should be in [0,x) – how many actually are there) normalized by N ” –Star discrepancy: D*(P) := sup{ | Δ(x,P) | : x ∈ [0,1] s }  Measure of how evenly P is distributed in [0,1) s

Benjamin Doerr Reminder: Star Discrepancy  Application: Numerical Integration – Given f : [0,1] s → R – Compute/estimate ∫ [0,1] s f(x) dx !  Hope: ∫ [0,1] s f(x) dx ≈ (1/N) ∑ i f(p i )  Koksma-Hlawka inequality: | ∫ [0,1] s f(x) dx - (1/N) ∑ i f(p i ) | ≤ V(f) D*(P) – V(f): Variation in the sense of Hardy and Krause  Low star discrepancy = good integration

Benjamin Doerr Reminder: Star Discrepancy  How good?

Benjamin Doerr Reminder: Star Discrepancy  Very good! There are N-point sets P with D*(P) ≤ C s (log N) s-1 / N  “More points = drastically reduced integration error”  Really? Note: All constants ‘C’ may be different. They never depend on N. If they depend on s, I call them ‘C s ’.

Benjamin Doerr Reminder: Star Discrepancy  Very good! – There are N-point sets P with D*(P) ≤ C s (log N) s-1 / N  “More points = drastically reduced integration error”  Really? No!

Benjamin Doerr Problem: Only good for many points! – Increasing for N ≤ e 10 (more points = worse integration?) – ≥ 1 (trivial bound), if N ≤ – ≥ D*(random point set), if N ≤ 10 2∙10 Need for “small” low-discrepancy point sets!

Benjamin Doerr Motivation, Outline  Previous slides: –O((log N) s-1 /N) bounds only good for many points  many: at least exponential in dimension. –Otherwise: Random points have better guarantees.  Plan for this talk: –Be inspired by random points –...and use this to construct better point sets  Note: Almost all ugly details omitted in this talk! – For many technicalities, the sharpest bounds and more results see the full paper (MCMAppl, to appear).

Benjamin Doerr Previous Work (1)  Heinrich, Novak, Wasilkowski, Woźniakowski (Acta Arith., 2002): – There are point sets with D*(P) ≤ C (s/N) 1/2  randomized construction  Talagrand inequality – Good bounds for N polynomial in s –  Existential result only, implicit constants not known

Benjamin Doerr Previous Work (2)  We build on previous results by D., Gnewuch, Srivastav (JoC ‘ 05, MCQMC ’ 06): – D*(P) ≤ C (s/N) 1/2 (log N) 1/2, C small – via randomized rounding:  discrepancy guarantee holds with high probability – derandomization:  deterministic construction of P in run-time (CN) s+2  computes the exact star discrepancy on the way – wait for Michael ’ s talk (next talk) to see how difficult computing the star discrepancy can be...

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )  Compute the ‘ fair ’ number x B of points for each box B: x B = N vol(B)

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )  Compute the ‘ fair ’ number x B of points for each box B: x B = N vol(B)  Round these numbers to integers y B

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )  Compute the ‘ fair ’ number x B of points for each box B: x B = N vol(B)  Round these numbers to integers y B such that for all aligned corners C, y C := ∑ B⊆C y B is close to x C := ∑ B⊆C x B x C =12.25 y C =12

Benjamin Doerr Rounding Approach  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )  Compute the ‘ fair ’ number x B of points for each box B: x B = N vol(B)  Round these numbers to integers y B such that for all aligned corners C, y C := ∑ B⊆C y B is close to x C := ∑ B⊆C x B. Then put y B points in B arbitrarily

Benjamin Doerr Classical Rounding Theory  Let x 1,..., x n be numbers, N:=||x|| 1 and I 1,..., I m ⊆ {1,...,n}.  Randomized Rounding: – If x i is an integer, y i := x i – If not, then y i := ⌈x i ⌉ with probability equal to the fractional part of x i and y i := ⌊x i ⌋ otherwise  Theorem: With probability 1-ε, we have for all 1 ≤ k ≤ m (*) | ∑ i ∈ I k y i - ∑ i ∈ I k x i | ≤ (0.5 N log(2m/ε)) 1/2  Derandomization: A rounding (y i ) satisfying (*) with ε=1 can be computed deterministically in time O(mn).  Experiment: Derandomization yields smaller rounding errors.

Benjamin Doerr Rounding Approach (continued)  Task: Put N = 16 points in the unit cube nicely  Partition the cube into small rectangles ( “ boxes ” )  Compute the ‘ fair ’ number x B of points for each box B: x B = N vol(B)  Round these numbers to integers y B such that for all aligned corners C, | ∑ B⊆C y B - ∑ B⊆C x B | ≤ (0.5 N log(2 #boxes)) 1/

Benjamin Doerr New Result:  The same can be done in a component-by-component way: – Compoment-by-compoment: Given an (s-1)-dimensional low- discrepancy point set, add an s th component to each point. – Adjust the randomized-rounding approach accordingly.  Advantage: – Fewer variables to be rounded in each iteration. – Total run-time (over all s iterations) roughly N (s+3)/2 instead of N s+2.  Surprise: Discrepancy increases only by a factor of s. – Roughly C s 3/2 N -1/2 log(N) 1/2 instead of C s 1/2 N -1/2 log(N) 1/2  That ’ s OK, because we can now compute N 2 points in roughly the same time as needed for N points before.

Benjamin Doerr Summary and Conclusion  Result: Component-by-Component derandomized construction is much faster and yields only slightly higher discrepancies compared to “ all at once ”.  Outlook: Could also be useful if components are of different importance. E.g., do the expensive derandomization only for few components.  Open problem: Come up with something really efficient.... (instead of N Cs ). Merci/Thanks!