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Adaptive Methods for Cubature
Fred J. HickernellDepartment of Applied Mathematics, Illinois Institute of Technology
[email protected] mypages.iit.edu/~hickernell
Thanks to Lan Jiang, Tony Jiménez Rugama, Jagadees Rathinavel,and the rest of the the Guaranteed Automatic Integration Library (GAIL) team
Supported by NSF-DMS-1522687
For more details see H. (2017+), H. et al. (2017+), and Choi et al. (2013–2015)
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Problemanswer(µ) =?, µ =
ż
Rdf(x)ν(dx) P Rp, pµn =
nÿ
i=1wif(xi)
Given εa, εr, choose n and {answer dependent on pµn to guarantee∣∣answer(µ)´ {answer∣∣ ď max(εa, εr |answer(µ)|) (with high probability)
adaptively and automatically
If µ P [pµn ´ errn, pµn + errn] (with high probability), then the optimal andsuccessful choice is
{answer =ans´max(εa, εr |ans+|) + ans+ max(εa, εr |ans´|)
max(εa, εr |ans+|) + max(εa, εr |ans´|)
where ans˘ :=
"
supinf
*
µ P [pµn ´ errn, pµn + errn]answer(µ)
provided|ans+ ´ ans´|
max(εa, εr |ans+|) + max(εa, εr |ans´|)ď 1 (H. et al., 2017+)
2/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Problemanswer(µ) =?, µ =
ż
Rdf(x)ν(dx) P Rp, pµn =
nÿ
i=1wif(xi)
Given εa, εr, choose n and {answer dependent on pµn to guarantee∣∣answer(µ)´ {answer∣∣ ď max(εa, εr |answer(µ)|) (with high probability)
adaptively and automatically
E.g.,
option price =
ż
Rdpayoff(x)
e´xTΣ´1x/2
(2π)d/2 |Σ|1/2 dx, Σ =
(min(i, j)T/d
)d
i,j=1
Gaussian probability =
ż
[a,b]
e´xTΣ´1x/2
(2π)d/2 |Σ|1/2 dx Genz (1993)
=
ż
[0,1]d´1f (x)dx
Sobol’ indexj =
ş
[0,1]2d
[output(x)´ output(xj : x
1´j)]output(x 1)dxdx 1
ş
[0,1]d output(x)2 dx´[ş
[0,1]d output(x)dx]2
Bayesian estimatej =
ş
Rd βj prob(data|β) probprior(β)dβş
Rd prob(data|β) probprior(β)dβ
If µ P [pµn ´ errn, pµn + errn] (with high probability), then the optimal andsuccessful choice is
{answer =ans´max(εa, εr |ans+|) + ans+ max(εa, εr |ans´|)
max(εa, εr |ans+|) + max(εa, εr |ans´|)
where ans˘ :=
"
supinf
*
µ P [pµn ´ errn, pµn + errn]answer(µ)
provided|ans+ ´ ans´|
max(εa, εr |ans+|) + max(εa, εr |ans´|)ď 1 (H. et al., 2017+)
2/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Problemanswer(µ) =?, µ =
ż
Rdf(x)ν(dx) P Rp, pµn =
nÿ
i=1wif(xi)
Given εa, εr, choose n and {answer dependent on pµn to guarantee∣∣answer(µ)´ {answer∣∣ ď max(εa, εr |answer(µ)|) (with high probability)
adaptively and automatically
If µ P [pµn ´ errn, pµn + errn] (with high probability), then the optimal andsuccessful choice is
{answer =ans´max(εa, εr |ans+|) + ans+ max(εa, εr |ans´|)
max(εa, εr |ans+|) + max(εa, εr |ans´|)
where ans˘ :=
"
supinf
*
µ P [pµn ´ errn, pµn + errn]answer(µ)
provided|ans+ ´ ans´|
max(εa, εr |ans+|) + max(εa, εr |ans´|)ď 1 (H. et al., 2017+)
2/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Berry-Esseen Stopping Rule for IID Monte Carlo
µ =
ż
Rdf (x)ν(dx)
pµn =1n
nÿ
i=1f (xi), xi
IID„ ν
Need µ P [µ̂n ´ errn, µ̂n + errn]with high probability
P[|µ´ µ̂n| ď errn] « 99%for Φ
(´?
