10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 36 TA Led Final Review BVP Finite Differences or Method of Lines C Forward Upwind Central difference formulas x 2C Central difference like x 2 Understand when to use the different formulas Boundary Condition Flux D C x Reaction per surface area moles m2 s boundary m2 s Internal Flux mol m3 m A The flux is the same for these two arrows B can solve even if A and B are not known Partition function coefficient Figure 1 The flux is the same for arrows at A and B Method of Lines Solve a differential equation Initial Condition gradient stiff in y directon Sparse Discretize 123 x y along line i 2 N 1 C x 2 C3 C1 2x Boundary Condition may need to use shooting method Figure 2 Example problem good for method of lines If this is the B C C C1 C 2 x 1 x Use this additional equation with rest to solve for C1 D A E 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 Models vs Data y f x y1 f x1 y2 f x2 yn f xn Assumption 1 y distributed normally around y 2 x are known exactly P y Figure 3 A normal distribution y y WANT 1 Find the best 2 Is the model consistent 3 Error bars on parameters Assume model is exact xi yi y i f xi data will be distributed around model x1 y1 x2 y2 xn yn y i f x i 2 2 2 P yi exp y i f x i 2 1 P y exp exp 2 2 2 i 1 2 N N FIT Max P y y Min i 1 N y i 1 i 2 f x i f x i 2 i k A exp Ea RT ln k ln A y xn Ea R 1 T T Linear in parameters ln k ln A Ea R 1 T x x x y xn n rows measurements m parameters 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 36 Page 2 of 6 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 S V D x Umxm mxnVnxn N vi y v i i 1 i N Sample variance guess for s2 y i 1 i y 2 N dim y is mean y f x If non linear use optimization methods For correctness compare s to Quantitatively use 2 chi squared 2 N i 1 y i f x i 2 Transform to z 2 Goodness of fit area under curve 2min to P y 1 y y z 0 mean of 0 1 N dim 2min 2 Figure 4 Usually we will accept a model with the integral greater than 5 but we would like it higher If 99 chance it is wrong reject Error Bars Difficult If linear in parameters and is known covariance 2 xTx 1 i min i z2 5 x T x i i 1 2 diagonal mxm matrix m parameters 0 025 2 5 Figure 5 Chi squared distribution min i from 2 Non linear xTx i i point that bounds error on min i xi j f x i j 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Find xi j Lecture 36 Page 3 of 6 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 In MATLAB use nlinfit nlparei 95 confidence 2 2 m 2min 2 1 1 m 1 Figure 6 Location of chi squared and 95 confidence interval in 1 2 space 2 additional degrees of freedom let 1 2 vary 2 12 min2 If unknown use student t distribution based on s Report T being N dim normal student t broader as N increases student t approaches normal distribution Figure 7 Comparison of normal and Student t distributions yi you want to calculate is known yi is to be measured Average value of parameter m yi N x N xTx N xTx 1 2 N Global Optimization Convex function H 0 Hessian Matrix is positive definite Figure 8 Example of a convex function Only 1 minimum Non convex 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 36 Page 4 of 6 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 Figure 9 An example of a non convex function Branch and bound Professor Barton Non convex function guarantees global minimum Divide domain Bound from above Underestimate below Find mimina Bound again Split Figure 10 An illustration of the branch and bound algorithm If new upper bound is lower than the lower bound use other region can stop considering that section Multistart Take a bunch of initial guesses and then run local minimization No guarantee 100 points 6 variables 1006 calculations 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 36 Page 5 of 6 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 Simulated annealing E 0 2 Dihedral angle Figure 11 The energy varies with dihedral angle Start at high temperature decrease T eventually can sample wells once the point is caught in a minimum Genetic Algorithms Hybrid system integer variables and continuous variables Sample space by allowing function values to live die replicate switch values etc Monte Carlo Metropolis Monte Carlo Gillespie Kinetics Monte Carlo Stochastics Look at homework solutions to 10 and 11 Additional Topics Fourier Transforms and operator splitting may make a showing 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 36 Page 6 of 6 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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