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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References Adaptive Algorithms for Stochastic Computation Fred J. Hickernell Department of Applied Mathematics, Illinois Institute of

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Adaptive Algorithms for Stochastic Computation

Fred J. Hickernell

Department of Applied Mathematics, Illinois Institute of Technology hickernell@iit.edu mypages.iit.edu/~hickernell Joint work with Tony Jim´ enez Rugama Tony, Yuhan Ding, and Xuan Zhou will present posters on Tuesday afternoon Supported by NSF-DMS-1115392 Many thanks to the organizers

September 15, 2014

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 1 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Some Problems in Stochastic Computation

µ “ EpY q “? µ “ ErfpXqs “ ż

Rd fpxq ̺pxq dx “?

µ “ Ppa ď Y ď bq “? p “ PpY ď µq, µ “? financial risk, statistical physics, photon transport, . . .

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Some Problems in Stochastic Computation

µ “ EpY q « ˆ µnptYiuq :“ 1 n

n

ÿ

i“1

Yi µ “ ErfpXqs “ ż

Rd fpxq ̺pxq dx

µ “ Ppa ď Y ď bq “? p “ PpY ď µq, µ “? financial risk, statistical physics, photon transport, . . . IID sampling, low discrepancy sampling, error bounds, tractability, multi-level (Richtmyer, 1951; Niederreiter, 1992; Sloan and Joe, 1994; Hickernell, 1998; Dick and Pillichshammer, 2010; Novak and Wo´ zniakowski, 2010; Dick et al., 2014; Giles, 2014), . . .

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Some Problems in Stochastic Computation

µ “ EpY q « ˆ µnptYiuq :“ 1 n

n

ÿ

i“1

Yi µ “ ErfpXqs “ ż

Rd fpxq ̺pxq dx

µ “ Ppa ď Y ď bq “? p “ PpY ď µq, µ “? Given a tolerance ε how do we choose n adaptively to make |µ ´ ˆ µn| ď ε (with high probability)? financial risk, statistical physics, photon transport, . . . IID sampling, low discrepancy sampling, error bounds, tractability, multi-level (Richtmyer, 1951; Niederreiter, 1992; Sloan and Joe, 1994; Hickernell, 1998; Dick and Pillichshammer, 2010; Novak and Wo´ zniakowski, 2010; Dick et al., 2014; Giles, 2014), . . . guaranteed, adaptive Monte Carlo (Hickernell et al., 2014; Jiang and Hickernell, 2014), trapezoidal rule (Clancy et al., 2014), quasi-Monte Carlo (Hickernell and Jim´ enez Rugama, 2014; Jim´ enez Rugama and Hickernell, 2014), GAIL (Choi et al., 2013–2014)

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 2 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Recent Results

§ µ “ EpY q “? (Hickernell et al., 2014), for somewhat different view see

(Bayer et al., 2014)

§ Compute a highly probable upper bound on true variance, C2ˆ

σ2

nσ, using nσ

IID samples.

§ Use a Berry-Esseen inequality (finite sample Central Limit Theorem) to find n

such that Pp|µ ´ ˆ µn| ď εq ě 99%.

§ Guaranteed for random variables in the cone

  • f bounded kurtosis

ErpY ´ µq4s{σ4 ď κmaxpnσ, Cq

§ Computational cost n — pσ{εq2 where σ is unknown.

§ µ “ EpY q “? for Bernoulli Y (Jiang and Hickernell, 2014)

§ Can find n that guarantees that Pp|µ ´ ˆ

µn| ď εaq ě 99% or Pp|µ ´ ˆ µn| ď εr |µ|q ě 99%.

§ µ “

şb

a fpxq dx “? (Clancy et al., 2014)

§ Can find n that guarantees that the trapezoidal rule with n trapezoids, ˆ

µn, gives |µ ´ ˆ µn| ď ε.

