1. 1 Design of experiment for computer simulations Let X = (X 1,…,X p )  R p denote the vector of input values chosen for the computer program Each X j is continuously adjustable between a lower and an upper limit, or 0 and 1 after transformation Let Y = (Y 1,…,Y q )  R q denote the vector of q output quantities Y = f(X), X  [0,1] p Important considerations: The number of input p The number of output q The speed with which f can be computed They are deterministic, not stochastic Why a statistical approach is called for?

2. 2 Design of experiment for computer simulations Conventional one-factor-at-a-time approach It may miss good combinations of X because it doesn’t fully explore the design space. It is slow, especially when p is large It may be misleading when interactions among the components of X are strong Randomness is required in order to generate probability or confidence intervals Introducing randomness by modeling the function f as a realization of a Gaussian process Introducing randomness by taking random input points

3. 3 Goals in computer experiments Optimization Standard optimization methods (e.g. quasi-Newton or conjugate gradients) can be unsatisfactory for computer experiments as they usually require first and possibly second derivatives of f Standard methods also depend strongly on having good starting values Computer experimentation is useful in the early stages of optimization where one is searching for a suitable starting value, and for searching for several widely separated regions for the predictor space that might all have good Y values

4. 4 Goals in computer experiments Visualization – being able to compute a function f at any given X doesn’t necessarily imply that one “understands” the function Computer simulation results can be used to help identify strong dependencies Approximation If the original program f is exceedingly expensive to evaluate, it may be approximated by some very simple function, holding adequately in a region of interest, though not necessarily over the entire domain of f Optimization may be done using large number of runs of the simple function

5. 5 Approaches to computer experiments There are two main statistical approaches to computer experiments One is based on Bayesian statistics Another is a frequentist one based on sampling techniques It is essential to introduce randomness in both approaches Frequentist approach For a scalar function Y = f(X), consider a regression model Y = f(X)  Z(X)  The coefficients  can be determined by least squares method with respect to some distribution F on [0,1] p  LS = (  Z(X)’Z(X)dF) -1  Z(X)’f(X)dF The quality of the approximation may be assessed globally by the integrated mean squared error  (Y – Z(X)  ) 2 dF

6. 6 Frequentist experimental design Assume the region of interest is the unit cube [0,1] p, p = 5 Grids (choose k different values for each of X 1 through X p and run all k p combinations) – works well but completely impractical when p is large. In situations where one of the responses Y k depends very strongly on only one or two of the inputs X j the grid design leads to much wasteful duplication

7. 7 Frequentist experimental design Good lattice points (based on number theory)

8. 8 Frequentist experimental design Latin hypercubes

9. 9 Frequentist experimental design Randomized orthogonal arrays

10. 10 Example – critical specimen size study

11. 11 W critical = f(t, h; E,  y,  0, e; k) 19 mm (ANSI/AWS) 25 mm (MIL) 35 mm (ISO) 76 mm (ANSI/AWS) 102 mm (MIL) 45 m m (IS O) 25 mm (MI L) 105 mm (ISO) 19 mm (ANSI/A WS) Specimen size requirements for tensile shear tests of 0.8 mm gauge steel sheets.

12. 12 P D Peak Load E Energy Maximum Displacement

13. 13 3.60 3.55 3.50 3.45 3.40 3.35 3.30 3.25 Peak Load (kN)

14. 14 Table 1. Ranges selected for computer simulation. t (mm)h (mm)E (GPa)  y (MPa)  0 (MPa) e (%)k 0.5~2.00.1~1.5190~200205~172550~2002~651.0~3.0 Table 2. Design matrix and simulation results. Ru n v1v1 t (mm) v2v2 h (mm) v3v3 E (MPa ) v4v4 kv5v5  y (MPa) v6v6 e (%) v7v7  uts (MPa) W criti cal (mm) 110.541010 0.881212 196.7 6 1414 2.595607.352111515 785.2922.6 240.8150.472190.8 8 11.068875.596257982.9427.5 330.721212 1.058194.4 11 2.241616 1590.8 8 52211645.2 9 37.3 471.0720.221313 197.3 5 31.291212 1233.2 4 8321414 1402.3 6 31.7 581.1610.141515 198.5 3 1313 2.471414 1412.0 61 4261510.5 9 33.8 650.901414 1.217193.8 2 81.881717 1680.2 9 1515 561616 1867.0 5 38.7 720.631 0.961414 197.9 4 61.653428.531717 635518.2419.1 860.9930.311190.2 9 1616 2.827786.181414 531010 920.0027.1 991.2590.809195.0 0 92.009965.0093591090.0 0 40.4 101212 1.511515 1.291717 199.7 1 21.181 1143.8 2 41881260.0 0 44.1 111616 1.8770.644192.0 6 1212 2.351515 1501.4 7 181313 1661.7 6 58.2 121313 1.6040.391 196.1 8 1010 2.121249.713152312.9524.2 131010 1.341717 1.463191.4 7 51.534517.947291212 669.4135.7 141 1.431616 1.385192.6 5 1515 2.716696.761010 394777.6439.2 151515 1.7860.551010 195.5 9 71.762339.121313 491717 534.7135.0 161414 1.691313 1.131616 199.1 2 1717 2.941010 1054.4 1 1212 461 1197.0 6 49.6 171717 1.9680.726193.2 4 41.411313 1322.6 5 1616 6031394.7 1 58.9

15. 15 Coupon Size Determination Simulation A two level full factorial would require 2 7 = 128 runs In the computer experiment, N levels of each variable can be chosen (based on the number of variables n). N is also the total number of runs needed. N = 17 for seven (7) variables in the example The computer simulation results are used to create the dependence of critical specimen size on the variables by Kriging regression method

16. 16 W critical,1 = 13.4044613 + 18.5987839 t W critical,2 = -6.0291481 + 18.5839362 t + 0.0146654  y + 6.6251147 h W critical,3 = 45.6391799 + 18.5849834 t + 0.0146654  y + 21.8791238 h + 28.3945601 e + 0.0811080 (  uts -  y ) - 0.0003401 E - 9.5332611 h 2 - 0.2280655 e (  uts -  y )