monte carlo and quasi-monte carlo methodsmeshfree-methods-seminar/presentations/talk_20140521... ·...

Choosing a Problem Simulation Problems Integration Problems Open Problems and Ongoing Work References

Monte Carlo and Quasi-Monte Carlo Methods

Fred J. Hickernell

Department of Applied Mathematics, Illinois Institute of [email protected] mypages.iit.edu/~hickernell

Joint work with Aleks Borkovskiy, Sou-Cheng Choi, Siyuan Deng, Lan Jiang, LluısAntoni Jimenez Rugama, Yuewei Liu, and Art Owen

Key: PhD student, MS student

With lots of good feedback by this Meshfree Methods Seminar

Supported by NSF-DMS-1115392

May 21, 2014

[email protected] (Quasi-)Monte Carlo IIT, 5/21/2014 1 / 33

mailto:[email protected]

[email protected]

http://mypages.iit.edu/~hickernell

mypages.iit.edu/~hickernell


How to Choose a Research Problem

How to Choose a Research Problem

A good research problem should

§ Interest you,

§ Be significant and of interest to others,

§ Have a manageable size for the time you have to spend,

§ Have a solution you can expect to publish and/or can present as a conferencetalk or poster,

§ Be within your skill set, and

§ Be one that allows for possible collaboration.

Moreover, you should have good communication with your advisor.



Guaranteed Simulation

Guaranteed, Automatic, Adaptive Simulation

option priceprobability of success

pixel intensity

option payoff under random scenariorandom success or fail, i.e., 1 or 0intensity of random incident light ray

µ “ EpY q “ ?, where Y „ complicated

« µn :“1

n

nÿ

i“1

Yi, Y1, Y2, . . . IID

this is called a simple Monte Carlo method

n “ ? based on Y1, Y2, . . .

What conditions are required on Y ?






pixel intensity



« µn :“1

n

nÿ

i“1

Yi, Y1, Y2, . . . IID


n “ ? based on Y1, Y2, . . .

Er|µ´ µn|2s “varpY q

n, but this criterion is too weak, and

and varpY q is unknown







pixel intensity



« µn :“1

n

nÿ

i“1

Yi, Y1, Y2, . . . IID


n “ ? based on Y1, Y2, . . .

Want Pr|µ´ µn| ď εs ě 99% guaranteed for user-specified ε




Heuristic Simulation

Heuristic, Automatic, Adaptive Simulation

§ Given an nσ and a C ą 1, estimate varpY q by an inflated sample variance:

σ2 “C2

nσ ´ 1

nσÿ

i“1

pYi ´ µσq2, µσ “

1

nσ

nσÿ

i“1

Yi

§ Use the Central Limit Theorem to determine how many samples are neededto estimate the mean:

nµ “

S

ˆ

2.58σ

ε

˙2W

§ Estimate the mean independently of the sample variance:

µ “1

nµ

nµÿ

i“1

Yi`nσ

and hope thatPr|µ´ µ| ď εs ě 99%



Heuristic Simulation

Heuristic, Automatic, Adaptive Simulation

§ Given an nσ and a C ą 1, estimate varpY q by an inflated sample variance:

σ2 “C2

nσ ´ 1

nσÿ

i“1

pYi ´ µσq2, how good is this? µσ “

1

nσ

nσÿ

i“1

Yi

§ Use the Central Limit Theorem (only good for nÑ8) to determine howmany samples are needed to estimate the mean:

nµ “

S

ˆ

2.58σ

ε

˙2W

§ Estimate the mean independently of the sample variance:

µ “1

nµ

nµÿ

i“1

Yi`nσ

and hope thatPr|µ´ µ| ď εs ě 99%



Guarantee the Variance Estimate

Guarantee the Variance

The sample variance, v is an unbiased estimate of σ2 “ EpY ´ µq2.

