Singular Value Decomposition Ordinary Differential Equations Trapezoidal Rule: Simpson’s Error Romberg Method (Richardson Extrapolation) ODE Solvers Explicit Euler Runge Kutta Stiff differential equation Predictor-Corrector Method DAE Optimization Sequential Quadratic Programming (SQP) Boundary Value Problems10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #26: TA led Review Singular Value Decomposition Amxn = Umxm Smxn VnxnT U-1 = UT V-1 = VTTnVSU⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡00021σσσ% A = U·S·VT σ1 ~ 0 UT·A = (UTU)S·VT S-1 = ⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡0011001321%σσσ 1/σ1 Æ ∞ (treat it as zero) UT·A = I·S·VT V·S-1UT·A = I·S·VTV·S-1 A·x = b x = A-1b A-1 = V·S-1UT x = V·S-1UT·b Ordinary Differential Equations dxnm/dt = Fn(x) x = x(to) + ()∫ttodttxF )( Trapezoidal Rule: to t1 t2f3 f2 f1 Error = O(∆t3)xN Î O(∆t2) Figure 1. Integration by the Trapezoidal Rule. Simpson’s Error Error = O(∆t5)xN Î O(∆t4) 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].Romberg Method (Richardson Extrapolation) Ftrue = ...)()(~)(20+Δ+∫tCNFdttfntt Ftrue = ...2)2(~)(20+⎟⎠⎞⎜⎝⎛Δ+∫tCNFdttfntt Ftrue = ()( )3)(1)2(4tONFNF Δ+− t t0 nf(t) 10.34, Numerical Methods Applied to Chemical Engineering Lecture 26 Prof. William Green Page 2 of 5 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]. 33 N Æ ∆t 2N Æ ∆t/2ODE Solvers Explicit Euler xn+1 = xn + F(xn)h + O(h2) Runge Kutta xn xn+1 Figure 2. Integration by the Romberg Method. Figure 3. Runge Kutta Integration of differential equations. Runge-Kutta-order 5 Æ 6 function evaluations per time steps use intermediate value to calculate Ode45 uses R-K 6 function evaluation Stiff differential equation x = a·e-t + b·e-1000000t { rate of time change are 1,000,000 times differentFigure 4. Example solution to differential equation. must use many time steps; if big steps are used, you will oscillate around the solution. d x= −c ⋅ x dt∆t < 2/λmax Must use Implicit Method Predictor-Corrector Method Figure 5. Predictor-corrector method. DAE 10.34, Numerical Methods Applied to Chemical Engineering Lecture 26 Prof. William Green Page 3 of 5 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]. ()⎛d x⎞M X⎜ ⎟=F()X ⎝dt⎠⎡⎤⎢⎥=f(x⎢⎥⎢⎥⎣0 0 0⎦) “ode23t,” “ode15i” Optimization minx f(x) g(x) = 0 i = 1...neh(x) ≥ 0 i = 1...ni CONSTRAINTS polynomial tn-k tn-k+1 tn tn+1If no constraints: Gradient Method: Vf =⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛∂∂∂∂∂∂nxfxfxf21 If no gradient given: fminsearch Figure 6. Gradient method contours. Conjugate gradient method As you get closer to the minimum, Newton’s Method gives good convergence: Vxn = -Hn-1·Vfn H = ⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂∂∂∂∂∂221212212212nnnxfxxfxxfxxfxf#" No Constraints: Broyden-Fletcher-Goldfarb-Shanno Method (BFGS) With Constraints Vf Æ Vgi(x) Lagrangian Vf = ∑=∇eniiixg1)(λ L(x,λ) = f∑=∇eniiixg1)(λ VxL = 0 VλL = 0 L(x,λ) = f – Σλigi(x) + Σ(1/2μi)[gi(x)]2 KKT 10.34, Numerical Methods Applied to Chemical Engineering Lecture 26 Prof. William Green Page 4 of 5 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].L(x,λ,k) = f(x) - ∑∑==−ieniiiniiixhkxg11)()(λVL = 0 gi(x) = 0 hi(x) ≥ 0 ki ≥ 0 hjkj = 0 Sequential Quadratic Programming (SQP) minx f(x) cm(x) – sm = 0 for equality constraints: sm = 0 for nonequality constraints: sm ≥ 0 Boundary Value Problems xxxOxxxSzyxDzvyvxvnnnxnnnnxzyxnnΔ+−=∂∂Δ+−−=∂∂+⎥⎦⎤⎢⎣⎡∂∂+∂∂+∂∂=∂∂+∂∂+∂∂+−−−22)()(112211222222φφφφφφφφφφφφφφ Types of BC: φ(xo) = φo; 024321=Δ−+−=∂∂xxoxnφφφφ 10.34, Numerical Methods Applied to Chemical Engineering Lecture 26 Prof. William Green Page 5 of 5 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
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