1

Efficient experimentation for nanostructure synthesis using Sequential Minimum Energy Designs (SMED)

V. Roshan Joseph+, Tirthankar Dasgupta* and C. F. Jeff Wu+

+ ISyE, Georgia Tech

*Statistics, Harvard

2

Statistical modeling and analysis for robust synthesis of nanostructures

Dasgupta, Ma, Joseph, Wang and Wu (2008), Journal of The American Statistical Association, to appear.

Robust conditions for synthesis of Cadmium Selenide (CdSe) nanostructures derived New sequential algorithm for fitting

multinomial logit models. Internal noise factors considered.

Fig 4: CdSe nanostructuresFig 4: CdSe nanostructures

3

Fitted quadratic response surfaces & optimal conditions

4

The need for more efficient experimentation

A 9x5 full factorial experiment was too expensive and time consuming.

Quadratic response surface did not capture nanowire growth satisfactorily (Generalized R2 was 50% for CdSe nanowire sub-model).

5

What makes exploration of optimum difficult?

Complete disappearance of morphology in certain regions leading to large, disconnected, non-convex yield regions.

Multiple optima.

Expensive and time-consuming experimentation 36 hours for each run Gold catalyst required

6

“Actual” contour plot of CdSe nanowire yield Obtained by averaging yields

over different substrates.

Large no-yield (deep green region).

Small no-yield region embedded within yield regions.

Scattered regions of highest yield.

7

How many trials needed to hit the point of maximum yield ?

Pre

ssur

e

Temperature

8

How many trials ? Let’s try one factor at-a-time !

Temperature

Pre

ssur

e

Could not find optimumAlmost 50% trials wasted (no yield)Too few data for statistical modeling

9

A 5x9 full-factorial experiment

Yield = f(temp, pressure)

17 out of 45 trials wasted (no morphology)!

Pre

ssur

e

10

Why are traditional methods inappropriate ?

Need a sequential approach to keep run size to a minimum.

Fractional factorials / orthogonal arrays Large number of runs as number of levels increase. Several no-morphology scenarios possible. Do not facilitate sequential experimentation.

Response Surface Methods Complexity of response surface. Categorical (binary in the extreme case) possible.

11

The Objective

To find a design strategy that

Is model-independent, Can “carve out’’ regions of no-morphology

quickly, Allows for exploration of complex response

surfaces, Facilitates sequential experimentation.

12

What if design points are positively charged particles ?

q1

q2

E = Kq1q2 / d

Charge inversely proportional to yield,e.g., q = 1-yield

Pre

ssur

e

= 0.6

= 1.0

Y = 40%

Y = 0

13

What position will a newly introduced particle occupy ?

q1

q2

Pre

ssur

e

= 0.6

= 1.0

Total Potential E

nergy Minimized !!

14

The key idea

Pick a point x. Conduct experiment at x and observe yield y(x). Assign charge q(x) inversely proportional to y(x)

How quickly will you reach the optimum ? Once you reach there, how will you know that THIS IS IT ?

Use y(x) to update your knowledge about yields at various points in the design space (How ?)

Pick the next point as the one that minimizes the total potential energy in the design space.

15

The SMED algorithm

16

The next design point

17

Charge at unselected points

18

Choice of tuning constants

PROPOSITION : There exists a value of (inverse of the maximum yield pg) for which the algorithm will stick to the global optimum, once it reaches there.

In practice, pg will not be known. The constant determines the rate of

convergence. Both and will be estimated iteratively.

19

Performance with known

20

Performance with known (Contd.)

21

Performance with known (Contd.)

Initial point = (0.55,0.50) Initial point = (0.77,0.50)

22

Criteria for estimators of and

.increases) as local more(Search (ii)

,/1 ,/1 (i)

:tsRequiremen

maximum. Global :

iteration,th the tillyield maximum Observed :

iteration,th after the and of Estimators :,

)(

)(

nγ

pp

p

np

n

n

nngn

g

n

nn

23

Iterative estimation of and

constant a is where,/1

)/1log(

0 where,)1(

1

)()(

na

aγ

apap

n

nn

nnnn

n

24

Improved SMED for random response

Instead of an interpolating function, use a smoothing function to predict yields (and charges) at unobserved points.

Update the charges of selected points as well, using the smoothing function.

Local polynomial smoothing used. Two parameters:

nT (threshold number of iterations after which smoothing is started).

(smoothing constant; small local fitting).

25

Improvement achieved for r = 5

Last row gives the performance of the standard algorithm. Modified algorithm

significantly improves the number of times the global optimum is reached,

does worse with respect to no-yield points (higher perturbation).

26

Summary A new sequential space-filling design SMED

proposed. SMED is model independent, can quickly “carve out”

no-morphology regions and allows for exploration of complex surfaces.

Origination from laws of electrostatics. Algorithm for deterministic functions. Modified algorithm for random functions. Performance studied using nanowire data, modified

Branin (2 dimensional) and Levy-Montalvo (4 dimensional) functions.

27

Predicting the future

What the hell! I don’t want to use this stupid strategy for experimentation !

Use my SMED !

Image courtesy : www.cartoonstock.com

Nano

Stat

28

29

Advantages of space filling designs

LHD (McCay et al. 1979), Uniform designs (Fang 2002) are primarily used for computer experiments.

Can be used to explore complex surfaces with small number of runs.

Model free. No problems with categorical/binary data. CAN THEY

BE USED FOR SEQUENTIAL EXPERIMENTATION ? CARVE OUT REGIONS OF NO-MORPHOLOGY QUICKLY?

30

Sequential experimentation strategies for global optimization

SDO, a grid-search algorithm by Cox and John (1997) Initial space-filling design. Prediction using Gaussian Process Modeling. Lower bounds on predicted values used for sequential selection

of evaluation points. Jones, Schonlau and Welch (1998)

Similar to SDO. Expected Improvement (EI) Criterion used. Balances the need to exploit the approximating surface with the

need to improve the approximation.

31

Why they are not appropriate Most of them good for multiple optima, but

do not shrink the experimental region fast.

Algorithms that reduce the design space (Henkenjohann et al. 2005) assume connected and convex failure regions.

Initial design may contain several points of no-morphology.

Current scenario focuses less on convergence and more on quickly shrinking the design space.

32

Some performance measures for n0 - run designs.

33

Performance with estimated and with 30-run designs

34

First 20 iterations (out of 30) with estimated and

35

Contour plots of estimated p(x) (=y/r) where y ~ binomial(r,p(x))

36

Performance of the algorithm with random response

Result of 100 simulations with = 1.25, starting point = (0,0).

The last row represents the case of deterministic response and first three random response.

Concern: as r decreases, the number of cases in which the global optimum is identified reduces drastically.