n errn /(1.2σ̂))= 0.005
by the Central Limit Theorem
where σ̂2 is the sample variation3/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Berry-Esseen Stopping Rule for IID Monte Carlo
µ =
ż
Rdf (x)ν(dx)
pµn =1n
nÿ
i=1f (xi), xi
IID„ ν
Need µ P [µ̂n ´ errn, µ̂n + errn]with high probability
P[|µ´ µ̂n| ď errn] ě 99%for Φ
(´?
n errn /(1.2σ̂nσ))
+ ∆n(´?
n errn /(1.2σ̂nσ), κmax) = 0.0025by the Berry-Esseen Inequality
where σ̂2nσ is the sample variation using an independent sample, and provided
that kurt(f (X)) ď κmax(nσ) (H. et al., 2013; Jiang, 2016)
3/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]df (x)dx
pµn =1n
nÿ
i=1f (xi), xi Sobol’ or lattice
Need µ P [µ̂n ´ errn, µ̂n + errn]
Express µ´ µ̂n in terms of the Fourier coefficients of f . Assuming that thesecoefficients do not decay erratically, the discrete transform,
rfn,κ(n´1κ=0, may be
used to bound the error reliably (H. and Jiménez Rugama, 2016; Jiménez Rugama andH., 2016; H. et al., 2017+):
|µ´ µ̂n| ď errn := C(n, `)2`´1ÿ
κ=2`´1
∣∣∣rfn,κ∣∣∣
4/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]df (x)dx
pµn =1n
nÿ
i=1f (xi), xi Sobol’ or lattice
Need µ P [µ̂n ´ errn, µ̂n + errn]
Express µ´ µ̂n in terms of the Fourier coefficients of f . Assuming that thesecoefficients do not decay erratically, the discrete transform,
rfn,κ(n´1κ=0, may be
used to bound the error reliably (H. and Jiménez Rugama, 2016; Jiménez Rugama andH., 2016; H. et al., 2017+):
|µ´ µ̂n| ď errn := C(n, `)2`´1ÿ
κ=2`´1
∣∣∣rfn,κ∣∣∣
4/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]df (x)dx
pµn =1n
nÿ
i=1f (xi), xi Sobol’ or lattice
Need µ P [µ̂n ´ errn, µ̂n + errn]
Express µ´ µ̂n in terms of the Fourier coefficients of f . Assuming that thesecoefficients do not decay erratically, the discrete transform,
rfn,κ(n´1κ=0, may be
used to bound the error reliably (H. and Jiménez Rugama, 2016; Jiménez Rugama andH., 2016; H. et al., 2017+):
|µ´ µ̂n| ď errn := C(n, `)2`´1ÿ
κ=2`´1
∣∣∣rfn,κ∣∣∣4/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Bayesian Cubature—f Is Random
µ =
ż
Rdf (x)ν(dx)
pµn =nÿ
i=1wi f (xi)
Need µ P [µ̂n ´ errn, µ̂n + errn]with high probability
Assume f „ GP(0,C). Choose the wi to integrate the best estimate of f given thedata txi, f (xi)u
ni=1 (Diaconis, 1988; O’Hagan, 1991; Ritter, 2000; Rasmussen and
Ghahramani, 2003)
P[|µ´ µ̂n| ď errn] = 99% for errn = an expression involving C and txi, f (xi)uni=1
A de-randomized interpretation exists (H., 2017+)5/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Gaussian Probability
µ =
ż
[a,b]
exp(´ 1
2tTΣ´1t
)a
(2π)d det(Σ)dt Genz (1993)
=
ż
[0,1]d´1f (x)dx
For some typical choice of a, b, Σ, d = 3, εa = 0; µ « 0.6763
Worst 10% Worst 10%εr Method % Accuracy n Time (s)
IID Monte Carlo 100% 8.1E4 1.8E´21E´2 Sobol’ Sampling 100% 1.0E3 5.1E´3
Bayesian Lattice 100% 1.0E3 2.8E´3
IID Monte Carlo 100% 2.0E6 3.8E´11E´3 Sobol’ Sampling 100% 2.0E3 7.7E´3
Bayesian Lattice 100% 1.0E3 2.8E´3
1E´4 Sobol’ Sampling 100% 1.6E4 1.8E´2Bayesian Lattice 100% 8.2E3 1.4E´2
6/10
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
Sobol’ IndicesY = output(X), where X „ U[0, 1]d; Sobol’ Indexj(µ) describes how muchcoordinate j of input X influences output Y (Sobol’, 1990; 2001):
Sobol’ Indexj(µ) :=µ1
µ2 ´ µ23, j = 1, . . . , d
µ1 :=
ż
[0,1)2d[output(x)´ output(xj : x
1´j)]output(x 1)dxdx 1
µ2 :=
ż
[0,1)doutput(x)2 dx, µ3 :=
ż
[0,1)doutput(x)dx.
