reliable error estimation for monte carlo and quasi...
TRANSCRIPT
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Reliable Error Estimation forMonte Carlo and Quasi-Monte Carlo SimulationWhen do you have enough samples to stop?
Fred J. Hickernell
Department of Applied Mathematics, Illinois Institute of [email protected] mypages.iit.edu/~hickernell
Joint work with Sou-Cheng Choi (NORC, U Chicago), Jiang Lan (IIT),Lluıs Antoni Jimenez Rugama (IIT), and Art Owen (Stanford)
Supported by NSF-DMS-1115392
Thank you for your kind invitation to visit!
October 7, 2014
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 1 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 2 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Monte Carlo method “ statistical sampling with computersoriginated at LANL ca. 1947 (Eckhardt, 1987)
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 2 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Monte Carlo method “ statistical sampling with computersoriginated at LANL ca. 1947 (Eckhardt, 1987)
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
n “ ? based on Y1, Y2, . . .eyeball it? as much time as possible?Central Limit Theorem confidence interval?
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 2 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Monte Carlo method “ statistical sampling with computersoriginated at LANL ca. 1947 (Eckhardt, 1987)
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
n “ ? based on Y1, Y2, . . .under certain reasonable conditions on Y ?
(H. et al., 2014; Jiang and H., 2014)
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 2 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Heuristic, Adaptive Simulation Based on the CLTCLT Adaptive Algorithm. Given an IID random generator for Y1, Y2, . . ., anabsolute error tolerance, εa, and parameters, nσ P N and C ą 1,
Step 1. Estimate the Variance. Compute the sample variance based on nσsamples of Y :
σ2 “1
nσ ´ 1
nσÿ
i“1
pYi ´ µσq2, µσ “
1
nσ
nσÿ
i“1
Yi,
and take C2σ2 as your estimate of varpY q.
Step 2. Estimate the Mean. Use the Central Limit Theorem to determine howmany samples are needed to estimate the mean, and estimate the meanindependently of the sample variance:
n “
S
ˆ
2.58Cσ
εa
˙2W
, µn “1
n
nσ`nÿ
i“nσ`1
Yi.
Hope that Pr|µ´ µn| ď εas ě 99%[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 3 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Heuristic, Adaptive Simulation Based on the CLTCLT Adaptive Algorithm. Given an IID random generator for Y1, Y2, . . ., anabsolute error tolerance, εa, and parameters, nσ P N and C ą 1,
Step 1. Estimate the Variance. Compute the sample variance based on nσsamples of Y :
σ2 “1
nσ ´ 1
nσÿ
i“1
pYi ´ µσq2, µσ “
1
nσ
nσÿ
i“1
Yi,
and take C2σ2 How good is this? as your estimate of varpY q.
Step 2. Estimate the Mean. Use the Central Limit Theorem (only good fornÑ8) to determine how many samples are needed to estimate the mean, andestimate the mean independently of the sample variance:
n “
S
ˆ
2.58Cσ
εa
˙2W
, µn “1
n
nσ`nÿ
i“nσ`1
Yi.
Hope that Pr|µ´ µn| ď εas ě 99%[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 3 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Guaranteed Adaptive Monte Carlo Simulation
Adaptive Algorithm meanMC g. Given an IID random generator for Y1, Y2, . . .,an absolute error tolerance, εa, and parameters, nσ P N and C ą 1,
Step 1. Bound the Variance. Compute the sample variance based on nσsamples of Y . We know that PrC2σ2 ě varpY qs ě 99.5% forkurtpY q :“ ErpY ´ µq4s{ varpY q2 ď κmaxpnσ,Cq by Cantelli’s inequality.
Step 2. Estimate the Mean. Use a Berry-Esseen Inequality to determine thesample size, n, needed for the sample mean by solving
Φ`
´?nεa{pCσq
˘
looooooooomooooooooon
CLT part
`∆np?nεa{pCσq, κmaxq
looooooooooooomooooooooooooon
Berry-Esseen extra part
ď 0.0025.
Then compute the sample mean, µn, of an independent sample of size n.
Theorem. (H. et al., 2014) If kurtpY q ď κmaxpnσ,Cq, then we must havePr|µ´ µn| ď εas ě 99%. The computational cost is nσ ` n “ OpvarpY q{ε2aq withhigh probability, even though varpY q is unknown a priori.
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 4 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Numerical Examples for meanMC g
Asian Geometric Mean Call Optiond “ 1, 2, 4, . . . , 64 time steps
cd
ż
Rde´‖x‖2
cosp‖x‖qdx “ 1
d “ 1, 2, 3, 4 Keister (1996)
« 100% success « 100% success§ Sample size for bounding varpY q should be ě 103 to get a reasonable κmax
§ meanMC g is conservative, see Bayer et al. (2014) for a heuristic alternative§ Cannot yet handle relative error tolerances
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 5 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
n “ ? based on Y1, Y2, . . .under certain reasonable conditions on Y ?
