Mlesa Design Variable Scaling 50 Mles Design Variable Scaling 505 In aircraft design, we are combining variables of very Consider the transformation x= Ly, where L is an arbitrary different magnitudes nonsingular transformation matrix eg. aircraft range~10° If at any iteration in the algorithm, x= ly (using exact arithmetic), then the algorithm is said to be scale invariant wing span-10'm this property will not hold skin thickness 10-3m The conditioning of the Hessian matrix at x gives us Need to non-dimensionalize and scale variables to information about the scaling of the design variables be of similar magnitude in the region of interest When H(x) is ill-conditioned, J(x) varies much more rapidly along some directions than along others Want each variable to be of similar weight during the optimization The ill-conditioning of the Hessian is a form of bad scaling since similar changes in xl do not cause similar changes in J e Massachusetts Institute of Technology- Prof de Weck and Prof Willcox e Massachusetts Insttute of Technology. Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics Engineering Systems Division and Dept of Aeronautics and Astronautics MIlesd Design Variable Scaling Objective Function Scaling We saw that if H(x*)is ill-conditioned(x(H)>>1), then the In theory, we can multiply J(x)by any constant or add change in the objective function due to a small change in x a constant term and not affect the solution will vary radically depending on the direction of perturbation In practice, it is generally desirable to have J-o(1)in J(x)may vary so slowly along an eigenvector associated with the region of interest a near-zero eigenvalue that changes that should be Algorithms can have difficulties if J(x) is very small significant are lost in rounding error everywhere, since convergence is usually tested using some small quantity Ve would like to scale our design variables so that K(H-1 Inclusion of a constant term can also cause difficulties In practice this may be unachievable(often we dont know H) since the error associated with the sum may reflect the size of the constant rather than the size of (x) Often, a diagonal scaling is used where we consider only the e.g. min x,2+x2 VS. min x,2+x2+1000 diagonal elements of H(x )and try to make them close to e Massachusetts Institute of Technology- Prof de Weck and Prof Willcox e Massachusetts Insttute of Technology. Prof de Weck and Prof Willcox Engineening Systems Division and Dept of Aeronautics and Astronautics Engineering Systems DiMsion and Dept of Aeronautics and Astronautics25 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Design Variable Scaling Design Variable Scaling • In aircraft design, we are combining variables of very different magnitudes • e.g. aircraft range ~ 106 m wing span ~ 101 m skin thickness ~ 10-3 m • Need to non-dimensionalize and scale variables to be of similar magnitude in the region of interest • Want each variable to be of similar weight during the optimization 26 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Design Variable Scaling Design Variable Scaling • Consider the transformation x = Ly, where L is an arbitrary nonsingular transformation matrix • If at any iteration in the algorithm, xk = Lyk (using exact arithmetic), then the algorithm is said to be scale invariant • In practice, this property will not hold • The conditioning of the Hessian matrix at x* gives us information about the scaling of the design variables • When H(x) is ill-conditioned, J(x) varies much more rapidly along some directions than along others • The ill-conditioning of the Hessian is a form of bad scaling, since similar changes in ||x|| do not cause similar changes in J 27 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Design Variable Scaling Design Variable Scaling • We saw that if H(x*) is ill-conditioned (κ(H)>>1), then the change in the objective function due to a small change in x will vary radically depending on the direction of perturbation • J(x) may vary so slowly along an eigenvector associated with a near-zero eigenvalue that changes that should be significant are lost in rounding error • We would like to scale our design variables so that κ(H)~1 • In practice this may be unachievable (often we don’t know H) • Often, a diagonal scaling is used where we consider only the diagonal elements of H(x0) and try to make them close to unity 28 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Objective Function Scaling Objective Function Scaling • In theory, we can multiply J(x) by any constant or add a constant term, and not affect the solution • In practice, it is generally desirable to have J~O(1) in the region of interest • Algorithms can have difficulties if J(x) is very small everywhere, since convergence is usually tested using some small quantity • Inclusion of a constant term can also cause difficulties, since the error associated with the sum may reflect the size of the constant rather than the size of J(x) e.g. min x12+x22 vs. min x12+x22 +1000