v “1

nσ ´ 1

nσÿ

i“1

pYi ´ µnσ q2, µnσ “

1

nσ

nσÿ

i“1

Yi, Y1, Y2, . . . IID „ ρ

Ervs “ σ2, varpvq “σ4

nσ

ˆ

κ´nσ ´ 3

nσ ´ 1

˙

, κ :“EpY ´ µq4

σ4

Cantelli’s Inequality (Lin and Bai, 2010, 6.1e) guarantees that an inflated samplevariance bounds the variance from above with uncertainty α,

σ2 :“ C2v, Probpσ2 ě σ2q ě 1´ α, C ą 1, e.g., α “ 0.5%

provided that the kurtosis of the integrand, κ, is not too large, i.e.,

κ ďnσ ´ 3

nσ ´ 1`

ˆ

αnσ1´ α

˙ˆ

1´1

C2

˙2

“: κmaxpα, nσ,Cq.



Guarantee the Variance Estimate

Guarantee the Variance

σ2 “C2

nσ ´ 1

nσÿ

i“1

pYi ´ µnσ q2,

Probpσ2 ě σ2q ě 1´ α

if κ ď κmaxpα, nσ,Cq

10−3

10−2

10−1

100

100

101

102

103

104

105

κ

max

α

nσ = 30

nσ = 100

nσ = 1000

nσ = 10000

nσ = 100000

C “ 1.5



Guarantee the Mean

Guarantee the Mean

The Central Limit Theorem gives an asymptotic result for fixed z ě 0:

Prob

„∣∣∣∣EpY q ´ 1

n

nÿ

i“1

Yi`nσlooooomooooon

µ

∣∣∣∣ ď zσ?n

Ñ 1´ 2Φp´zq as nÑ8

A non-uniform Berry-Esseen Inequality (Nefedova and Shevtsova, 2012) gives ahard upper bound:

Prob

„

|µ´ µ| ď zσ?n

ě 1´ 2

ˆ

Φp´zq `18.1139κ3{4

?n

p1` |z|q´3

˙

This guarantees that Prob r|µ´ µ| ď εs ě 1´ α if the sample size is large enough:

n ě NBpε{σ, α, κq :“ min

#

m P N : Φ`

´ε?m{σ

˘

`18.1139κ3{4

?m p1` ε

?m{σq

3 ďα

2

+

—σ2

ε2as

ε

σÑ 0



meanMC g

meanMC g (Hickernell et al., 2014)

To evaluate EpY q given input f , ρ, ε, α, nσ, C, and Nmax:

§ Compute α “ 1´?

1´ α, and the maximum kurtosis allowed,κmaxpα, nσ,Cq.

§ Overestimate the variance: σ2 “C2

nσ ´ 1

nσÿ

i“1

pYi ´ µnσ q2.

§ Choose the new sample size, nµ “ min pmax pnσ, NBpε{σ, α, κmaxqq , Nmaxq,for the sample mean.

§ Finally, compute the sample mean: µ “1

nµ

nµÿ

i“1

Yi`nσ .

Then Prob r|EpY q ´ µ| ď εs ě 1´ α provided κ ď κmax and n ă Nmax.

(See (Bayer et al., 2014) for a similar method.)



Guarantee the Cost (Sample Size, Time)

Guarantee the Time (Sample Size)

Cantelli’s inequality also tells us that the estimated variance, σ2, will notoverestimate the true variance, σ2, by much, and so the number of functionvalues needed is not unnecessarily large:

costpε, meanMC g, σq “ supf :κďκmax

minNtProbrnσ ` n ď N s ě 1´ βu

ď nσ `maxpnσ, NBpε{pσγq, α, κ3{4maxqq —

σ2

ε2,

γ :“ C

$

&

%

1`

d

ˆ

α

1´ α

˙ˆ

1´ β

β

˙ˆ

1´1

C2

˙2,

.

-

1{2

.

Cost depends on σ2 “ varpY q, but the algorithm does not need to know σ2.



