Problem Solving Models v. Data Stochastics10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #35: Problem Solving Summary and Review. Problem Solving Well-Posed Problems • In reality, you define the problem Example: At Exxon, Professor Green was told that the “lubricant gums up in the engine.” Had to take ill-posed problem and transform to a well-posed problem. • Could solve kinetics • Could solve thermodynamics RECOGNIZE WHAT TYPE OF PROBLEM Æ Rewrite equations in standard form • Algebraic equations o Linear o Non-linear • Differential eq uations o ODE Initial Value Problems Boundary Value Problems o PDE • Optimization • Stochastic Simulations Estimate SOLUTION • REALITY CHECK! • Set constraints for optimization (i.e. least-squares) • Good initial guess • At least think about UNITS! Write some MATLAB Æ Run Computer Æ SOLUTION • (OR) Error or warning message Check if solution works!! • is reasonable to spend as much time checking solution as obtaining it o e.g. have two different programs written by two different people • How important is it that you are right? 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].• Even if solution works, does not guarantee solution will happen Sensitivity Analyses – check for ill-conditioning • Numbers from numerical solution will not match experiment because input parameters have uncertainty • Usually at least 1% error in measurements • If )parameters(input (solution)∂∂ sensitive to 1% error, ill-conditioning M·x ≈ b Æ use SVD • once you get a solution, do Taylor expansion, linear equations, check condition number cond(M) Models v. Data Stochastics • Metropolis Monte Carlo • Gillespie Kinetic Monte Carlo Ydata(x) Ymodel(x10.34, Numerical Methods Applied to Chemical Engineering Lecture 35 Prof. William Green Page 2 of 3 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]. ,θ,q) min χ2(θ) is χ2(θbest) small enough? find θbest not going to adjust θ2θ1θbest(θ1best±δθ1, θ2best±δθ2) χ2 = 5 χ2 = 2 χ2 = 14 Figure 1. Diagram showing search for best θ. Start with messy equations……… differential equations 1) Discretize Æ F(x) = 0 2) Taylor series Æ Linearize10.34, Numerical Methods Applied to Chemical Engineering Lecture 35 Prof. William Green Page 3 of 3 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]. J · ∆x = -F Solve linear equations. Then may discretize a different way to see whether we get the same answer. Newton’s method has best convergence close to minimum. Computer Programming (Key to MATLAB) • Reusability (avoid writing too much code) • HEADER to function/program: inputs/outputs/function description (what it
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