Two Topics in Parametric Integration Applied to …...Heinrich (2001): MLMC for Parametric...

Two Topics in Parametric IntegrationApplied to Stochastic Simulation

in Industrial Engineering

Jeremy Staum

Department of Industrial Engineering and Management SciencesNorthwestern University

September 15th, 2014

Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering

Outline

1 Simulation Metamodeling

introduction and overview

2 Multi-Level Monte Carlo Metamodeling

with Imry Rosenbaumhttp:

//users.iems.northwestern.edu/~staum/MLMCM.pdf

3 Generalized Integrated Brownian Fields for SimulationMetamodeling

with Peter Salemi and Barry L. Nelsonhttp://users.iems.northwestern.edu/~staumGIBF.pdf


http://users.iems.northwestern.edu/~staum/MLMCM.pdf

http://users.iems.northwestern.edu/~staum/MLMCM.pdf

http://users.iems.northwestern.edu/~staumGIBF.pdf

MCQMC / IBC Application Domain

Industrial Engineering & Operations Research

using math to analyze systems and improve decisions

Stochastic Simulation: production, logistics, financial, . . .

integration: µ = E[Y ] =∫Y (ω) dω

parametric integration: approx. µ def. by µ(x) = E[Y (x)]

optimization: minµ(x) : x ∈ X


What is Stochastic Simulation Metamodeling?

Stochastic simulation model example

Fuel injector production line

System performance measure µ(x) = E[Y (x)]

x: design of production lineY (x): number of fuel injectors produced

Simulating each scenario (20 replications) takes 8 hours

Stochastic simulation metamodeling

Simulation output Y (xi ) at xi , i = 1, . . . , n

Predict µ(x) by µ(x), even without simulating at x

µ(x) is usually a weighted average of Y (x1), . . . , Y (xn)


Overview of Multi-Level Monte Carlo (MLMC)

Error in Stochastic Simulation Metamodeling

prediction µ(x) =k∑

i=1

wi (x)Y (xi )

variance: Var[µ(x)] caused by variance of simulation output

interpolation error (bias): E[µ(x)] 6= µ(x)


Main Idea of Multi-Level Monte Carlo

Ordinary Monte Carlo

to reduce variance: large number n of replicationsper simulation run (design point)

to reduce bias: large number k of design points (fine grid)

very large computational effort kn

Multi-Level Monte Carlo

to reduce variance: coarser grids, many replications each

to reduce bias: finer grids, few replications each

less computational effort / better convergence rate


Our Contributions

Theoretical

mix and match or expand (from Heinrich’s papers):derive desired conclusions under desired assumptions

to suit IE goals and applications

Practical

algorithm design (based on Giles)

Experimental

show how much MLMC speeds up realistic examples in IE


Heinrich (2001): MLMC for Parametric Integration

Assumptions

Approximate µ given by µ(x) =∫

Ω Y (x, ω) dω over x ∈ X .

X ⊆ Rd and Ω ⊆ Rd2 : bounded, open, Lipschitz boundary.

With respect to x, Y has weak derivatives up to r th order.

Y and weak derivatives are Lq-integrable in (x, ω).

Sobolev embedding condition: r/d > 1/q.

Measure error as (∫

Ω ‖µ− µ‖pq dω)1/p, where p = min2, q.

Conclusion: There is a MLMC method with optimal rate.

MLMC attains the best rate of convergence in C , the number ofevaluations of Y . The error bound is proportional to

C−r/d if r/d < 1− 1/p

C 1/p−1 logC if r/d = 1− 1/p

C 1/p−1 if r/d > 1− 1/p.


Assumptions

Smoothness: assume r = 1

Stock option, Y (x , ω) = maxxR(ω)− K , 0Queueing: waiting time Wn+1 = maxWn + Bn − An+1, 0Inventory: Sn = minIn + Pn,Dn, In+1 = In − Sn

Parameter Domain

Assume X ⊂ Rd is compact (not open).

Heinrich and Sindambiwe (1999), Daun and Heinrich (2014)

If X were open, we would have to extrapolate.

No need to approximate unbounded µ near a boundary of X .