(H. et al., 2014; Jiang and H., 2014)
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
using X1,X2, . . . more even than IID(H. and Jimenez Rugama, 2014)(Jimenez Rugama and H., 2014)
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 6 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Quasi-Monte Carlo Cubature Using Digital Nets∣∣∣∣∣ż
r0,1qdfpxqdx
looooooomooooooon
µ“ErfpXqs
´1
2m
2m´1ÿ
i“0
fpziq
∣∣∣∣∣ ď εa
ď
8ÿ
λ“1
∣∣fλ2m∣∣ ď pωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qproof ď εa
digital net nodes
§ n “ 2m “? samples neededto satisfy error tolerance
§ Highly stratified sampling
§ Can be random (Owen, 2000)
§ Sobol’ sequences are apopular choice (Dick and
Pillichshammer, 2010)
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 7 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Quasi-Monte Carlo Cubature Using Digital Nets∣∣∣∣∣ż
r0,1qdfpxqdx
looooooomooooooon
µ“ErfpXqs
´1
2m
2m´1ÿ
i“0
fpziq
∣∣∣∣∣ ď D`
tziu2m´1i“0
˘
V pfqloooooooooomoooooooooon
(Niederreiter, 1992; H., 1998)
ď εa
ď
8ÿ
λ“1
∣∣fλ2m∣∣ ď pωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qproof ď εa
digital net nodes
§ n “ 2m “? samples neededto satisfy error tolerance
§ D`
tziu2m´1i“0
˘
requires atleast Opn2q “ Op22mqoperations
§ V pfq is impractical tocalculate because it involvesLp-norms of mixed partialderivatives of f
0 0.25 0.5 0.75 10
0.25
0.5
0.75
1
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 7 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Quasi-Monte Carlo Cubature Using Digital Nets∣∣∣∣∣ż
r0,1qdfpxqdx
looooooomooooooon
µ“ErfpXqs
´1
2m
2m´1ÿ
i“0
fpziq
∣∣∣∣∣ ď 8ÿ
λ“1
∣∣fλ2m∣∣
ďpωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qproof ď εa
digital net nodes Walsh coefficients in dual net
Walsh functions & coefficients
fpxq “8ÿ
κ“0
p´1qxkpκq,xy fκ
fm,κ :“1
2m
2m´1ÿ
i“0
p´1qxkpκq,ziyfpziq
“
8ÿ
λ“0
fκ`λ2m aliasing
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 7 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Quasi-Monte Carlo Cubature Using Digital Nets∣∣∣∣∣ż
r0,1qdfpxqdx
looooooomooooooon
µ“ErfpXqs
´1
2m
2m´1ÿ
i“0
fpziq
∣∣∣∣∣ ď 8ÿ
λ“1
∣∣fλ2m∣∣ ď pωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qproof ď εa
digital net nodes Walsh coefficients in dual net
pS`,mpfq :“2`´1ÿ
κ“t2`´1u
8ÿ
λ“1
∣∣fκ`λ2m∣∣,qSmpfq :“
8ÿ
κ“2m
∣∣fκ∣∣S`pfq :“
2`´1ÿ
κ“t2`´1u
∣∣fκ∣∣rS`,mpfq :“
2`´1ÿ
κ“t2`´1u
∣∣fm,κ∣∣Ð can get 100
101
102
103
104
10−15
10−10
10−5
100
κ
|fκ|
err ≤ S(0, 12)S(12)S(8)
conditions: pS`,mpfq ď pωpm´ `qqSmpfq, qSmpfq ď qωp`qSm´`pfq.
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 7 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Guaranteed Adaptive Quasi-Monte Carlo Simulation
Have
∣∣∣∣∣ż
r0,1qdfpxqdx
looooooomooooooon
µ“ErfpXqs
´1
2m
2m´1ÿ
i“0
fpziq
looooooomooooooon
µm
∣∣∣∣∣ ď pωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qloooooooooooomoooooooooooon
errpmq
.
We want to find m and µm that guarantees |µ´ µm| ď εa.
Adaptive Algorithm cubSobol g. Given a black-box function evaluator, f , anda tolerance, εa, fix ` and initalize m ą `.
Step 1. Compute the data-based error bound, errpmq.Step 2. If errpmq is small enough such that errpmq ď εa, then return µm.Step 3. Otherwise, increase m by one, and return to Step 1.
