1. Multiple Shooting: No One Injured !
Got the title worked in…
September 20, 2018September 20, 2018
Multiple Shooting, MTH422

2. 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

3. Newton’s Method Expanded (n-D)
To Solve the System F(x)=0, F:RnRn
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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. Multiple Shooting: An Example Part 2
This means we need to solve the BVP:
September 20, 2018September 20, 2018
Multiple Shooting, MTH422

18. Multiple Shooting: An Example, Initial Guess
September 20, 2018September 20, 2018
Multiple Shooting, MTH422

19. Multiple Shooting: An Example, First Iteration
Note how quickly the function smooths out.
September 20, 2018September 20, 2018
Multiple Shooting, MTH422

20. 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

21. Multiple Shooting: An Example, Initial Guess and Final Iteration
Simple Comparison.
September 20, 2018September 20, 2018
Multiple Shooting, MTH422

22. Multiple Shooting Any questions?
If all goes well there shouldn’t be any questions.
September 20, 2018September 20, 2018
Multiple Shooting, MTH422