Numerical Example

Asian Geometric Mean Call OptionY “ the discounted payoff of an option, where the stock is monitored d times

µ “ EpY q “ fair price of the option

d “ # of time steps “ 1, 2, 4, . . . , 64

ε “ 0.05, C “ 1.1

nσ “ 213 “ 8192, κmax “ 2.24 nσ “ 218 “ 262144, κmax “ 40.1



Integration As Simulation

Multivariate Integration ÐÑ Simulation

Suppose that one wants to compute the integralż

Rdgpxqdx.

where g is some known function. Write

gpxq “ fpxqρpxq

where ρ is a probability density function. There are many ways to do this. Thenwe may think of Y “ fpXq for where X „ ρ, and then

µ “ EpY q “ ErfpXqs “ż

Rdfpxq ρpxqdx “

ż

Rdgpxqdx

µ “1

n

n´1ÿ

i“0

Yi “1

n

n´1ÿ

i“0

fpziq, tziu8i“0 “„” ρ

The algorithm cubMC g uses meanMC g to compute such [email protected] (Quasi-)Monte Carlo IIT, 5/21/2014 11 / 33


Quasi-Monte Carlo — Can We Get the Answer Faster?

Quasi-Monte Carlo — Better Ways to Sample than IID

You want µ “ EpY q “ż

r0,1sdfpxqdx, where Y “ fpXq, X „ Ur0, 1sd. How do

you best choose the zi to generate Yi “ fpziq and µ? E.g., n “ 64, d “ 4:

randombunches &

gaps ofpoints

x1

x2

x1

x2

Latinhypercubeexcellentmarginals

orthogonalarraygood

marginals,good low

orderinteractions

x1

x2

x1

x2

gridgood highorderinteractions



Quasi-Monte Carlo — Can We Get the Answer Faster?

Sobol’ Points — Having It All

You want µ “ EpY q “ż

r0,1sdfpxqdx, where Y “ fpXq, X „ Ur0, 1sd. How do

you best choose the zi to generate Yi “ fpziq and µ? E.g., n “ 64, d “ 4:

§ all the right tiles have theright numbers of points

§ Sobol’ points (Sobol’, 1967)comprise the best knownfamily digital nets (Dick andPillichshammer, 2010;Lemieux, 2009)

§ extensible in n (powers of 2)and d

§ can be scrambled (Owen,2000)

x1

x2



Scrambling Digital Nets

Sobol’ and Other Digital Nets May Be RandomlyScrambledTo facilitate error estimation and remove bias, Sobol’ sets may be randomlyscrambled while preserving their equi-distribution structure. This expressed as arandom transformation T : r0, 1qd Ñ r0, 1qd applied to each sample point (Owen,1995; 1997a; 1997b).

Initial Level 1





Level 1 Level 2





Level 2 Level 3





Level 3 Level 10





Level 10 Final





Initial Final



Error Analysis of Quasi-Monte Carlo Methods


The goal is to approximate expectations by sample averages:

µ :“ ErfpXqs “

ż

Rdfpxq ρpxqdx « µn :“

1

n

n´1ÿ

i“0

fpziq (sample average)

For a fixed design, tziun´1i“0 , the error is (Hickernell, 1998; 1999)

H is a Hilbert space supfPH

‖f‖Hď1

|µ´ µn|with reproducing kernel K

f is a zero-mean stochastic process b

Ef |µ´ µn|2with covariance kernel K

,

/

/

/

/

.

/

/

/

/

-

“ Dptziun´1i“0 q

Unfortunately, this does not help us build an automatic algorithm because wecannot easily compute ‖f‖H.