output(x) = ´x1 + x1x2 ´ x1x2x3 + ¨ ¨ ¨+ x1x2x3x4x5x6 (Bratley et al., 1992)
εa = 1E´3, εr = 0 j 1 2 3 4 5 6n 65 536 32 768 16 384 16 384 2 048 2 048
Sobol’ Indexj 0.6529 0.1791 0.0370 0.0133 0.0015 0.0015{Sobol’ Indexj 0.6528 0.1792 0.0363 0.0126 0.0010 0.0012
Sobol’ Indexj(pµn) 0.6492 0.1758 0.0308 0.0083 0.0018 0.00397/10
Thank you
Slides available atwww.slideshare.net/fjhickernell/siam-cse-2017-talk
Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
References I
Bratley, P., B. L. Fox, and H. Niederreiter. 1992. Implementation and tests of low-discrepancysequences, ACM Trans. Model. Comput. Simul. 2, 195–213.
Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, Ll. A. Jiménez Rugama, X. Tong, Y. Zhang, and X. Zhou.2013–2015. GAIL: Guaranteed Automatic Integration Library (versions 1.0–2.1).
Cools, R. and D. Nuyens (eds.) 2016. Monte Carlo and quasi-Monte Carlo methods: MCQMC,Leuven, Belgium, April 2014, Springer Proceedings in Mathematics and Statistics, vol. 163,Springer-Verlag, Berlin.
Diaconis, P. 1988. Bayesian numerical analysis, Statistical decision theory and related topics IV,Papers from the 4th Purdue symp., West Lafayette, Indiana 1986, pp. 163–175.
Genz, A. 1993. Comparison of methods for the computation of multivariate normal probabilities,Computing Science and Statistics 25, 400–405.
H., F. J. 2017+. Error analysis of quasi-Monte Carlo methods. submitted for publication,arXiv:1702.01487.
H., F. J., L. Jiang, Y. Liu, and A. B. Owen. 2013. Guaranteed conservative fixed width confidenceintervals via Monte Carlo sampling, Monte Carlo and quasi-Monte Carlo methods 2012, pp. 105–128.
H., F. J. and Ll. A. Jiménez Rugama. 2016. Reliable adaptive cubature using digital sequences,Monte Carlo and quasi-Monte Carlo methods: MCQMC, Leuven, Belgium, April 2014, pp. 367–383.arXiv:1410.8615 [math.NA].
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Problem IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Numerical Examples References
References II
H., F. J., Ll. A. Jiménez Rugama, and D. Li. 2017+. Adaptive quasi-Monte Carlo methods. submittedfor publication, arXiv:1702.01491 [math.NA].
Jiang, L. 2016. Guaranteed adaptive Monte Carlo methods for estimating means of randomvariables, Ph.D. Thesis.
Jiménez Rugama, Ll. A. and F. J. H. 2016. Adaptive multidimensional integration based on rank-1lattices, Monte Carlo and quasi-Monte Carlo methods: MCQMC, Leuven, Belgium, April 2014,pp. 407–422. arXiv:1411.1966.
O’Hagan, A. 1991. Bayes-Hermite quadrature, J. Statist. Plann. Inference 29, 245–260.
Rasmussen, C. E. and Z. Ghahramani. 2003. Bayesian Monte Carlo, Advances in Neural InformationProcessing Systems, pp. 489–496.
Ritter, K. 2000. Average-case analysis of numerical problems, Lecture Notes in Mathematics,vol. 1733, Springer-Verlag, Berlin.
Sobol’, I. M. 1990. On sensitivity estimation for nonlinear mathematical models, Matem. Mod. 2,no. 1, 112–118.
. 2001. Global sensitivity indices for nonlinear mathematical models and their monte carloestimates, Math. Comput. Simul. 55, no. 1-3, 271–280.
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