Quasi-Monte Carlo on Simplices - kinjalbasu.comkinjalbasu.com/talks/indus_aff.pdfQuasi-Monte Carlo on Simplices ... Department of Statistics Stanford University Joint work with Prof

IntroductionBackground

ConstructionGeneralizations

Quasi-Monte Carlo on Simplices

Kinjal Basu

Department of StatisticsStanford University

Joint work with Prof. Art Owen

27th February 2015

Kinjal Basu QMC on Simplices



Overview

1 IntroductionQuasi-Monte CarloMotivation

2 BackgroundDiscrepancy

3 ConstructionTriangular van der Corput pointsTriangular Kronecker LatticeResults

4 GeneralizationsHigh Dimensional AnaloguesResults




Quasi-Monte CarloMotivation

Introduction

Problem : Numerical Integration over [0, 1]d

µ =∫

[0,1]d f (x)dx µn = 1n

∑ni=1 f (xi )

Monte Carlo : O(n−1/2).

quasi-Monte Carlo : O(n−1 log(n)d−1).

Figure: MC and two QMC methods in Unit Square





Quasi-Monte Carlo

Koksma-Hlawka inequality

|µn − µ| 6 D∗n(x1, . . . , xn)× VHK (f )

The local discrepancy of x1, . . . , xn at a ∈ [0, 1]d is

δ(a; x1, . . . xn) =1

n

n∑i=1

1xi∈[0,a) − vol([0, a))

The star discrepancy of x1, . . . , xn ∈ [0, 1]d is

D∗n(x1, . . . , xn) = supa∈[0,1]d

|δ(a; x1, . . . xn)|





Discrepancy

Figure: Local Discrepancy at two points a, b





Motivation for Triangles - Human Biology

(σ2A, σ

2D , σ

2N) ∈ ∆

Integrate out σ2A + σ2

D + σ2N .

Left with a problem of averagingover the triangle.

Each point in the triangle requiredre-computation of a large dataSVD.

Dimension is low. Cost per point ishigh → Quadrature method

Phenotype

AdditiveDominance

Environmental Noise

σ2Aσ2

D

σ2N

Effects Effects

Figure: Explaining aphenotype by genes’ effects





Motivation for Triangles - Computer Graphics

Figure: Triangular Mesh for Graphical Rendering [Source: Wikipedia]





Prior Work

Recent work relating to general spaces by Aistleitner et al.,(2012), Brandolini et. al (2013)

Mapping by special functions/transformation from [0, 1]d

Pillards and Cools (2005). No bound on variation.

Several notions of discrepancy on the triangle/simplex but noexplicit constructions. Pillards and Cools (2005), Brandoliniet. al (2013).




Discrepancy

General Notions of Discrepancy

The signed discrepancy of P at the measurable setS ⊆ Ω ⊂ Rd is

δN(S ;P,Ω) = vol(S ∩ Ω)/vol(Ω)− AN(S ;P)/N.

The absolute discrepancy of points P for a class S ofmeasurable subsets of Ω is

DN(S;P,Ω) = supS∈S

DN(S ;P,Ω),

whereDN(S ;P,Ω) = |δN(S ;P,Ω)|.

Standard QMC works with Ω = [0, 1)d and takes for S the setof anchored boxes [0, a) with a ∈ [0, 1)d .




Discrepancy

Discrepancy due to Brandolini et. al. (2013)

A C

B

F ELb

La

D

Ta,b,C

a

b

Figure: Sets for Parallel Discrepancy




Triangular van der Corput pointsTriangular Kronecker Lattice

Triangular van der Corput construction

van der Corput sampling of [0, 1] the integern =

∑k>1 dkb

k−1 in base b > 2 is mapped to

xn =∑

k>1 dkb−k .

Points x1, . . . , xn ∈ [0, 1) have a discrepancy of O(log(n)/n).

Our situation : 4-ary expansion.






n > 0 in a base 4 representation n =∑

k>1 dk4k−1 wheredk ∈ 0, 1, 2, 3

0

12

3

AB

C

0001 02

03 101112

13

202122

23

303132

33

Figure: A labeled subdivision of ∆(A,B,C ) into 4 and then 16congruent subtriangles. Next are the first 32 triangular van der Corputpoints followed by the first 64. The integer labels come from the base 4expansion.






Computation : T = ∆(A,B,C )

T (d) =

∆(B+C

2 , A+C2 , A+B

2

), d = 0

∆(A, A+B

2 , A+C2

), d = 1

∆(B+A

2 ,B, B+C2

), d = 2

∆(C+A

2 , C+B2 ,C

), d = 3.