Discrepancy as Goodness-of-Fit

The discrepancy of a design, tziun´1i“0 is the distance between its empirical

distribution, Ftziu

n´1i“0

, and a specified target distribution, F (Hickernell, 1998;

1999):

Dptziun´1i“0 q “

∥∥∥F ´ Ftziun´1i“0

∥∥∥M

think Kolmogorov-Smirnov statistic

“

"ż

RdˆRdKpx, tqdpF ´ F

tziun´1i“0qpxqdpF ´ Feqptq

*1{2

“

"ż

RdˆRdKpx, tq ρpxqρptqdxdt

´2

n

n´1ÿ

i“0

ż

RdKpzi, tq ρptqdt`

1

n2

n´1ÿ

i,j“0

Kpzi, zjq

+1{2

provided that the norm is induced by an inner product. Here K is a symmetric,positive definite kernel, and ρ is the probability density corresponding to F .




Centered Discrepancy ExampleTypically integration domains are boxes, target distributions have independentmarginals, and kernels has product form. E.g., for the uniform distribution onr0, 1sd, a typical kernel is (Hickernell, 1998)

Kpx, tq “dź

`“1

„

1`1

2

ˆ∣∣∣∣x` ´ 1

2

∣∣∣∣` ∣∣∣∣t` ´ 1

2

∣∣∣∣´ |x` ´ t`|˙

‖f‖2H “∣∣∣∣f ˆ1

2, . . . ,

1

2

˙∣∣∣∣2 ` ∥∥∥∥ BfBx1

ˆ

¨,1

2, . . . ,

1

2

˙∥∥∥∥2L2

` ¨ ¨ ¨

`

∥∥∥∥ B2f

Bx1Bx2

ˆ

¨, ¨,1

2, . . . ,

1

2

˙∥∥∥∥2L2

` ¨ ¨ ¨ `

∥∥∥∥ Bdf

Bx1 ¨ ¨ ¨ Bxd

∥∥∥∥2L2

Dptziun´1i“0 q “

#

ˆ

13

12

˙d

´2

n

n´1ÿ

i“0

dź

`“1

«

1`1

2

˜∣∣∣∣xi` ´ 1

2

∣∣∣∣´ ∣∣∣∣xi` ´ 1

2

∣∣∣∣2¸ff

`1

n2

n´1ÿ

i,j“0

dź

`“1

„

1`1

2

ˆ∣∣∣∣xi` ´ 1

2

∣∣∣∣` ∣∣∣∣xj` ´ 1

2

∣∣∣∣´ |xi` ´ xj`|˙

+1{2



Sobol Points

Sobol’ Points

Let ‘ denote binary digit by digit addition modulo 2:

1

8‘

5

8“ 20.001‘ 20.101 “ ‘20.100 “

1

2, 1‘ 5 “ 0012 ‘ 1012 “ 1002 “ 4

Sobol’ points tziu satisfy

z0 “ 0, zi ‘ z` “ zi‘` @i, ` P N0

z1 ‘ z5 “ p20.100, 20.100q

‘ p20.101, 20.001q

“ p20.001, 20.101q

“ z4 “ z1‘5

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

x1

x2

z1

z5

z4



Walsh Functions

Walsh FunctionsThe base-2 Walsh function, walshp¨, ¨q : pk,xq ÞÑ p´1qxk,xy , is defined in terms ofthe a bilinear function x¨, ¨y : Nd0 ˆ r0, 1qd Ñ t0, 1u:

xk,xy “ xpk1, . . . , kdq, px1, . . . , xdqy :“dÿ

j“1

xkj , xjy mod 2

xkj , xjy “ xp¨ ¨ ¨ kj1kj0q2, 20.xj1xj2 ¨ ¨ ¨y :“ kj0xj1 ` kj1xj2 ` ¨ ¨ ¨ mod 2

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(0,x)

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(1,x)

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(2,x)

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(3,x)

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(4,x)

0 0.25 0.5 0.75 1

−1

−0.5

0

0.5

1

x

walsh(5,x)



Error Bounds via Fourier-Walsh Expansions

Sobol’ Cubature Error via Fourier-Walsh Expansions

The Fourier-Walsh expansion of an integrand is given by

fpxq “ÿ

kPNd0

p´1qxk,xy fpkq, where fpkq :“

ż

r0,1qdfpxqp´1qxk,xy dx.