Theorem. (H. and Jimenez Rugama, 2014) For integrands satifying the
conditions cubSobol g succeeds proof , and the computational cost isOprm` $pfqs2mq “ Oprlogpnq ` $pfqsnq, for some
m ď min
"
m1 :r1` pωp`qqωp`qspωpm1qqωp`qSm1´`pfq
1´ pωp`qqωp`qď εa
*
proof more proof
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 8 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Numerical Examples for meanMC g and cubSobol g
Asian Geometric Mean Call Option, d “ 1, 2, 4, . . . , 64
meanMC g cubSobol g
« 100% success « 100% success
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 9 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Numerical Examples for meanMC g and cubSobol g
cd
ż
Rde´‖x‖2
cosp‖x‖qdx “ 1 Keister (1996)
d “ 1, . . . , 4 d “ 1, . . . , 4
meanMC g cubSobol g
« 100% success « 100% success
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 10 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Numerical Examples for meanMC g and cubSobol g
cd
ż
Rde´‖x‖2
cosp‖x‖qdx “ 1 Keister (1996)
d “ 1, . . . , 4 d “ 1, . . . , 19
meanMC g cubSobol g
« 100% success « 95% success
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 10 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Observations on meanMC g and cubSobol g
§ These algorithms require Y or f to lie in a , which describes how
nasty they are allowed to be. Bigger ùñ more robust algorithms.
§ We focus on of random variables or functions (instead of other
shapes) because our problems are homogeneous, and our error bounds arepositively homogeneous.
§ Lyness (1983) warned against adaptive algorithms that use C |µn ´ µn1 | as anerror estimate (stop when the change is small). We avoid this type error ofestimate, but it is prevalent in popular adaptive algorithms.
§ There are rather general sufficient conditions under which adaption providesno advantage (Bahadur and Savage, 1956 ; Traub et al., 1988, Chapter 4,
Theorem 5.2.1 ; Novak, 1996). To violate those conditions we consider
nonconvex of random variables or integrands.§ cubSobol g accommodates hybrid error requirements,|µ´ µn| ď maxpεa, εr |µ|q, and meanMC g will soon.
§ These algorithms are featured in our Guaranteed Automatic IntegrationLibrary (GAIL) code.google.com/p/gail/ (Choi et al., 2013–2014).
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 11 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Stopping the Simulation When the Error is Small Enoughoption price
probability of process failurepixel intensity
option payoff under random scenariorandom failure or not, 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
Want Pr|µ´ µn| ď εas ě 99% guaranteed for user-specified εa
What about Y “ fpXq, µ “
ż
r0,1qdfpxqdx “?
|µ´ µn| ď maxpεa, εr |µ|qY “ the limit of Y pdq as dÑ8, e.g.,
discretizing an SDE with d steps, orthe dth time step of a Markov Chain
p “ PpY ď µq, µ “? quantiles
µ “ ErY pxq |Y px1q, . . . , Y pxnqs “? kriging
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 12 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
References I
Bahadur, R. R. and L. J. Savage. 1956. The nonexistence of certain statistical procedures innonparametric problems, Ann. Math. Stat. 27, 1115–1122.
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.
Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, and Y. Zhang. 2013–2014. GAIL: GuaranteedAutomatic Integration Library (versions 1, 1.3).
Dick, J. and F. Pillichshammer. 2010. Digital nets and sequences: Discrepancy theory andquasi-Monte Carlo integration, Cambridge University Press, Cambridge.
Eckhardt, R. 1987. Stan Ulam, John von Neumann, and the Monte Carlo method, Los AlamosScience, 131–136.
H., F. J. 1998. A generalized discrepancy and quadrature error bound, Math. Comp. 67,299–322.
H., 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.
H., F. J. and Ll. A. Jimenez Rugama. 2014. Reliable adaptive cubature using digital sequences.submitted for publication.
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 13 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
References II
Jiang, L. and F. J. H. 2014. Guaranteed conservative confidence intervals for means of Bernoullirandom variables. in preparation.
Jimenez Rugama, Ll. A. and F. J. H. 2014. Adaptive multidimensional integration based onrank-1 lattices. in preparation.
Keister, B. D. 1996. Multidimensional quadrature algorithms, Computers in Physics 10,119–122.
Lyness, J. N. 1983. When not to use an automatic quadrature routine, SIAM Rev. 25, 63–87.
Niederreiter, H. 1992. Random number generation and quasi-Monte Carlo methods,CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
Novak, E. 1996. On the power of adaption, J. Complexity 12, 199–237.
Owen, A. B. 2000. Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo, MonteCarlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo, pp. 86–97.