§ Guaranteed for integrands in the cone

Varpf 1q ď τ f 1 ´ rfpbq ´ fpaqs{pb ´ aq1

§ Computational cost n —

a Varpf 1q{ε where Varpf 1q is unknown.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 3 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Digital Net Cubature Error via Walsh Expansions

  • ż

r0,1qd fpxq dx ´ 1

2m

2m´1

ÿ

i“0

fpziq

  • ď

8

ÿ

λ“1

  • ˆ

fλ2m

  • ď p

ωpmq˚ ωpℓqr Sm´ℓ,mpfq 1 ´ p ωpℓq˚ ωpℓq

proof

digital net nodes Walsh coefficients in dual net Walsh functions & coefficients fpxq “

8

ÿ

κ“0

p´1qxkpκq,xy ˆ fκ ˜ fm,κ :“ 1 2m

2m´1

ÿ

i“0

p´1qx˜

kpκq,ziyfpziq

“

8

ÿ

λ“0

ˆ fκ`λ2m aliasing

0.25 0.5 0.75 1 0.25 0.5 0.75 1

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Digital Net Cubature Error via Walsh Expansions

  • ż

r0,1qd fpxq dx ´ 1

2m

2m´1

ÿ

i“0

fpziq

  • ď

8

ÿ

λ“1

  • ˆ

fλ2m

  • ď p

ωpmq˚ ωpℓqr Sm´ℓ,mpfq 1 ´ p ωpℓq˚ ωpℓq

proof

digital net nodes Walsh coefficients in dual net Walsh functions & coefficients fpxq “

8

ÿ

κ“0

p´1qxkpκq,xy ˆ fκ ˜ fm,κ :“ 1 2m

2m´1

ÿ

i“0

p´1qx˜

kpκq,ziyfpziq

“

8

ÿ

λ“0

ˆ fκ`λ2m aliasing

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Digital Net Cubature Error via Walsh Expansions

  • ż

r0,1qd fpxq dx ´ 1

2m

2m´1

ÿ

i“0

fpziq

  • ď

8

ÿ

λ“1

  • ˆ

fλ2m

  • ď p

ωpmq˚ ωpℓqr Sm´ℓ,mpfq 1 ´ p ωpℓq˚ ωpℓq

proof

digital net nodes Walsh coefficients in dual net p Sℓ,mpfq :“

2ℓ´1

ÿ

κ“t2ℓ´1u 8

ÿ

λ“1

  • ˆ

fκ`λ2m

  • ,

q Smpfq :“

8

ÿ

κ“2m

  • ˆ

fκ

  • Sℓpfq :“

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˆ

fκ

  • r

Sℓ,mpfq :“

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˜

fm,κ

  • 10

10

1

10

2

10

3

10

4

10

−15

10

−10

10

−5

10 κ | ˆ fκ| error ≤ ˆ S0,12(f) ˇ S12(f) S8(f) Cone conditions: p Sℓ,mpfq ď p ωpm ´ ℓqq Smpfq, q Smpfq ď ˚ ωpℓqSm´ℓpfq.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 4 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Adaptively Attaining Absolute or Relative Error Tolerances

Have

  • ż

r0,1qd fpxq dx

looooooomooooooon

µ

´ 1 2m

2m´1

ÿ

i“0

fpziq looooooomooooooon

ˆ µm

  • ď p

ωpmq˚ ωpℓqr Sm´ℓ,mpfq 1 ´ p ωpℓq˚ ωpℓq loooooooooooomoooooooooooon

errpmq

. We want to find m and ˜ µm that guarantees |µ ´ ˜ µm| ď maxpεa, εr |µ|q. Algorithm cubSobol g. Given tolerances εa and εr, fix ℓ and initalize m ą ℓ. Step 1. Compute the data-based error bound, errpmq, and ˆ µm. Step 2. If errpmq is small enough such that errpmq ď 1 2rmaxpεa, εr |ˆ µm ´ errpmq| ` maxpεa, εr |ˆ µm ` errpmq|s, then return the shrinkage estimator ˜ µm “ ˆ µm ` 1 2rmaxpεa, εr |ˆ µm ´ errpmq| ´ maxpεa, εr |ˆ µm ` errpmq|s. Step 3. Otherwise, increase m by one, and return to Step 1.

