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Bill Martin Mathematical Sciences and Computer Science Worcester Polytechnic Institute. The Story of (T,M,S)-Nets. Caveats, etc. Many photos borrowed from the web (sources available on request)
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Bill Martin Mathematical Sciences and Computer Science Worcester Polytechnic Institute The Story of (T,M,S)-Nets
Caveats, etc. • Many photos borrowed from the web (sources available on request) • This talk focuses only on the combinatorics; there is a lot more activity that I won’t talk about • WPI is looking for graduate students and visiting faculty
Pre-History • Quadrature rules • Numerical simulation • Global optimization
Quasi-Random is not Random • Random • Pseudo-random (should fool an observer) • Quasi-Random: entirely deterministic, but has some statistical properties that a random set “should” have
Some Ways to Sample the Cube • Random (Monte Carlo) • Lattice rules • Latin hypercube sampling • (T,M,S)-nets
Evenly Sampling the Unit Cube • A set N of N points inside [0,1)s • An interval E = [0,a1)x[0,a2)x . . . x[0,as) “should” contain Vol(E) |N | of these points • The star discrepancy of a set N of N points in [0,1)s is the supremum of | |N E| / N - Vol(E) | taken over all such intervals E. Call it D*(N ) U
Koksma-Hlawka Inequality J. Koksma E. Hlawka
Elementary Intervals For any given shape (d1,d2,. . .,ds), the unit cube is partitioned into bm elementary intervals of this shape, each being a translate of every other.
(T,M,S)-Nets Harald Niederreiter Working on low discrepancy sequences, quasi-randomness, pseudo-random generators, applications to numerical analysis, coding theory, cryptography Expertise in finite fields and number theory
(T,M,S)-Nets Niederreiter (1987), generalizing an idea of Sobol’ (1967)
Using Latin Squares Two MOLS(3) yield an orthogonal array of strength two
Latin Squares to (0,2,2)-net Replace alphabet by {0,1,…,b-1} (here, base b=3)
Latin Squares to (0,2,2)-net Insert decimal points to obtain a (0,2,2)-net in base 3
The Resulting (T,M,S)-Net (0,2,2)-net in base 3
Su Doku! Now fill in with cosets of the linear code
Vienna, Austria 1980s Madison, Wisconsin 1995
Generalized Orthogonal Arrays Mark Lawrence, Chief Risk Officer, Australia and New Zealand Banking Group
Generalized Orthogonal Arrays • In an orthogonal array of strength t, all entries are chosen from some fixed alphabet {0,1,. . .,b-1} • In any t columns, every possible t-tuple over the alphabet (there are qt of these) appears equally often • So the total number of rows is l.bt where l is the replication number • If this hold for a set of columns, then it also holds for all subsets of that set • Now specify a partial order on the columns and require this only for lower ideals in this poset of size t or less
Vienna, Austria 1980s Salzburg, Austria 1995
Ordered Orthogonal Arrays Wolfgang Ch. Schmid and Gary Mullen • Introduced OOA concept • Proved equivalence to (T,M,S)-nets • constructions • bounds
