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MIT 10 34 - Lecture #36: TA-Led Final Review

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BVP: Finite Differences or Method of Lines Method of Lines Models vs. Data Error Bars – Difficult Global Optimization Multistart: Simulated annealing Genetic Algorithms Stochastics Additional Topics10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #36: TA-Led Final Review. BVP: Finite Differences or Method of Lines xC∂∂= Forward/Upwind/Central difference formulas 22xC∂∂= Central difference-like Understand when to use the different formulas. Boundary Condition (Flux) DboundaryxC∂∂=Reaction per surface area [moles/m2·s] [m2/s] Internal Flux [(mol/m3)/m] A B The flux is the same for these two arrows 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 along line i = 2, …, N-1 xCCxC2132−=∂∂ Sparse Discretize y 1 2 3 x Boundary Condition –may need to use shooting method Initial Condition gradient stiff in y-directon Figure 2. Example problem good for method of lines. If this is the B.C.: xCCxCΔ−=∂∂121 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,θ) y1 = f(x1,θ) y2 = f(x2,θ) | yn = f(xn,θ) Assumption: 1) y distributed normally around yˆ 2) x are known exactly P(y) σ yˆ yˆ y 10.34, Numerical Methods Applied to Chemical Engineering Lecture 36 Prof. William Green 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]. WANT: 1) Find the best θ 2) Is the model consistent? 3) Error bars on parameters θ Assume model is exact xi  yi f(=iyˆxi,θ)  data will be distributed around model x1  y1 x2  y2 … xn  ynP(yi) ∝ ()⎥⎥⎦⎤⎢⎢⎣⎡−−222),(expσθiixfy P(y) ∝ ()()⎥⎦⎤⎢⎣⎡−−∝⎥⎥⎦⎤⎢⎢⎣⎡−−∑∏==NiiiNiiixfyxfy122122),(21exp2),(expθσσθ FIT: Max P(y)  Min ()∑=−Niiixfy12),(θ k = A·exp(-Ea/RT) ln k = ln A - Ea/R (1/T) Linear in parameters ln k, ln A, Ea/Ry = xn·θ  θ = [xTx]-1xT·yxn: n rows (measurements), m parameters Figure 3. A normal distribution.S.V.D.: x = UmxmΣmxnVnxn θ = ∑=⎟⎟⎠⎞⎜⎜⎝⎛⋅Niiiivyv1σ Sample variance guess for σ: s2 = )dim()(12θ−−∑=NyyNii y is mean y, f(x,θ) If non-linear, use optimization methods. For correctness, compare s to σ. Quantitatively, use χ2 (chi squared) χ2 = ()∑=−Niiixfy122),(σθ Transform to z σ P(y) yˆ yˆ y 1 0 mean of 0, σ = 1 z [N-dim(θ)] χ2min χ2Goodness of fit: area under curve χ2min to ∞σ P(y) yˆ yˆ y Error Bars – Difficult If linear in parameters and σ is known, covariance(θ) = σ2[xTx]-1 (diagonal mxm matrix) θi = θmin,i±z2,5σ[xTx]i,i-1/2 m = # parameters θmin,i point that bounds from χ2 error on θmin,i0.025 2.5% 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. Figure 5. Chi-squared distribution. Non-linear: σ[xTx]i,i xi,j = jixfθθ∂∂ ),( Find xi,j 10.34, Numerical Methods Applied to Chemical Engineering Lecture 36 Prof. William Green 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].IFI yσxFn MATLAB, use nlinfit, nlparei 10.34, Numerical Methods Applied to Chemical Engineering Lecture 36 Prof. William Green 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]. igure 6. Location of chi-squared and 95% confidence interval in θ1-θ2 space. ∆χ2 ≡ [χ12 – χmin2] ν = 2 additional degrees of freedom: let θ1, θ2 vary f σ unknown, use student t distribution based on s. normal student tbroader Report T(χ,ν), ν being N-dim. θ as N increases, student t approaches normal distribution i = θ ( you want to calculate θ) is known, yi is to be measured. Average value of parameter: θm = (Σyi)/N = xN⎥⎦⎤⎢⎣⎡Tx = [ = N σ[x]⎥⎦⎤⎢⎣⎡Tx]-1/2  σ/√N Global Optimization Convex function – H ≥ 0 (Hessian Matrix is positive definite) Figure 7. Comparison of normal and Student-t distributions.igure 8. Example of a convex function. Only 1 minimum Non-convex: θ2,m θ2 95% confidenceχ2min θ1,m θ1 θ1 θ210.34, Numerical Methods Applied to Chemical Engineering Lecture 36 Prof. William Green 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]. Branch and boundProfessor Barton – Non convex function guarantees global minimum Split Divide domain Bound from above Underestimate below Find mimina. Bound again… Figure 9. An example of a non-convex function. 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.Simulated annealing E 0 2π φ 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.


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