  • Theorem. For integrands satifying the cone

conditions cubSobol g succeeds

proof , and the computational cost is Oprm ` $pfqs2mq, for some

m ď mintm1 : r1 ` p ωpℓq˚ ωpℓqsp1 ` εrqp ωpm1q˚ ωpℓqSm1´ℓpfq ď maxpεa, εr |µ|qr1 ´ p ωpℓq˚ ωpℓqsu

proof more proof hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 5 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Observations about cubSobol g

  • Theorem. For integrands satifying the cone

conditions cubSobol g succeeds, and the computational cost is Oprm ` $pfqs2mq, for some m ď mintm1 : r1 ` p ωpℓq˚ ωpℓqsp1 ` εrqp ωpm1q˚ ωpℓqSm1´ℓpfq ď maxpεa, εr |µ|qr1 ´ p ωpℓq˚ ωpℓqsu

§ The error bound and stopping criterion are based on integrand values, not

norms of the integrand which are hard to compute.

§ The algorithm does not know the decay rate of the Walsh coefficients but

takes advantage of fast decay.

§ Because the algorithm is adaptive, upper bound on computational cost is

based on characteristics of the integrand, but these need not be known.

§ For relative error only (εa “ 0) the computational cost goes to 8 as µ Ñ 0. § The extra cost for not knowing µ or the true Walsh coefficients comes in the

form of the constant multiple r1 ` p ωpℓq˚ ωpℓqsp1 ` εrq on the left and r1 ´ p ωpℓq˚ ωpℓqs on the right.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 6 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Numerical Examples

Asian Geometric Mean Call Option d “ 1, 2, 4, . . . , 64 Keister’s (1996) Example ż

Rd e´x2 cospxq dx

d “ 1, . . . , 19 « 99% success « 96% success

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 7 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

They Said It Couldn’t Be Done!

§ Lyness (1983) warned against adaptive algorithms that use C |ˆ

µn ´ ˆ µn1| as an error estimate. We avoid error estimates of this type, but they are prevalent in existing adaptive algorithms.

§ There are rather general sufficient conditions under which adaption provides

no advantage (Bahadur and Savage, 1956; Traub et al., 1988, Chapter 4, Theorem 5.2.1; Novak, 1996). To violate those conditions we consider nonconvex cones

  • f random variables or integrands.

§ We focus on cones

(instead of other shapes) because our problems are homogeneous and our error bounds are positively homogeneous.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 8 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Why Not Replications to Estimate Error?

  • ż

r0,1qd fpxq dx ´ ˆ

µn

  • ď C

g f f e 1 R ´ 1

R

ÿ

r“1

pYr ´ ˆ µnq2, ˆ µn :“ 1 R

R

ÿ

r“1

Yr IID Replications Yr “ R n

n{R´1

ÿ

i“0

fpzprq

i q, where tzprq i u are independent

randomizations. Internal Replications Yr “ R n

rn{R´1

ÿ

i“pr´1qn{R

fpziq. Want R small to take advantage of low discrepancy, but need R large to ensure that sample variance of Yr represents error (Deng, 2013; Hickernell et al., 2014).

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 9 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Guaranteed Automatic Integration Library (GAIL) code.google.com/p/gail/

§ Version 1.3 (Choi et al.,

2013–2014) includes integral g.m, meanMC g.m, cubMC g.m, funappx g.m

§ Version 2.0 (exp. fall 2014)

should include the following:

§ cubSobol g.m,

cubLattice g.m

§ meanBernoulli g.m § meanMC g.m, cubMC g.m

with absolue/relative error

§ funappx g.m with local

adaption

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 10 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Future Work

§ Connections between our cone

conditions and familiar spaces of integrands, such as Korobov spaces

§ Lower bound on the computational complexity § Multilevel or multivariate decomposition method algorithms § Quasi-Monte Carlo for probabilities § Quantiles

