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Multiple Shooting: No One Injured !
Got the title worked in… September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method in Review (1-D)
Approximates xn given f and initial guess x0 Quick refresher on Newton’s method in 1-dimension, which relies on the previous point and knowledge of the (continuous) function. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method Expanded (n-D)
To Solve the System F(x)=0, F:RnRn We use Xk+1=xk-(F’(xk))-1F(xk) Where F’(xk) := J(xk) J(xk)=Jacobian matrix of F at xk F(x) must map a given number of points to the same number of points. Newton’s method is present just rewritten. Explain “:=“ operator September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Jacobian matrix of xk September 20, 2018September 20, 2018
Explain that the Jacobian is merely a matrix of the partial derivatives of x sub k’s points. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method Expanded Part2
In practice xk+1=xk-F’(xk))-1F(xk) is never computed. Use J(xk)(xk+1-xk)=-F(xk) instead, which is of the form Ax=b. Can be written: J(xk)h=-F(xk),xk+1=xk+h Which is a linear system. Explain that (F’(xk))^ -1 is very computationally expensive. Show how it can be rewritten. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method: An Example
Solve the nonlinear system using Newton’s method: f1: x+y+z=3 f2: x2+y2+z2=5 f3: ex+xy-xz=1 Where F(x,y,z)=(x+y+z-3, x2+y2+z2-5, ex+xy-xz-1) Straight forward, note the function labels fx… September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method: An Example Part 2
Compute the Jacobian: Generic 3X3 Jacobian matrix written out to help understanding. F1->x+y+z-3=0 F2->xx+yy+zz-5=0 F3->e^x+xy-xz-1=0 September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method: An Example Part 3
Newton’s Method becomes: (xk+1,yk+1,zk+1)=(xk,yk,zk)+(h1,h2,h3) (should that be x sub k?) September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Newton’s Method: An Example Part 4
If (x0, y0, z0) = (0.2, 1.4, 2.6) This method converges Quadratically to the unique point p, such that F(p) = 0 ||xk+1-x*|| <= C||xk-x*||2 where x* is the exact solution, so ||errork+1|| <= C||errork||2 Reaches (0, 1, 2) in 5 iterations! We must provide an initial guess. Newton’s method converges very quickly. X* is the original function’s value at the specified point. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Convergence of Newton’s Method
The error at each iteration is as follows: Error ( ||h|| ) * 10-1 * 10-2 * 10-4 * 10-8 * 10-15 Error of the actual solution, which we do not have. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting Setup Part 1
x’ = f(t,x) and g(x(a), x(b)) = 0 Which is a Boundary Value Problem (BVP) and can be rewritten as: x’- f(ty, x) = 0 and g(x(a), x(b)) = 0 Simple arithmetic rewrite. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting Setup Part 2
or F(x) = 0 This is a nonlinear system of equations. These are functional, not algebraic. Note that “0” is a short hand for the 1X2 matrix of zeros. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: Newton’s Method Part 1
F’(xk+1(t))(xk+1(t)-xk(t)) = -F(xk(t)), which is again of the form Ax=b F’(xk+1(t)) is a very general version of the derivative, called a Frechét Derivative. Frechet Derivative is a derivative based on functions instead of variables. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: Newton’s Method Part 2
If we take ω to be an arbitrary function we can produce: September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: Newton’s Method Part 3
We can make a similar case for H(x(t)) Next: G’(xk)(xk+1 - xk)=-G(xk), and similar for H. ω’ = fx(t, x)ω – (x’ – f – f(t, ω)) Baω(a) + Bbω(b) = -g(x(a), x(b)) ω’ is of the form ω’ = Aω + q Quasilinearization September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example
Compute a periodic solution (with period τ) of the system: x’ = f(x, λ) x1’ = 10(x2-x1) x2’ = λx1 – x2 –x1x3 x3’ = x1x2 – (8/3)x3 For λ = 24.05 September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example Part 2
This means we need to solve the BVP: September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example, Initial Guess
September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example, First Iteration
Note how quickly the function smooths out. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example, plots
Look at the similarity between k=2 and k=6. Note how quickly individual points converge to the solution. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting: An Example, Initial Guess and Final Iteration
Simple Comparison. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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Multiple Shooting Any questions?
If all goes well there shouldn’t be any questions. September 20, 2018September 20, 2018 Multiple Shooting, MTH422
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