Domain of Integration

Ω ⊆ Rd2 is not important; d2 does not appear in theorem.


Changing Perspective

Measure of Error

Use p = q = 2 to get Root Mean Integrated Squared Error(∫Ω

∫X (µ(x)− µ(x))2 dx dω

)1/2

Sobolev Embedding Criterion with r = 1, q = 2

r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??

Why We Don’t Need the Sobolev Embedding Condition

Assume the domain X is compact.

Assume Y (·, ω) is (almost surely) Lipschitz continuous.

Conclude Y (·, ω) is (almost surely) bounded.



Measure of Error



)1/2


r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??







Measure of Error



)1/2


r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??






Our Assumptions

On the Stochastic Simulation Metamodeling Problem

X ⊂ Rd is compact

Y (x) has finite variance for all x ∈ X|Y (x′, ω)− Y (x, ω)| ≤ κ(ω)‖x′ − x‖, a.s., and E[κ2] <∞.

On the Approximation Method and MLMC Design

µ(x) =∑N

i=1 wi (x)Y (xi ) where each wi (x) ≥ 0 and

Total weight on points xi far from x gets close to 0.

Total weight on points xi near x gets close to 1.

Thresholds for “far”/“‘near” and “close to”are O(N−1/2φ) as number N of points increases.

Examples: piecewise linear interpolation on a grid;nearest-neighbors, Shepard’s method, kernel smoothing


Approximation Method Used in Examples

Kernel Smoothing

µ(x) =N∑i=1

wi (x)Y (xi )

weight wi (x) is

0 if xi is outside the cellcontaining xotherwise, proportional toexp(−‖x− xi‖)

weights are normalized tosum to 1


Our Conclusions

MLMC Performance

As number N of points used in a level increases,

Errors due to bias and refinement variance are like O(N−1/φ).

Example: nearest-neighbor approximation on grid, φ = d/2

Computational Complexity (based on Giles 2013)

To attain RMISE < ε, the required number of evaluations of Y isO(ε−2(1+φ)) for standard Monte Carlo and for MLMC it is

O(ε−2φ) if φ > 1

O((ε−1(log ε−1))2) if φ = 1

O(ε−2) if φ < 1.


Sketch of Algorithm (based on Giles 2008)

Goal: add levels until target RMISE < ε is achieved.

1 INITIALIZE level ` = 0.2 SIMULATE at level `:

1 Run level ` simulation experiment with M0 replications.2 Observe sample variance of simulation output.3 Choose number of replications M` to control variance; run

more replications if needed.

3 TEST CONVERGENCE:1 Use Monte Carlo to estimate the size of the refinement ∆µ`,∫

X (∆µ`(x))2 dx.2 If refinements are too large compared to target RMISE,

increment ` and return to step 2.

4 CLEAN UP: Finalize number of replications M0, . . . ,M` tocontrol variance; run more replications at each level if needed.


Asian Option Example, d = 3

MLMC up to 150 times better than standard Monte Carlo


Inventory System Example, d = 2

MLMC was 130-8900 times better than standard Monte Carlo


Conclusion on Multi-Level Monte Carlo

Celebration

Multi-Level Monte Carlo worksfor typical IE stochastic simulation metamodeling too!

Future Research

Handle discontinuities in simulation output.

Combine with good experiment designs.Grids are not good in high dimension.


Introduction: Generalized Integrated Brownian Field

Kriging / Interpolating Splines

Pretend µ is a realization of a Gaussian random field M withmean function m and covariance function σ2.

Kriging predictor:

µ(x) = m(x) + σ2(x)Σ−1(Y −m) = m(x) +∑i

βiσ2(x, xi )

σ2(x) is a vector with ith element σ2(x, xi )Σ is a matrix with i , jth element σ2(xi , xj)Y −m is a vector with ith element Y (xi )−m(xi )

Stochastic Kriging / Smoothing Splines

µ(x) = m(x) + σ2(x)(Σ + C )−1(Y −m) = m(x) +∑i

βiσ2(x, xi )

