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Water wave analysis with nonlinear Fourier transforms Peter Prins p.j.prins@tudelft.nl 7 September 2018 This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and

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Water wave analysis with nonlinear Fourier transforms

Peter Prins p.j.prins@tudelft.nl 7 September 2018

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 716669).

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Nonlinear water waves

“ I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and

  • vertook it still rolling on at a rate of some eight or nine miles an hour, preserving its
  • riginal figure some thirty feet long and a foot to a foot and a half in height. Its height

gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation. “

John Scott Russell (1808-1882)

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https://youtu.be/w-oDnvbV8mY

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SLIDE 4 4

https://youtu.be/D14QuUL8x60?t=51s

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SLIDE 5 5

Wave equations

  • Water surface elevation:
  • Linear wave equation:
  • Korteweg-De Vries equation (KdV):
  • Many other non-linear wave equations

exist. utt − uxx = 0 − ut + 6uux + uxxx = 0

u = u(x, t)

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SLIDE 6 6

Why a Fourier transform?

Hard calculations ☹ u(t) y(t) Easy calculations " Fourier Transform Inverse Fourier Transform

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SLIDE 7 7

Why a Fourier transform?

| Kexp(j!0t) = H(j!0)exp(j!0t) u(t) = Z ∞

−∞

U(j!) exp(j!t) d! y(t) = Ku(t) = Z ∞

−∞

U(j!) Kexp(j!t) d! = Z ∞

−∞

U(j!) H(j!) | {z }

Y(jω)

exp(j!t) d!

Mvi = ivi w = X

i

civi y = Mw = X

i

ciMvi = X

i

ciivi

y(t) = ↵0u(t) + ↵1 du

dt + ↵2 d2u dt2 =: (↵0 + ↵1t + ↵2tt)

| {z }

K

u(t)

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SLIDE 8 8

Solitons overtaking: No superposition

⬆ Source: http://lie.math.brocku.ca/~sanco /solitons/kdv_solitons.php ➡ Adapted from: https://carretero.sdsu.edu/ teaching/M-639/lectures/ nonlinear_waves.html

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SLIDE 9 9

Lax pair

  • A and L are linear operators.
  • The eigenvalues of L are constant in time iff

u(x,t) satisfies the KdV equation.

  • ut + 6uux + uxxx = 0

− − A := 4xxx + 6ux + 3ux L := xx + u

− Lt + LA − AL = 0

⇕

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SLIDE 10 10

Schrödinger eigenvalue problem

  • The Non-linear Fourier transform of an

input u(x,t0) w.r.t. the KdV equation, is the eigenvalue problem of L.

  • It can be shown that

– All are real. – All <0 are an eigenvalue (cf. ordinary Fourier transform) – Some isolated >0 may be eigenvalues (cf. matrix eigenvalues)

" λ " λ " λ

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SLIDE 11 11

Schrödinger eigenvalue problem

  • δxx + u(x)
  • w(x) = λw(x)

wxx(x) =

  • λ − u(x)
  • w(x)

d dx " w(x) wx(x) # = " 1 λ − u(x) # · " w(x) wx(x) #

d dx w(x) = A(x) w(x) d dx w(x) = ¯

Aw(x) ⇒ w(x) = exp ¯ Ax

  • · w(0)
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SLIDE 12 12

Staircase approximation

  • 6
  • 4
  • 2
2 4 6

x

0.2 0.4 0.6 0.8 1

u(x,t0)

¯ ui := u(xi) ¯ Ai = " 1 λ − ¯ ui # w

  • xD + ε/2, λ
  • = exp

¯ AD(λ) · ε

  • |

{z }

GD(λ)

· · · exp ¯ A2(λ) · ε

  • |

{z }

G2(λ)

· exp ¯ A1(λ) · ε

  • |

{z }

G1(λ)

·w

  • x1 − ε/2, λ
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SLIDE 13 13

Computational complexity

  • D gridpoints in x => (D-1) matrix

multiplications.

  • D gridpoints in => Repeat the

calculation D times.

  • Hence complexity O(D^2).
  • We have improved it to O(D log^2(D)),

see next slide.

" λ

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SLIDE 14 14

Fast Nonlinear Fourier Transform

Source: Wahls, Sander, and H. Vincent Poor. "Fast numerical nonlinear Fourier transforms." IEEE Transactions on Information Theory 61.12 (2015): 6957-6974.

= p1

↘ ↗

p1p2 p2 ↘ p1p2p3p4 p3 ↗

↘ ↗

p3p4 ↘ p4 p p5

↘ ↗

p5p6 ↗ p6 ↘ p5p6p7p8 p7 ↗

↘ ↗

p7p8 p8

Gi(λ) ≈ " p11i(z) p12i(z) p21i(z) p22i(z) # p·i(z) = c0 + c1z + c2z2 + . . . z = z(λ)

Main benefit: No need to repeat for every . We can fill out the resulting 4 polynomials for every value of we need.