Traub, J. F., G. W. Wasilkowski, and H. Wozniakowski. 1988. Information-based complexity,Academic Press, Boston.
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 14 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Proof of Cubature Error bound
S`pfq “2`´1ÿ
κ“t2`´1u
∣∣fκ∣∣ “ 2`´1ÿ
κ“t2`´1u
∣∣∣∣∣fm,κ ´ 8ÿ
λ“1
fκ`λ2m
∣∣∣∣∣ď
2`´1ÿ
κ“t2`´1u
∣∣fm,κ∣∣looooooomooooooon
rS`,mpfq
`
2`´1ÿ
κ“t2`´1u
8ÿ
λ“1
∣∣fκ`λ2m ∣∣loooooooooooomoooooooooooon
pS`,mpfq
ď rS`,mpfq ` pωpm´ `qqωpm´ `qS`pfq
S`pfq ďrS`,mpfq
1´ pωpm´ `qqωpm´ `qprovided that pωpm´ `qqωpm´ `q ă 1
∣∣∣∣∣ż
r0,1qdfpxqdx´
1
2m
2m´1ÿ
i“0
fpziq
∣∣∣∣∣ ď 8ÿ
λ“1
fλ2m “ pS0,mpfq
ď pωpmqqωp`qS`pfq ďpωpmqqωp`qrS`,mpfq
1´ pωpm´ `qqωpm´ `qback
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 15 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Proof that the Absolute/Relative Error Criterion is Met
Define the average and half the difference of the error criterion at the lower andupper bounds for µ:
∆m,˘ :“1
2rmax pεa, εr |µm ´ errpmq|q ˘max pεa, εr |µm ` errpmq|qs
Then if µm “ µm `∆m,´ and |µ´ µm| ď errpmq ď ∆m,`, it follows that
0 “ ˘pµm ´ µm ´∆n,´q ď ∆n,` ´ errpmq
ùñ µm `∆n,´ ´∆n,` ` errpmq ď µm ď µm `∆n,´ `∆n,` ´ errpmq
ùñ µm ` errpmq ´max pεa, εr |µm ` errpmq|q ď µm ď
µm ´ errpmq `max pεa, εr |µm ´ errpmq|qùñ µ´max pεa, εr |µ|q ď µm ď µ`max pεa, εr |µ|q
since b ÞÑ b˘maxpεa, εr |b|q is non-decreasing
ùñ |µ´ µm| ď max pεa, εr |µ|q
back
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 16 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Bounding errpmq Required in Terms of µ and Tolerances
Note that for |µ´ µm| ď errpmq it follows that
maxpεa, εr |µm ` signpµmq errpmq|q ě maxpεa, εr |µ|qmaxpεa, εr |µm ´ signpµmq errpmq|q
“ maxpεa, εr |µ´ µ` µm ´ signpµmq errpmq|qě maxpεa, εr |µ|q ´ εr |´µ` µm ´ signpµmq errpmq|ě maxpεa, εr |µ|q ´ 2εr errpmq,
which implies that
∆`,m ě maxpεa, εr |µ|q ´ εr errpmq
Therefore, if errpmq satisfies the inequality
errpmq ďmaxpεa, εr |µ|q
1´ εr
then the error condition of errpmq ď ∆`,m must be met. back
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 17 / 18
Simulation Problems Adaptive Monte Carlo Adaptive Quasi-Monte Carlo Discussion References
Bounding errpmq in Terms of S`pfq
rS`,mpfq “2`´1ÿ
κ“t2`´1u
∣∣fm,κ∣∣ “ 2`´1ÿ
κ“t2`´1u
∣∣∣∣∣fκ ` 8ÿ
λ“1
fκ`λ2m
∣∣∣∣∣ď
2`´1ÿ
κ“t2`´1u
∣∣fκ∣∣looooomooooon
S`pfq
`
2`´1ÿ
κ“t2`´1u
8ÿ
λ“1
∣∣fκ`λ2m ∣∣loooooooooooomoooooooooooon
pS`,mpfq
ď r1` pωpm´ `qqωpm´ `qsS`pfq
which implies that
errpmq “pωpmqqωp`qrSm´`,mpfq
1´ pωp`qqωp`qď
pωpmqqωp`qr1` pωp`qqωp`qsSm´`pfq
1´ pωp`qqωp`qalways.
Therefore, we know that errpmq ď ∆`,m must be satisfied when
pωpmqqωp`qr1` pωp`qqωp`qsSm´`pfq
1´ pωp`qqωp`qď
maxpεa, εr |µ|q1´ εr
. back
[email protected] Reliable (Q)MC Simulation Northwestern U, 10/7/2014 18 / 18