M Gradient Vector 16888 ESD.J7 Consider a function J(x),X=[ The gradient of x at a point x is a vector of length n X V(x)=0 ax Each element in the vector is evaluated at the point x C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
12 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Gradient Vector Gradient Vector Consider a function J ( x), x=[ x1,x2,..., xn ] The gradient of J ( x) at a point x 0 is a vector of length n: 0 1 0 0 2 0 ( ) ( ) ( ) ( ) n J x J J x J x ª º w « » w« » « » w « » w« » « » « » w « » « » w¬ ¼ x x x x # Each element in the vector is evaluated at the point x 0
Mest Hessian Matrix 16888 ESD.J7 Consider a function√(x),X=X1x2…,xn The second derivative of (x at a point x is a matrix of size n×n OXOx a Ox ax n Hx)≡V2J(x) ax, ax n Each element in the matrix is evaluated at the point Xo C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Hessian Matrix Hessian Matrix Consider a function J ( x), x=[ x1,x2,..., xn ] The second derivative of J ( x) at a point x 0 is a matrix of size n u n: 22 2 2 1 12 1 2 0 20 1 2 2 2 2 1 () () n n n JJ J x xx xx J x x J J J xx x ª º ww w « » w ww ww « » « » w « » w w « » { « » « » « » « » w w « » « » ww w ¬ ¼ Hx x " " % # # % # %# Each element in the matrix is evaluated at the point x 0
Mlesd Gradients Hessian Example 16888 ESD.J7 J(x)=3x1+xX2+x×2+6x2x3 C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Gradients & Hessian Example Gradients & Hessian Example 2 3 1 12 3 2 3 J x xx x x x () 3 6 x
Mest Taylor Series 16888 E77 Consider scalar case f(2)=f(2)+(z-2)+ 1 df z-z)2+ aZ 2 dz. When function depends on a vector T (x)=√(x")+VJ(×")(x-x) n nx +-(X-X XX 1×n n×n nx C Massachusetts Institute of Technology - Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Taylor Series Taylor Series Consider scalar case: 0 0 2 0 0 02 2 1 () ( ) ( ) ( ) z 2 z df d f fz fz z z z z dz dz " When function depends on a vector: 0 00 0 0 () ( ) ( ) ( ) 1( ) ( )( ) 2 T T JJ J ª º ¬ ¼ 0 x x x xx x x Hx x x " 1 un n u 1 1 un n n u n u 1