" λ

" λ

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SLIDE 15 15
  • We show a method to make the

Fast Non-linear Fourier Transform (FNFT) as accurate as a classical (slow) NFT, without increasing the computational complexity. Introduction

  • The Fast Non-linear Fourier Transform (FNFT)
  • Preliminaries
  • Improvement: More accurate exponential splitting
  • Example 1: Rectangular signal
  • Examples 2 & 3: Smooth signals
−−−− − − − −
  • ()
() = sech () =
  • ( +
)ε
  • −
− − − − − −
  • ζ
|(ζ)| [−] ;
  • ˆ
(ζ) − (ζ)
  • [−]
  • References
  • Fast Nonlinear Fourier Transform
  • We show a method to make the
Fast Non-linear Fourier Transform (FNFT) as accurate as a classical (slow) NFT, without increasing the computational complexity. Introduction ∂ ∂(, ) + (, ) ∂ ∂(, ) + ∂ ∂(, ) = ,
  • The Fast Non-linear Fourier Transform (FNFT)
(ζ,)ε (ζ) (ζ) ζ O ⎛ ⎜ ⎝ log ⎞ ⎟ ⎠ Preliminaries () −∞ ∞ ∂ ∂() = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −ζ (, ) − ζ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ·() = ⎛ ⎝ζ, () ⎞ ⎠ ·() , (, ) = () → ±∞ ζ () (∞) = ⎡ ⎢ ⎢ ⎣ −ζ − ζ ⎤ ⎥ ⎥ ⎦ε · · · ⎡ ⎢ ⎢ ⎣ −ζ − ζ ⎤ ⎥ ⎥ ⎦ε· ⎡ ⎢ ⎢ ⎣ −ζ − ζ ⎤ ⎥ ⎥ ⎦ε·(−∞), := ⎛ ⎜ ⎝( + )ε ⎞ ⎟ ⎠ ε ζ O ⎛ ⎝ ⎞ ⎠ ∂ ∂(, ) + ∂ ∂(, ) + |(, )| (, ) =
  • − −∗(, )
Improvement: More accurate exponential splitting
  • ε (ζ)
  • (ζ, ) = (ζ) + () =
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −ζ
  • ζ
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ε ε ε ε (ζ) = ζε/ ε
  • †
ε =
  • εε
  • ε + O(ε) ; †
  • ε =
  • ε
  • ε
  • ε
  • ε −
εε + O(ε) ; †
  • ε =
  • ε
  • ε
  • ε
  • ε
  • ε −
  • εε
  • ε + O(ε) ; †
  • ε =
  • ε
  • ε
  • ε
  • ε
  • ε
  • ε −
  • ε
  • ε
  • ε
  • ε +
  • εε + O(ε) ;
  • ε =
  • ε
⎛ ⎜ ⎜ ⎝
  • ε
  • ε
⎞ ⎟ ⎟ ⎠
  • ε
  • ε −
  • ε
  • ε
  • ε
  • ε
  • ε +
  • εε
  • ε + O(ε) ;
  • ε =
  • ε
⎛ ⎜ ⎜ ⎝
  • ε
  • ε
⎞ ⎟ ⎟ ⎠
  • ε −
  • ε
⎛ ⎜ ⎜ ⎝
  • ε
  • ε
⎞ ⎟ ⎟ ⎠
  • ε +
  • ε
  • ε
  • ε
  • ε −
  • εε + O(ε) ;
  • ε =
  • ε
⎛ ⎜ ⎜ ⎝
  • ε
  • ε
⎞ ⎟ ⎟ ⎠
  • ε
  • ε−
  • ε
⎛ ⎜ ⎜ ⎝
  • ε
  • ε
⎞ ⎟ ⎟ ⎠
  • ε
  • ε+
  • ε
  • ε
  • ε
  • ε
  • ε−
  • εε
  • ε+O(ε) .
ε = (+)ε = ∞ Σ =
  • !( + )ε = + ( + )ε +
( + )ε + ( + )ε + . . .
  • Example 1: Rectangular signal
−−−− − − − −
  • ()
() = rect () =
  • ( +
)ε
  • Examples 2 & 3: Smooth signals
−−−− − − − −
  • ()
() = sech () =
  • ( +
)ε
  • −−−−−− − −
  • ()
() = . sech () =
  • ( +
)ε
  • −
− − − − − −
  • ζ
|(ζ)| [−] ;
  • ˆ
(ζ) − (ζ)
  • [−]
  • −
− − − − − −
  • ζ
|(ζ)| [−] ;
  • ˆ
(ζ) − (ζ)
  • [−]
  • −
− − − − − −
  • ζ
|(ζ)| [−] ;
  • ˆ
(ζ) − (ζ)
  • [−]
  • < .%
  • References
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SLIDE 16 16

Research group

Numerics for Control & Identification

Prof.dr.ir. Michel Verhaegen

Non-linear Fourier Transforms

  • Ir. Peter Prins
  • Ir. Shrinivas

Chimmalgi Dr.Ir. Sander Wahls

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SLIDE 17 17

Find us on Github

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Recommended courses

  • SC42060: Nonlinear Systems Theory

– (Obligatory, 4 ECTS, Q2)

  • WI4046: Spectral Theory of Linear Operators

– (Highly recommended as a free elective, 6 ECTS, Q3)

  • WI4212: Advanced Numerical Methods

– (6 ECTS, Q3)

  • WI4260TU: Scientific Programming for Engineers

– (3 ECTS, Q3)

  • ME46040: Experimental Dynamics

– (3 ECTS, Q3 & Q4)

Experience in programming in C can be gained on the job.

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SLIDE 19 19

Thank you.