Sobol’ points integrate most Walsh functions perfectly, but some terribly:

1

2m

2m´1ÿ

i“0

p´1qxk,ziy “

#

1, k P PKm :“ tk P Nd0 : xk, ziy “ 0, i “ 0, . . . 2m ´ 1u,

0 k R PKm

So Sobol’ cubature error depends on the sizes of the Fourier-Walsh coefficients fork in the dual Sobol’ set PKm:

errpf, 2m, tziuq :“

∣∣∣∣∣ż

r0,1qdfpxqdx´

1

2m

2m´1ÿ

i“0

fpziq

∣∣∣∣∣ “∣∣∣∣∣ ÿ

kPPKmzt0u

fpkq

∣∣∣∣∣How do we reliably bound this error based on the fpziq?




Wavenumber Map

Define a bijective mapping k : N0 Ñ Nd0 such that for all m,λ P N0 andκ “ 0, . . . , 2m ´ 1,

kp0q “ 0, kpκ` λ2mq “ kpκq ‘ l for some l P PKm,e.g., kp29q “ kp5q ‘ l for some l P PK3 , but not for some l P PK4

One can express the Fourier-Walsh expansion for the integrand and the error as

fpxq “8ÿ

κ“0

p´1qxkpκq,xy fκ, fκ :“ fpkpκqq, errpf, 2m, tziuq “

∣∣∣∣∣ 8ÿλ“1

fλ2m

∣∣∣∣∣Large κ implies typically smaller fκ in a way that is made explicit on the nextslide. Moreover, the discrete Fourier-Walsh coefficients are aliased as follows:

fm,κ :“1

2m

2m´1ÿ

i“0

p´1qxkpκq,ziyfpziq fm,κ`λ2m “ fm,κ,

e.g., f3,29 “ f3,5, but generally f29 ‰ f5 and f4,29 ‰ f4,5




Cone Assumptions on Decay of Fourier Walsh Coefficients

100

101

102

103

104

10−15

10−10

10−5

100

κ

|fκ|

err ≤ S(0, 11)S(11)S(7)

100

101

102

103

104

10−15

10−10

10−5

100

κ

|fκ|

err ≤ S(0, 12)S(12)S(8)

Make cone assumptions on how the fκ decay. There exist non-increasing pω and qωsuch that for all 0 ď ` ď m

pSp`,mq :“2`´1ÿ

κ“t2`´1u

8ÿ

λ“1

∣∣fκ`λ2m ∣∣, qSpmq :“8ÿ

κ“2m

∣∣fκ∣∣ Sp`q :“2`´1ÿ

κ“t2`´1u

∣∣fκ∣∣pSp`,mq ď pωpm´ `qqSpmq, qSpmq ď qωp`qSpm´ `q @m ě `` `˚.




Error Bound in Terms of Discrete Fourier-Walsh Coeff.

Cone conditions:

pSp`,mq :“2`´1ÿ

κ“t2`´1u

8ÿ

λ“1

∣∣fκ`λ2m ∣∣, qSpmq :“8ÿ

κ“2m

∣∣fκ∣∣ Sp`q :“2`´1ÿ

κ“t2`´1u

∣∣fκ∣∣pSp`,mq ď pωpm´ `qqSpmq @`, qSpmq ď qωp`qSpm´ `q @m ě `` `˚.

Then we can bound the error as follows:

errpf, 2m, tziuq “

∣∣∣∣∣ 8ÿλ“1

fλ2m

∣∣∣∣∣ ď 8ÿ

λ“1

∣∣fλ2m ∣∣ “ pSp0,mq

ď pωpmqqSpmq ď pωpmqqωp`qSpm´ `q

ďpωpmqqωp`qrSpm´ `,mq

1´ pωp`qqωp`qproof

where

rSp`,mq :“2`´1ÿ

κ“t2`´1u

∣∣fm,κ∣∣, fm,κ :“1

2m

2m´1ÿ

i“0

p´1qxkpκq,ziyfpziq




cubSobol g—Guaranteed, Adaptive, Sobol’ CubatureGiven an error tolerance ε ą 0 and an integrand f , fix the lag ` P N and let

C “qωp`q

1´ pωp`qqωp`q, m “ `` `˚.

