10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 18 Optimization Sensitivity Analysis Introduction Boundary Value Problems BVPs Summary Optimization with Constraints minx f x such that cm x sm 0 sm 0 m 1 Ninequalities 2 minx f x c s sm 0 m Ninequalities penalty method second term c s 2 is optional KKT conditions at constrained local minimum Augmented Vxf m m Vxcm 0 Lagrangian cm sm 0 LA mcm 0 see book sm 0 m 1 Ninequalities sm 0 equalities f m c m cm sm F x 0 m cm s m c x s Newton SQP If everything is linear SIMPLEX i e many business problems g x 0 xN G x1 xN 1 Unconstrained trust region Newton type BFGS gigantic conjugate gradient In Chemical Engineering the problems often involve models with differential equations cost f x w Y t i i o return x Yi t f x i knobs what we need what we produce can adjust feed composition Need Jacobian of G with respect to Y need in stiff solver to solve dY G Y x dt Y t 0 Y 0 x Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Need gradient and f To use all of our methods we need to be able to compute Y i Yi t f f wi x x j x i j j how do you compute this Yi G Yn Gi x j t n Yn x j x j sensitivity of Yi tf to xj chain rule Yi t x j sij G G d sij i s nj i dt n Yn x j Vf for every x we get an Vf that can be used for optimization solve this with initial conditions Jacobian of G J Have n2 differential equations stiff linear in s Sensitivity Analysis Programs to do this DASPK SOLVE for s and f simultaneously DAEPACK DSL485 DASAC G G d sij i s nj i dt n Yn x j Initial Conditions What is sji t0 sji t0 0 most knobs sji t0 1 for adjustment of Y0 Professor Barton teaches an advanced course in optimization 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 18 Page 2 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Boundary Value Problems BVPs Conservation Laws t V v V JD S convection diffusion reaction JD V isotropic JD cV for steady state isotropic 0 V v cV2 S x Laplacian Boundary conditions Dirichlet boundary number von Neumann V boundary number or 0 Symmetry 0 x j x infinite rare to find exact adjust large finite number 104 approx x f x c N basis Basis function expansions approx c x n 1 n n m 1 N basis m x approx v c 2 approx s approx 0 x approx c B C x f x i approx approx xi called Residual xi mesh grid some interpolation Finite difference approximation to differential equation Dirichlet boundary x x1 2 0 x2 x0 x xi i 1 i 1 xi 1 xi 1 State how you did approximation because there are many ways to do it i 1 N B C von Neumann 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 18 Page 3 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY x Usual 2nd order polynomials given 0 x0 x x 0 x unknown x 0 f 1 2 x1 0 x2 0 x x known x0 unknown 2 f 1 2 x 2 x0 x1 x1 x 0 x0 x 2 x0 x0 0 x x0 2 9 9 x 2 x0 2 2 1 x1 x0 2 0 x2 x0 2 1 x1 x0 2 If x uniform 1 2 x x0 2 x 2 x0 1 2 2 x 2 1 2 2 x 2 for x1 x0 2 x0 x 2 x 0 2 x0 x 0 x0 4 1 2 3 This is how you find out B C with second order polynomial schemes and a finite difference approximation 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 18 Page 4 of 4 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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