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 11 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

References I

Bahadur, R. R. and L. J. Savage. 1956. The nonexistence of certain statistical procedures in nonparametric problems, Ann. Math. Stat. 27, 1115–1122. Bayer, C., H. Hoel, E. von Schwerin, and R. Tempone. 2014. On nonasymptotic optimal stopping criteria in monte carlo simulations on nonasymptotic optimal stopping criteria in Monte Carlo Simulations, SIAM J. Sci. Comput. 36, A869–A885. Choi, S.-C. T., Y. Ding, F. J. Hickernell, L. Jiang, and Y. Zhang. 2013–2014. GAIL: Guaranteed Automatic Integration Library (versions 1, 1.3). Clancy, N., Y. Ding, C. Hamilton, F. J. Hickernell, and Y. Zhang. 2014. The cost of deterministic, adaptive, automatic algorithms: Cones, not balls, J. Complexity 30, 21–45. Deng, S. 2013. An investigation of the quasi-standard error for quasi-monte carlo method, Master’s Thesis. Dick, J., F. Kuo, and I. H. Sloan. 2014. High dimensional integration — the Quasi-Monte Carlo way, Acta Numer., 1–153. Dick, J., F. Y. Kuo, G. W. Peters, and I. H. Sloan (eds.) 2014. Monte Carlo and quasi-Monte Carlo methods 2012, Springer-Verlag, Berlin. Dick, J. and F. Pillichshammer. 2010. Digital nets and sequences: Discrepancy theory and quasi-Monte Carlo integration, Cambridge University Press, Cambridge.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 12 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

References II

Genz, A. 1987. A package for testing multiple integration subroutines, Numerical integration: Recent developments, software and applications, pp. 337–340. Giles, M. 2014. Multilevel Monte Carlo methods, Monte Carlo and quasi-Monte Carlo methods 2012. Hickernell, F. J. 1998. A generalized discrepancy and quadrature error bound, Math. Comp. 67, 299–322. Hickernell, F. J., L. Jiang, Y. Liu, and A. B. Owen. 2014. Guaranteed conservative fixed width confidence intervals via Monte Carlo sampling, Monte Carlo and quasi-Monte Carlo methods 2012, pp. 105–128. Hickernell, F. J. and L. A. Jim´ enez Rugama. 2014. Reliable adaptive cubature using digital

  • sequences. submitted for publication.

Jiang, L. and F. J. Hickernell. 2014. Guaranteed conservative confidence intervals for means of Bernoulli random variables. in preparation. Jim´ enez Rugama, L. A. and F. J. Hickernell. 2014. Adaptive multidimensional integration based

  • n rank-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.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 13 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

References III

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. Novak, E. and H. Wo´

  • zniakowski. 2010. Tractability of multivariate problems Volume ii:

Standard information for functionals, EMS Tracts in Mathematics, European Mathematical Society, Z¨ urich. Richtmyer, R. D. 1951. The evaluation of definite integrals, and a Quasi-Monte-Carlo method based on the properties of algebraic numbers, Technical Report LA-1342, Los Alamos Scientific Laboratory. Sloan, I. H. and S. Joe. 1994. Lattice methods for multiple integration, Oxford University Press, Oxford. Traub, J. F., G. W. Wasilkowski, and H. Wo´

  • zniakowski. 1988. Information-based complexity,

Academic Press, Boston.

hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 14 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Proof of Cubature Error bound

Sℓpfq “

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˆ

fκ

  • “

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˜

fm,κ ´

8

ÿ

λ“1

ˆ fκ`λ2m

  • ď

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˜

fm,κ

  • looooooomooooooon

r Sℓ,mpfq

`

2ℓ´1

ÿ

κ“t2ℓ´1u 8

ÿ

λ“1

  • ˆ

fκ`λ2m

  • loooooooooooomoooooooooooon

p Sℓ,mpfq

ď r Sℓ,mpfq ` p ωpm ´ ℓq˚ ωpm ´ ℓqSℓpfq Sℓpfq ď r Sℓ,mpfq 1 ´ p ωpm ´ ℓq˚ ωpm ´ ℓq provided that p ωpm ´ ℓq˚ ωpm ´ ℓq ă 1