Step 1. Compute the sum of the (data-based) discrete Fourier-Walsh

coefficients rSpm´ `,mq.Step 2. If the error tolerance is satisfied, i.e.,

CpωpmqrSpm´ `,mq ď ε,

then return the Sobol’ cubature answer.Step 3. Otherwise, increase m by one, and return to Step 1.

Theorem. If f satisfies the cone conditions on its Fourier-Walsh coefficients, then∣∣∣∣żr0,1qd

fpxqdx´1

2m

2m´1ÿ

i“0

fpziq

∣∣∣∣ ď ε

at a computational cost of Oprm` $pfqs2mq, where $pfq is the cost of a functionevaluation and m ď mintm1 : Cpωpm1qr1` pωp`qqωp`qsSpm1 ´ `q ď εu proof .



Numerical Examples

Asian Geometric Mean Call Option

Y “ the discounted payoff of an option, where the stock is monitored d times

µ “ EpY q “ fair price of the option

d “ # of time steps “ 1, 2, 4, . . . , 64

IID, cubMC g Sobol’, cubSobol g



Numerical Examples

Genz and Keister Examples

These examples come from Genz (1987) and Keister (1996)

cubSobol g

‚ within budget‚ exceeded budget



Alternatives

Why Not Replications to Estimate Error?

∣∣∣∣∣ż

r0,1qdfpxqdx´ µn

∣∣∣∣∣ ď C

g

f

f

e

1

R´ 1

Rÿ

r“1

pYr ´ µnq2, µn :“1

R

Rÿ

r“1

Yr

IID Replications Yr “R

n

n{R´1ÿ

i“0

fpzprqi q, where tz

prqi u are independent

randomizations.

Internal Replications Yr “R

n

rn{R´1ÿ

i“pr´1qn{R

fpziq.

Want R small to take advantage of Sobol’ point evenness, but need R large toensure that sample variance of Yr represents error (Deng, 2013; Hickernell et al.,2014).



Monte Carlo Methods for Computing Expectations of Random Variables

Monte Carlo Methods for Computing Expectations ofRandom Variables (Lan Jiang, You?)

§ Sometimes we want to compute EpY q based on samples of a random variableY , such that

Probr|EpY q ´ µ| ď εs ď 99%.

§ The simplest way is to use µ “nÿ

i“1

Yi, where the Yi are IID.

§ The problem is how large to choose n (Hickernell et al., 2014). We focus on

the of random variables with kurtosis bounded by κmax.

§ The problem of Bernoulli Y is an important special case.

§ This work can also be used to evaluate multidimensional integrals. IfY “ fpXq, where X has probability density function ρ, then

EpY q “ż

Rdfpxq ρpxqdx.



Multilevel Monte Carlo Methods

Multilevel Monte Carlo Methods (Aleks Borkovskiy)

§ Sometimes we want to compute EpY q based on samples of a random variableY , but getting just one Y costs an infinite amount of time.

§ However, one can write Y p0q “ 0

Y “ pY p1q ´ Y p0qq ` pY p2q ´ Y p1qq ` ¨ ¨ ¨ ` pY pLq ´ Y pL´1qq ` pY ´ Y pLqq

where Y p`q only costs $p`q operations, and Y ´ Y pLq Ñ 0 as LÑ8

§ The multilevel idea is to use a lot of samples to approximate EpY p1q ´ Y p0qq,fewer samples to approximate EpY p2q ´ Y p1qq, . . .

§ How many samples are needed for each piece? Again should look at a

of random variables.



Quasi-Monte Carlo Methods for Computing Multidimensional Integrals

Quasi-Monte Carlo Methods for ComputingMultidimensional Integrals (Siyuan, Tony, You?)