This construction defines an infinite sequence of fT (i) ∈ T forintegers i > 0.

For an n point rule, take x i = fT (i − 1) for i = 1, . . . , n.





Discrepancy Results

Theorem 1: B and Owen (2014)

For an integer k > 0 and non-degenerate triangle Ω = ∆(A,B,C ),let P consist of x i = fΩ(i − 1) for i = 1, . . . ,N = 4k . Then

DPN (P; Ω) =

7

9, N = 1

2

3√N− 1

9N, else.





Discrepancy Results


Let Ω be a non-degenerate triangle and, for integer N > 1, letP = (x1, . . . , xN), where x i = fΩ(i − 1). Then

DPN (P; Ω) 6 12/

√N.





Triangular Kronecker Lattice

Fun fact: some numbers are more irrational than others

Definition 1

A real number θ is said to be badly approximable if there exists aconstant c > 0 such that n||nθ|| > c for every natural numbern ∈ N and || · || denotes the distance from the nearest integer.





Construction Algorithm

Given a target sample size N, an angle α such that tan(α) is badlyapproximable (such as 3π/8) and a target triangle ∆(A,B,C ),

Take integer grid Z2

Rotate anti clockwise by α

Shrink by√

2N

Remove points not in thetriangle.

(Optionally) add/subtractpoints to get exactly N points

Linearly map it onto the desiredtriangle ∆(A,B,C )









Shrink by√

2N












Shrink by√

2N












Shrink by√

2N












Shrink by√

2N












Shrink by√

2N




Angle 3pi/8

Angle 5pi/8

Angle pi/4

Angle pi/2

Triangular lattice points







Let N > 1 be an integer and let R be the triangle withco-ordinates (0, 0), (0, 1) and (1, 0). Let α ∈ (0, 2π) be an anglefor which tan(α) is a badly approximable. Following the abovealgorithm, if P is the final set of points,

DPN (P;R) ≤ C log(N)/N.






3π8

5π8

π4

π2

Triangular Lattice Points

Figure: Triangular lattice points for target N = 64. Domain is anequilateral triangle. Angles 3π/8 and 5π/8 have badly approximabletangents. Angles π/4 and π/2 have integer and infinite tangentsrespectively and do not satisfy the conditions for discrepancyO(log(N)/N).






Parallel discrepancy oftriangular lattice points forangle α = 3π/8 andvarious targets N. Thenumber of points wasalways N or N + 1. Thedashed reference line is1/N. The solid line islog(N)/N.

5 10 20 50 100 200 500

0.01

0.02

0.05

0.10

0.20

0.50

Grid rotated by 3pi/8 radians

Sample size N

Dis

crep

ancy




High Dimensional AnaloguesResults

Generalization to Product Spaces

Motivation - Graphical Rendering

The potential for light to leave one triangle and reach anotheris calculated by

I =

∫X 2

f (x1, x2)dx1dx2

where X is a triangle.





Methodology

Let τ : [0, 1]→ XConsider a randomized (t,m, s)-net in base b on [0, 1]s , sayx1, . . . , xN ∈ [0, 1]s .

Figure: Digital Nets in two dimensions





Methodology

Let xi = (x1i , . . . , x

si ).

pi = (τ(x1i ), . . . , τ(x si )) ∈ X s .

We approximate the integral by

I =1

n

n∑i=1

f (pi )





Result for Product of two triangles


Let f be a smooth function on X 2. Let x1, . . . , xn be the points ofa randomized (t,m, s) net in base b. If we approximate theintegral of f by I = 1

n

∑ni=1 f (τ(xi )), then

Var(I ) = O

(log n

n2

)





Result for General Product Spaces

Let X s =∏s

j=1Xj where each Xj has dimension dj ≤ d .Further, assume each Xj can be recursively partitioned suchthat the partitions have a sphericity constraint.

2^6 decomposition 3^3 decomposition 4^3 decomposition

Figure: Recursive splitting in a Triangle






Figure: Recursive splitting in a disc







Let f be a smooth function on X s . Let x1, . . . , xn be the points ofa randomized (t,m, s) net in base b. If we approximate the integralof f by I = 1

n

∑ni=1 f (τ(xi )), then under the above conditions,

Var(I ) = O

((log n)s−1

n1+2/d

)





Thank you. Questions?


Documents

Quasi-Monte Carlo on Simplices - kinjalbasu.comkinjalbasu.com/talks/indus_aff.pdfQuasi-Monte Carlo on Simplices ... Department of Statistics Stanford University Joint work with Prof