C = covariance matrix of noise, estimated from replications


Radial Basis Functions vs. Integrated Brownian Field

Radial Basis Functions

Basis Functions from r -Fold Integrated Brownian Field

(d) r = 0 (e) r = 1 (f) r = 2


Response Surfaces in IE Stochastic Simulation

(g) Credit Risk (h) Inventory


r -Integrated Brownian Field Br

Covariance function / reproducing kernel

σ2(x, y) =d∏

i=1

1

(r !)2

∫ 1

0(xi − ui )

r+(yi − ui )

r+ dui

Inner product

〈f , g〉 =

∫(0,1)d

(f ([r ···r ])(u))(g ([r ···r ])(u))du

Space

Tensor product of Sobolev Hilbert space H r (0, 1)with boundary conditions f (j)(0) = 0 for j = 0, . . . , r

What’s missing? polynomials of degree r


Removing Boundary Conditions: d = 1

Generalized integrated Brownian motion

Xr (x) =r∑

k=0

√θkZk

xk

k!+√θr+1Br (x)


σ2(x , y) =r∑

k=0

θkxkyk

(k!)2+ θr+1

∫ 1

0

(x − u)r+(y − u)r+(r !)2

du

Sobolev space H r (0, 1) , no boundary conditions

Inner product

〈f , g〉 =r∑

k=0

1

θk(f (k)(u))(g (k)(u)) +

1

θr+1

∫ 1

0(f (r)(u))(g (r)(u))du


Multidimensional, Without Boundary Conditions

Tensor-Product RKHS with Weights

Example of reproducing kernel for d = 2, r = 1

K (x, y) = θ00 + θ10x1y1 + θ20(x1 ∧ y1) + θ01x2y2 + θ02(x2 ∧ y2)

+θ11x1x2y1y2 + θ12x1y1(x2 ∧ y2)

+θ21(x1 ∧ y1)x2y2 + θ22(x1 ∧ y1)(x2 ∧ y2)

In general, one weight for each of∏d

i=1(ri + 2) subspaces.

Generalized Integrated Brownian Field


σ2(x, y) =d∏

i=1

(ri∑

k=0

θi ,kxki y

ki

(k!)2+ θi ,ri+1

∫ 1

0

(xi − ui )ri+(yi − ui )

ri+

(ri !)2dui

)

In general, number of weights is∑d

i=1(ri + 2).


Multidimensional, Without Boundary Conditions

Tensor-Product RKHS with Weights

Example of reproducing kernel for d = 2, r = 1

K (x, y) = θ00 + θ10x1y1 + θ20(x1 ∧ y1) + θ01x2y2 + θ02(x2 ∧ y2)

+θ11x1x2y1y2 + θ12x1y1(x2 ∧ y2)

+θ21(x1 ∧ y1)x2y2 + θ22(x1 ∧ y1)(x2 ∧ y2)

In general, one weight for each of∏d

i=1(ri + 2) subspaces.

Generalized Integrated Brownian Field


σ2(x, y) =d∏

i=1

(ri∑

k=0

θi ,kxki y

ki

(k!)2+ θi ,ri+1

∫ 1

0

(xi − ui )ri+(yi − ui )

ri+

(ri !)2dui

)

In general, number of weights is∑d

i=1(ri + 2).


Our Contributions

more parsimonious parametrization

makes maximum likelihood estimation easierand MLE search for r1, . . . , rd

GIBF has Markov property

d = 1: proofd > 1: conjecture

IE simulation examples

stochastic and deterministic simulationstandard and nonstandard information


Credit Risk Example, d = 2

Experiment design: 63 Sobol’ points,predictions in a smaller square

Factor by which MISE decreased using (1,1)-GIBF

Number of replications ∞ 100 25Noise level none low medium

Without gradient estimates 94 111 120With gradient estimates 81 83 69

(i) Credit risk surface (j) Gaussian (k) (1, 1)-GIBF


Conclusion on Generalized Integrated Brownian Field

Emancipating Simulation Metamodeling from Geostatistics

a new covariance function for kriging,designed for simulation metamodeling in engineering

Superior Practical Performance

4-120 times better than Gaussian covariance functionin 2-6 dimensional exampleswith or without gradient information


Thank You!


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