§ Sometimes we want to compute a multidimensional integralż

r0,1qdfpxqdx (perhaps after a suitable variable transformation).

§ Quasi-Monte Carlo methods choose tziu8i“0 where the first 1, 2, 4, 8, . . .

points are evenly distributed on r0, 1qd. Then∣∣∣∣∣ż

r0,1qdfpxqdx´

1

2m

2m´1ÿ

i“0

fpziq

∣∣∣∣∣ ď Dptziu8i“0,mqV pfq

where Dptziu8i“0,mq is the discrepancy (measure of unevenness) of the first

2m points, and V pfq is the variation (measure of roughness) of the integrand.§ We want to choose m to make∣∣∣∣∣

ż

r0,1qdfpxqdx´

1

2m

2m´1ÿ

i“0

fpziq

∣∣∣∣∣ ď ε.




Quasi-Monte Carlo Methods for ComputingMultidimensional Integrals (Siyuan, Tony, You?)

∣∣∣∣∣ż

r0,1qdfpxqdx´

1

2m

2m´1ÿ

i“0

fpziq

∣∣∣∣∣ ď Dptziu8i“0,mqV pfq ď ε for m “?

§ Some results doing an alternative error analysis using Fourier series.

§ Again focus on a of integrands.

§ Key hurdles:§ How well does it work in practice?§ Relating the of integrands to traditional sets of integrands.

§ Determining the lower bound on the cost of the best algorithm (lowercomplexity bound).




References I

Bayer, C., H. Hoel, E. von Schwerin, and R. Tempone. 2014. On nonasymptotic optimalstopping criteria in monte carlo simulations on nonasymptotic optimal stopping criteria inMonte Carlo Simulations, SIAM J. Sci. Comput. 36, A869–A885.

Deng, S. 2013. An investigation of the quasi-standard error for quasi-monte carlo method,Master’s Thesis.

Dick, J. and F. Pillichshammer. 2010. Digital nets and sequences: Discrepancy theory andquasi-Monte Carlo integration, Cambridge University Press, Cambridge.

Genz, A. 1987. A package for testing multiple integration subroutines, Numerical integration:Recent developments, software and applications, pp. 337–340.

Hickernell, F. J. 1998. A generalized discrepancy and quadrature error bound, Math. Comp. 67,299–322.

. 1999. Goodness-of-fit statistics, discrepancies and robust designs, Statist. Probab.Lett. 44, 73–78.

Hickernell, F. J., L. Jiang, Y. Liu, and A. B. Owen. 2014. Guaranteed conservative fixed widthconfidence intervals via Monte Carlo sampling, Monte Carlo and quasi-Monte Carlo methods2012, pp. 105–128.




References IIKeister, B. D. 1996. Multidimensional quadrature algorithms, Computers in Physics 10,119–122.

Lemieux, C. 2009. Monte Carlo and quasi-Monte Carlo sampling, Springer Science+BusinessMedia, Inc., New York.

Lin, Z. and Z. Bai. 2010. Probability inequalities, Science Press and Springer-Verlag, Beijingand Berlin.

Nefedova, Yu. S. and I. G. Shevtsova. 2012. On non-uniform convergence rate estimates in thecentral limit theorem, Theory Probab. Appl. 57, 62–97.

Owen, A. B. 1995. Randomly permuted pt,m, sq-nets and pt, sq-sequences, Monte Carlo andquasi-Monte Carlo methods in scientific computing, pp. 299–317.

. 1997a. Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34,1884–1910.

. 1997b. Scrambled net variance for integrals of smooth functions, Ann. Stat. 25,1541–1562.

. 2000. Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo, MonteCarlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo, pp. 86–97.

Sobol’, I. M. 1967. The distribution of points in a cube and the approximate evaluation ofintegrals, U.S.S.R. Comput. Math. and Math. Phys. 7, 86–112.


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