  • ż

r0,1qd fpxq dx ´ 1

2m

2m´1

ÿ

i“0

fpziq

  • ď

8

ÿ

λ“1

ˆ fλ2m “ p S0,mpfq ď p ωpmq˚ ωpℓqSℓpfq ď p ωpmq˚ ωpℓqr Sℓ,mpfq 1 ´ p ωpm ´ ℓq˚ ωpm ´ ℓq

back hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 15 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples 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 and upper bounds for µ: ∆m,˘ :“ 1 2 rmax pεa, εr |µm ´ errpmq|q ˘ max pεa, εr |ˆ µm ` errpmq|qs Then if ˜ µm “ ˆ µm ` ∆m,´ and |µ ´ ˆ µm| ď errpmq ď ∆m,`, it follows that 0 “ ˘ p˜ µm ´ ˆ µm ´ ∆n,´q ď ∆n,` ´ errpmq ù ñ ˆ µm ` ∆n,´ ´ ∆n,` ` errpmq ď ˜ µm ď ˆ µm ` ∆n,´ ` ∆n,` ´ errpmq ù ñ ˆ µm ` errpmq ´ max pεa, εr |ˆ µm ` errpmq|q ď ˜ µm ď ˆ µm ´ errpmq ` max pεa, εr |ˆ µm ´ errpmq|q ù ñ 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 ù ñ |µ ´ ˜ µm| ď max pεa, εr |µ|q

back hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 16 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Bounding errpmq Required in Terms of µ and Tolerances

Note that for |µ ´ ˆ µm| ď errpmq it follows that maxpεa, εr |ˆ µm ` signpˆ µmq errpmq|q ě maxpεa, εr |µ|q maxpεa, εr |ˆ µm ´ signpˆ µmq errpmq|q “ maxpεa, εr |µ ´ µ ` ˆ µm ´ signpˆ µmq errpmq|q ě maxpεa, εr |µ|q ´ εr |´µ ` ˆ µm ´ signpˆ µmq 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 ď maxpεa, εr |µ|q 1 ´ εr then the error condition of errpmq ď ∆`,m must be met.

back hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 17 / 18

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Problem Digital Net Cubature Error Bounds Attaining Error Tolerances Numerical Examples Discussion References

Bounding errpmq in Terms of Sℓpfq

r Sℓ,mpfq “

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˜

fm,κ

  • “

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˆ

fκ `

8

ÿ

λ“1

ˆ fκ`λ2m

  • ď

2ℓ´1

ÿ

κ“t2ℓ´1u

  • ˆ

fκ

  • looooomooooon

Sℓpfq

`

2ℓ´1

ÿ

κ“t2ℓ´1u 8

ÿ

λ“1

  • ˆ

fκ`λ2m

  • loooooooooooomoooooooooooon

p Sℓ,mpfq

ď r1 ` p ωpm ´ ℓq˚ ωpm ´ ℓqsSℓpfq which implies that errpmq “ p ωpmq˚ ωpℓqr Sm´ℓ,mpfq 1 ´ p ωpℓq˚ ωpℓq ď p ωpmq˚ ωpℓqr1 ` p ωpℓq˚ ωpℓqsSm´ℓpfq 1 ´ p ωpℓq˚ ωpℓq always. Therefore, we know that errpmq ď ∆`,m must be satisfied when p ωpmq˚ ωpℓqr1 ` p ωpℓq˚ ωpℓqsSm´ℓpfq 1 ´ p ωpℓq˚ ωpℓq ď maxpεa, εr |µ|q 1 ´ εr .

back hickernell@iit.edu Adaptive Algorithms for Stochastic Computation ICERM IBC & Stoch. Comp. 2014 18 / 18