Summary week 8

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Summary week 8

• Traveling salesman problem
• Optimize algorithms number of steps, sigma,
  different strategies
• Poor algorithm 160Mm (taking closest city)
• Good algorithm 130 Mm
• Shortest path on the website

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Final results

• 3 distances lt 128000 km in 800 seconds
• 79 configurations lt 129500 km

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Summary Week 8

• Integration
• closed form boundaries included
• open form less precise (1 order lower in number
  of steps/derivative)
• redefine integration variables in case of
  improper integrals.
• Higher accuracy by weighting the function values
  differently
• Basic 1st order accuracy
• Simpsons, adaptive stepsize, 4rd order
  (1/3,4/3,2/3,4/3,)

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Summary Week 8

• Romberg cancel even powers by doubling the
  number of grid points rapid convergation
• Euler-MacLaurin Summation
• Open formulas triple number of points
• Gaussian Quadrature
• Freedom to choose abscissa as well as weight
• Fastest convergation, typically within 20
  function calls)

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(16) Ordinary Differential Eqs.

• ordinary differential equations sets of
  first-order differential equations.
• new variables typically derivatives of each
  other. Generic diff. equation becomes coupled set
  of first order diff. eq.

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Example

• Suppose you want to calculate trajectories from a
  testmass that is on a spring, attached to a
  string
• mass m, string length R0, radius r, uncompressed
  spring length l0, angles theta and phi.
• There are nine dependent variables and 1
  independent (the time)

O
z
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variables, equations

• coordinates
• velocities
• accelerations

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Boundary conditions

• Crucial boundary conditions.
• simple case variables have certain values at
  given points.
• complicated non-linear algebraic relations among
  the variables.
• initial value problems
• yi specified at xs
• solution required at (sets of) xf
• two-point boundary value problems
• boundary conditions at more values of x
• typically, some at xs, remainder at xf
• e.g. grenade trajectory
• boundary initial speed, final destination
• required angle of canon

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Initial-value boundary problems

• Initial-value boundary problems
• Runge-Kutta
• Bulirsch-Stoer
• predictor-corrector
• Two-point boundary problems
• shooting
• relaxation

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Runge-Kutta

• step from xi to xf in small steps h
• Taylor expansion in series of steps
• one step differentials specified at beginning of
  step. O(h2) accuracy. (Eulers method)
• Better calculate derivative at midpoint

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midpoint- Runge Kutta
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Fourth-order Runge-Kutta
fourth-order RK superior to midpoint if h can
be chosen twice as big.
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Runge-Kutta

• quite robust. In NR supply function derivs() to
  calculate the right-hand side, as well as the
  derivatives at the initial step.
• Adaptive stepsize control precision of result
• overhead 11/8 calculations

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Embedded Runge-Kutta

• Alternative fifth-order RK requires 6 function
  evaluations. However, these same six evaluations
  can be used to make a 4th-order extrapolation as
  well. This then serves as error estimate
• Values for a,b,c from Cash and Karp
• stepsize tune

15
Modified midpoint method

• advance from x to xH with n substeps hH/n.
  Right-hand side evaluations n1
• midpoint method for intermediate z. One
  derivative per intermediate step, instead of 2
  for second-order RK. Error contains only even
  powers of h

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Bulirsch-Stoer

• modified midpoint fourth-order accuracy is
  obtained by
• Richardson extrapolation (Bulirsh-Stoer) a la
  Romberg integration. Cancel higher orders in
  error, fine when ODEs are smooth (Else use
  Runge-Kutta with adaptive stepsize).
• consider final answer as analytical function of
  parameter (stepsize h), probe it with some values
  h, fit the result for h-gt0
• use rational function extrapolation
• even error in powers of h

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Bulirsch-Stoer
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Stiff sets of equations

• Stiffness two or more very different scales.
• requires very small stepsize hltlt1/1000. Also for
  intermediate values!

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Stiff sets of equations

• Stability at a risk. Cure implicit differencing.
  
• Explicit
• Unstable if hgt2/c.
• Implicit calculate y at right-hand side
• Stable yn1 -gt 0 for h-gtinf.

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Implicit differencing

• Generalize for sets of equations
• This only goes to zero if largest eigenvalue of
  (1-Ch) smaller than 1.
• Implicit differencing
• Always stable (for eigenvalues larger than one)
  but requires matrix inversion.

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Implicit differencing

• In general matrix C contains functions of y
• This is the semi-implicit Euler method.
  Generalizations
• Runge-Kutta like Kaps-Rentrop methods
• Bulirsch-Stoer like Bader and Deuflhard

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Bader-Deuflhard

• Implicit form of midpoint rule
• use with the semi-implicit Euler step and a
  special initial and final step, this series again
  only involves even powers of h in the error.
  Method NRsimpr()

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exercise
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Two-point boundary value problems

• Obviously, harder to obtain result in this case
• Shooting method choose starting values at one
  boundary, guess the free parameters. Vary free
  parameters until the boundary eqs. at the second
  point are satisfied.
• Relaxation Replace the diff. equations by
  finite-difference equations on a mesh of points.
  Choose (arbitrary) parameters on each mesh point
  and gradually try to decrease the discrepancies.

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Shooting

• Starting at x1, there are N-n1n2 free
  parameters. Write method that calculates the n1
  parameters for an arbitrary vector V with n2
  components (V spans space of possible solutions)
• calculate y at x2, e.g. with Bulirsch-Stoer.
• define the difference
• Newton-Raphson

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• Shooting requires n21 integrations per trial.
  For linear problems only 1 trial needed. A
  second round may clean up result. Requires a
  function
• (load()) to calculate the initial values of y,
  given n2 free parameters,
• (score()) to calculate difference F,
• starting vector v(0..n2-1)
• function derivs to calculate the ODE integration.

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Shooting to midpoint

• Alternatively, one can shoot from x1 and x2 to a
  common midpoint.
• more complicated difference depends on 2
  vectors, with n1 and n2 parameters.
• preferable when endpoints are singular one can
  integrate away from it but not to it.

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Relaxation methods

• Replace ordinary differential equations by
  finite-difference equations
• M meshpoints, N equations NM parameters.
  Newton-Raphson NMNM matrix. However it is in
  block-diagonal form.
• Midpoints k coupled equations between k and k-1,
  2N parameters-gt N(M-1) parameters specified.
  remaining N parameters come from boundaries.
  Choose last n1 parameters at 0, first at M-1

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Relaxation

• initial guess yj,k. Calculate difference
• NM-1 relations for M points and N variables.
• remaining N relations from boundary
• E0 B(x0,y0), EM-1 B(xM-1 ,yM-1)
• difference delta write in matrix form
• matrix has following form

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relaxation, block diagonal matrix
5 variables 4 mesh points 3 boundarycond, at x0,
2 boundarycond at x3
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relaxation, gaussian elimination
target form. Can be obtained by block by
block diagonalization. MN2N steps.
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Relaxation

• Automated calculation of mesh points possible
• advantage for singular points.